Foundations of Infinitesimal Stochastic Analysis

STUDIES IN LOGIC AND

THE FOUNDATIONS OF MATHEMATICS VOLUME 119

Editors

J. BARWISE, Stanford D. KAPLAN, LosAngeles H. J. KEISLER, Madison P. SUPPES,Stanford A. S . TROELSTRA, Amsterdam

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD

FOUNDATIONS OF INFINHESIMAL STOCHASTIC ANKYSIS

K. D. STROYAN Mathematics Department The University of Iowa lo wa City, l owa 52242 U .S.A. and

Jose Manuel BAYOD Facultad de Ciencias Universidad de Santander Santander, Spain

1986

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD

0 ELSEVIER

SCIENCE PUBLISHERS B.V., 1986

AN rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the priorpermission of the copyright owner.

ISBN: 0 444 87927 7

Published by: ELSEVIER SCIENCE PUBLISHERS B.V. P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U.S.A.and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52Vanderbilt Avenue NewYork, N.Y. 10017 U.S.A.

Library of Congress Cataloging-inqublicstionh t r Stroyan, K. 0. Foundations of infinitesimal stochastic analysis. (Studies in logic and the foundations of mathematics ; V.

119)

Bibliography: p. Includes index. 1. Stochastic analysis. 2. Mathematical analysis, Nonstandard. .I. Bayod, Jose Manuel. 11. Title. 111. Series.

~ ~ 2 7 4 . 2 1986 . ~ ~ ISBN 0-444-87927-7

519.2

PRINTED IN THE NETHERLANDS

85-28540

viii

ACKNOWLEDGEMENTS This project has taken much longer than expected. Our final worry is that we will forget to thank one of the many people who offered us their help during the many years! We appreciate even the smallest suggestions because we know that a sum of infinitesimals can be infinite. Most of all we thank H. Jerome Keisler for his seminar notes, ideas, examples, criticism, preprints and encouragement. This book would not exist without his help. C. Ward Henson also gave us a great deal of help and an example. Jorg Flum, L. C.r Moore, Jr.. Robert M. Anderson, and Tom L. Lindstrom generously gave us detailed criticism of parts of early drafts. Douglas N. Hoover, Edwin Perkins. L . L. Helms. Nigel Cutland, J. E. Fenstad and Peter A. Loeb sent us their preprints and discussed the K. Jon Barwise, Juan Gatica. project with us at meetings. Eugene Madison, Robert H. Oehmke, John Birch, Constantin Drossos. Gonzalo Mendieta. David Ross. Vitor Neves, Anna Roque. Lee Panetta and others participated in seminars on various parts of the book. We thank all these people for their help and encouragement. It seems to us that the combined effort of many people is what has made this branch of Robinson's Theory of Infinitesimals blossom. Bayod thanks the Fulbright Foundation for partial support during the first part of the project and his colleagues at the University of Santander. who, by increasing their work load. Stroyan allowed him to take a two-semester leave in Iowa. gratefully acknowledges support, years ago, of the National Science Foundation of The United States for summer research that appears in parts of the book. Stroyan thanks The University of Iowa for its tolerance and sometimes generous support of his peculiar research interests. We thank Ada Burns for her superb typing of infinitely many drafts, revisions and corrections of author errors. We also thank Laurie Estrem f o r excellent typing of part of the next-tolast draft. We thank the staff of North-Holland in advance for the production task they are about to undertake. The final draft of this book was prepared with the excel lent new technical word processor T3 from TCI Software Research, Inc. and printed by them on an HP LaserJet+ printer. The series editors, Arjen Sevenster and others at NorthHolland have been patient and helpful in making arrangements for this book to appear in The Studies in Logic and The Foundations o f Mathematics. We are delighted that this book will appear in the same series as Abraham Robinson's classic book on infinitesimal s .

ix

W e dedicate this book to Jerry Keisler f o r h i s professional help and to our wives f o r their emotional support during the project.

to

Jev, Carol and Cristina

X

FOREWORD Bayod obtained support from the Fulbright Foundation to visit The University of Iowa for the 78-79 academic year in order to learn about Abraham Robinson's Theory of Infinitesimals (so-called "Non-Standard Analysis"). We agreed to focus our seminar on infinitesimal analysis of probability and measures because of exciting work of Keisler and Perkins then in progress and with hopes of further applications. We made careful notes, while, unknown to us, Keisler was doing the same thing with his students. The second draft of this book corpbined both sets of notes and comprised roughly the present chapters 0 to 4 . Chapter 0 contains all the 'nonstandard stuff' that our reader needs in order to learn about the applications in this book. The reader who is familiar with the basic principles of infinitesimal analysis can go directly to Chapter 1. Chapter 0 tries to give the beginner in infinitesimal analysis the working tools of the trade without proof. We feel that the logical principles such as Leibniz' (transfer) Principle and the Internal Definition Principle, together with Continuity Principles such as Robinson's Sequential Lemma, saturation and comprehension are the things our beginning reader should focus his attention on. Section (0.1) gives the definition of "all of classical Section (0.2) analysis" in the form of a "superstructure." explains the meaning of Leibniz' Principle and begins to show its usefulness. We believe that our reader can get a working knowledge of these tools of logic by working several of the exercises. Section (0.3) contains more explanation of some basic notions of infinitesimal analysis that are used throughout the book. Section ( 0 . 4 ) contains the important saturation property that we base our measure constructions upon. The approach to measures in this book was initiated by Peter A. Loeb [1975]. Most of the basic results are due to him. but we have given a more elementary new exposition based on inner and outer internally generated measures. This approach replaces the use of Caratheodory's extension theorem by direct elementary arguments. We have also added some fine points and examples not found in the literature. Section (1.1) deals with probability measures, while section (1.2) treats infinite measures. Since infinite measures cause extra technicalities, we have given a short outline for the reader who is anxious to (It appears right apply Loeb's construction to probability. after the Table of Contents.) C. Ward Henson [1979a.b] discovered the connections between Loeb. Bore1 and Souslin sets and first proved uniqueness of hyperfinite extension and the unbounded case. The remainder of Chapter 1 explains the relationship between *finite sums and integrals against

xi

Foreword

hyperfinite measures. Robert M. Anderson [1976. 19821 systematically investigated Radon measures, filled in some of the basics on S-integrability and studied product measures. S-integrability is the main ingredient needed to relate sums and integrals. In Chapter 2 we study the relation between Borel and hyperfinite measures. The basic idea there is to 'pull back with the standard part map'. The case of a completed Borel measure is technically easier to treat, so section (2.1) treats Lebesgue measure independently, while sections (2.2) and (2.3) treat naked sigma algebras and measures. This draws on the works of Anderson and Henson cited above. Anderson and Salim Rashid [1978] and Loeb [1979a] investigated weak standard parts of measures: section (2.4) presents a simple case of those results. Chapter 3 contains a Fubini-type theorem due to H. Jerome Keisler [1977] as well as a mixed Fubini-type theorem. Anderson's work above showed that a hyperfinite product measure extends the product of hyperfinite measures, while Douglas N. Hoover [1978] showed that that extension is strict. The Fubini theorem holds anyway. Chapter 4 has a basic treatment of distributions, laws and independence from the point of view of infinitesimal analysis. Even the "foundations" of stochastic analysis consist of more than measures. Anderson [1976] discussed Brownian motion, including path continuity and Ito's lemma, using Loeb's techniques. Loeb [1975] treated infinite coin tossing and a different approach to the Poisson process. Keisler's [1984] preprint investigated more general processes. (P. Greenwood and R. Hersh [1975] and Edward Nelson [1977] have some infinitesimal analysis of stochastic processes using different techniques.) We discussed these things in the spring of 1979. but only wrote rough notes including some extensions of this work. In the meantime we learned of other fundamental work of Tom L. Lindstrom [1980] and of Hoover and Edwin Perkins [1983]. Our treatment of Chapters 5, 6 and 7 relies most heavily on the work of Keisler. Hoover-Perkins and Lindstrom. Chapter 5 is devoted to 'path properties' of processes. Our treatment of paths with only jump discontinuities is a little different from Lindstrom-Hoover-Perkins'. We precede that with the less technical case of continuous paths and include a lot of details in both cases. Section (5.4) contains results and extensions of results from Keisler [1984] relating Loeb. Borel and Souslin sets on the product space [O.l] x R . Section (5.5) sketches how one makes the extension of these results to C 0 . m ) . The infintesimal analysis is very similar to [O,l], but since the classical metrics are rather technical we avoided a complete account. Chapter 6 gives the basic theory of how events evolve We measure-theoretically in time on a hyperfinite scheme. follow Keisler [1984] again, but extend his results to include "pre-visible measurability" as well as "progressive measurabi 1 i ty " A hyperfinite evolution scheme has

.

xii

Foreword

'measurability' and 'completeness' properties that make i t richer than an arbitrary 'filtered (or adapted) probability space' of the "general theory of processes." We prove that the error "adapted implies progressively measurable" is actually true. We show that previsibility arises from a left filtration. Hence, while we borrowed extensively from Lindstrom [1980] and especially Hoover and Perkins [1983] in later chapters, we adapted their results to Keisler's more specific combinatorial framework. We feel that this would be justified simply because of the concrete "liftings" we obtain. However, Hoover and Keisler [1982] show that this results in no loss of generality in a certain specific logical sense described briefly in the Afterword. The a i m of all our chapters is to give fundamental results needed to apply infinitesimal analysis to the study of stochastic processes. We drew the line of what we call "foundations" at "measure theory" but our reader should not take this seriously. We hope many people will work to extend the foundations of infinitesimal stochastic analysis as well as to give new applications of these methods in solving problems about stochastic processes. We doggedly adhered to a hyperfinite bias. That made certain things 'nice' and should not hinder our reader from learining and using other 'Loeb-space' techniques. Chapter 7 gives a hyperfinite treatment of semimartingale integrals. We have written i t at two levels. Sections (7.1). (7.2). (7.3) and the beginning of (7.7) treat the easily integrated case in complete detail. Section ( 7 . 4 ) shows how the more general local theory at least partly parallels the square integrable case. The remainder o f the chapter only outlines the main ideas of the best known contemporary theory. We hope that the statement of results and examples will act as a guide to further study. The survey article of Cutland [1983b] could be read by a beginner to get an overview of hyperfinite measure theory.

1

CHAPTER 0 :

PRELIMINARY CONSTRUCTIONS

(0.0) Motivation with a Finite Probability Experiment

Consider the probability experiment of tossing a f a i r c o i n n

times. We can represent each possible outcome as a sequence

of

H’s

T’s

and

.

These outcomes can be viewed as the

elements of the set:

nl= {H.T)~. Rolling a f a i r d i e

n

times could be viewed as elements of the

set:

R2= {I, 11,111.IV,V.VI}”. Coin tossing can be modeled in the die experiment by considering an even

outcome as

Sampling

n

“heads“ and

an

odd

outcome as

”tails“.

times from an urn with r e p l a c e m e n t a n d c o m p l e t e

m i x i n g after each sample can be modeled on the set

R3=

W

where the

is a set with

A

urn.

m

particular

w”.

elements representing the balls in

sequence

of

draws

from

the urn

is

W

on

represented by a function u E

Q3

u : (1 .*.*.n} +

This means that the first draw,

u(1) u(2)

W.

is the particular ball in the urn

is the ball on the second draw, and s o

on.

If

m

is even, coin tossing can be modeled inside

considering even outcomes as ”heads” and odd outcomes as “tails“.

This can be coded by any function

Rg by

Chapter

2

c: W + {H,T} #

(We use

#

with

,

[c

0: Preliminary Constructions

-1

=

(H)]

#

[c

-1

(T)].

for the finite cardinality function.)

[a]

The number

of sample sequences that have exactly half heads is n # [ u € R 3 : #[j : c(u(j)) = HI = 2 1 . (There will not be any unless m

that

If

is even.)

n

6

is even and we already assumed

divides

m

then the fair die

R 3 in a similar way.

experiment can be modeled inside

to have lots of divisibility we may as well assume that n m = n , for a single integer

and let

h

€

*IN

h

>

In order n = h

(Eventually we wi 1

1.

be an infinite integer.)

We wish to think of the index in the component functions (such as

above) as a time, s o that i t is more convenient to

u

represent our random experiment a slightly different way. means that. when we make

h

(and hence

n

and

m)

This

larger,

more sampling takes place in the same elapsed time. We first take a set of times

U = { A t , 2 A t , 3 A t . . . . , nAt}.

where

At

=

1 n .

Then we

take a finite set

W

with

m

elements and define our space of sample sequences to be the set of all functions from

(0.0.1)

Now i f

t=5

U

into

R = { w I

w

E R,

then

w ( 51)

W

w:U+W}=W

U

is the ball selected at the time

for the particular sequence of samples represented by

The idea of a m i x e d urn mentioned above means that each

w.

Section

ball

0.0: A Finite Experiment

in

W

sampling. sequence

is equally Thus

the

likely to be drawn at each

of

probability

each

AP(w) =

1 n m

The set

&

those sample sequences Ah€ E

Ah =

[RI

R

is called the a l g e b r a o f

For example, the event consisting of that contain exactly half heads is

w

given by

{a E R

:

#[t

:

n = H] = -} 2

c(o(t))

The p r o b a b i l i t y o f an e v e n t

A

€

8

is given by the finite A:

sum of the individual probabilities from

P[A]

=

Z[AP(A)

:

h E A]

For example, the probability o f the event the binomial coefficient

I,;,[

Ah

above is given by

over the number of ways that

one can assign heads or tails to a sample along

[ $1 2” n

(0.0.4)

sample

1 #

of all subsets o f

e v e n t s of the experiment.

(0.0.3)

individual

time o f

w E R , is the uniform amount

(0.0.2)

the set

3

=

T.

1

’

We could attach values to heads or tails corresponding to winning or losing a step in a game, for example,

Chauter

4

In

this

case

the

running

0: Preliminary Constructions

total

of

signed

winnings

is

the

a)

stochastic process (a random walk of step size

We have written very much about a simple example, because

we want to point out what sorts of mathematical involved.

First. we have an

element set into

W.

W.

and

the

R

U

e ement set R

Next, we have the set

Then we have a function

all subsets of

AP

n

P

entities are and an

defined on the set

summation

function

I[-

*I.

:

Moreover,

B

Finally, we have a function

UxR

:

.

by the summation function and another function. AB let

At

be an infinitesimal

6t

formulas to analyse Brownian motion. will

require

more

of

E

in terms of the (constant uniform) function

computing an event defined in terms of the function

{H,T} .

U

of functions from

P

associate certain simple combinatorial formulas with

into

m

than

just

c +

we

when

from

R

W

given

We want to

and use the same kinds of Our infinitesimal analysis

extending

numbers

to

include

infinitesimals. but will also require extensions of functions, functions of functions (including summation), sets of these, and combinatorial formulas relating to them all. Abraham allows us

Robinson's

to enlarge

infinitesimals

and

contemporary

Theory of

Infinitesimals

"all of classical analysis" infinite

numbers

as

functions. sets, sets of functions, and s o on.

well

to include as

certain

This extension

0.0: A Finite Experiment

Section

procedure

satisfies

a

precise

5

transfer

principle

akin

to

Leibniz’ old idea that what holds for ordinary numbers, curves, etc.,

also

holds

of

formulation

for

the

the

ideal

principle

extensions.

uses

formal

a

Robinson’s language

for

precision, but in practice we need only care with quantifiers and some training in the limitations of the formal transfer.

LEIBNIZ’ TRANSFER PRINCIPLE (heuristic form) A property only

Q

is true in cLassicaL anaLysis i f and

* transform

if its

*Q

is

true

in

infinitesimal

anaLysis.

The precise section

formulation of Appendix

(0.2).

constructions

needed

for

the transfer principle

2

the

sketches

some

formulation.

chapter is t o show the reader how to

use

is in

set-theoretic

The aim

of

this

Leibniz’ Principle-

proofs can be found in the references given in the appendices.

For now you should think of

*

as a mapping defined on all the

objects of classical analysis (in an informal sense).

For example.

*IR

is a set extending the set of real numbers,

in the following sense. fixed number

r E IR.

*

restriction of 0

R

=

{r

property

€

*IRI of

Nevertheless,

(3

to

for example,

R

to the set s €

two

IR)[r =

numbers

*

is defined on each

*0. * 1.

* A , *r.

+

binary functions

and

*+

because

being

unequal

and the relation and

**

The

maps bijectively onto the set

*s ] } ,

*

preserves

to

each

is a proper subset of the set

OR

binary functions

The mapping

IR

<

and a relation

on

*<

other.

*IR. R on

the

The extend

*IR.

6

Chapter

field.

*O. *1, *+,

(*IR,

The tuple

The

*<)

*a,

of

extensions

0: Preliminary Constructions

is a real closed ordered

fixed numbers

properties (such as being an identity), s o

IR

copy of the

on

U

in

*IR.

IR

and the

extensions via

values

y E

is an isomorphic

on the field operations and order.

*'s

For example, -1

satisfying

sine and cosine have natural

<

y

* sin(x)

<

+1

takes on all the

while still satisfying

on the functions) for all

x

in

Summation extends to the same type of operation and

in

the trig identities (with

*IR.

basic

which satisfy the same first order properties

as the originals.

*IR

lR

their

For simplicity of notation we usually drop

Transcendental functions like

*

retain U

*

general

*'s

preserves types or levels of sets.

However, above

the level of numbers there are restrictions on the extensions. The general rule for deciding how an extended quantity acts is to examine the transform under the classical

quantity.

"*- transforms .

We

call

*

map of properties of the

the

resulting

properties

"

*-TRANSFORMS (heuristic form) Y o u j u s t put

**s

on e u e r y t h i n g !

A statement in classical analysis has a *-transform given by extending

each fixed set, function, or point used

statement via the

*

map.

must have specific bounds.

all positive

E IR

s

every element every subset of

A

"

The quantifiers in such a statement 'Specific bounds' means that

is O.K.. but "for all s

in the power set of

H

"

is not.

in the

IN

''

"

"for

is not, "for

is O.K., but "for

The *-transform of a property like

Section

0.0: A Finite Experiment

continuity is referred to as Not

all

the

*continuity.

properties

analysis arise by transfer.

7

of

the

extension

of

classical

Otherwise the extension would be

isomorphic to the original and thus not contain infinitesimals. The

properties

of

"internal"

sets

and

functions

arise

by

transfer, whereas "external" sets and functions are not subject to

transferred

properties.

This

important

distinction

is

characteristic of contemporary infinitesimal analysis and is the main thing this chapter tries to uncover.

8

(0.1)

Superstructures

We want an object big enough to include "all of classical analysis" and since this might space

we

begin

could

.

IR

containing

with

take

Xo

indiuiduaLs

(or

later include some probability

fixed

a

but

arbitrary

ground

set

=

S

.

IR

U

The elements

set-theoretic

If

have no elements.

x

€

Xo

"atoms" ,

or

then y

€

we

must

"urelements") ;

be

they

x is not defined. (In

r € IR

particular, we shall think of numbers

Xo

of

XO S

For example, i f we were given a space

as "points" and

not as sets of sequences of rationals or as Dedekind cuts, etc.) Let

D(A)

denote the power set of all subsets of a set

A.

We form a sequence of sets inductively as follows:

inductively for

n

E

R4

n

(0.1.1) DEFINITION: The

3 = [X

set

n

n E

a]

Xo .

structure o u e r the ground set a r e called indiuiduals while

is

called

the

super-

The elements o f

Xo

Z \ Xo

are

the eLements o f

called entities.

Notice that We call

3

X

C

1 -

X2 C

--*

and

Xo f l

Xn = 0 , for

n 2 1.

a structure because we can think of analysis as

coded in the theory of

€

and

=

restricted to the entities and

Section

0.1:

individuals. (I , E

3

9

SuDerstructures

Strictly

speaking

the superstructure is a

triple

=).

The following examples illustrate why a superstructure is big enough to be considered a model of "all of classical analysis."

(b)

D E IR.

sin(-) E I:Any

real

valued

function

with

domain,

may be viewed as a set of ordered pairs of real numbers,

hence, sin(*) E X3 E I.

X3,

(c)

C[O,l]

E I:Every function defined on [O.l] belongs to

so

C[O,l]

E

X4

I. These

remarks

apply

many

other

e m , the space of all bounded

classical function spaces such as

sequences, which may be viewed as a set of into

to

functions from

IN

IR.

(d)

I l - l l m E I:The uniform norm on

from an element of

(e)

X4

into

IR,

E I:The topology

T

belongs to

so i t

T

=

C[O.l]

belongs to

{U E C[O,l]

is a function X6.

I U is Il.llm-open}

X5.

Notice that a sequence that f o r each

n.

(xn I n E LN)

defined in such a way

xn E Xm+l\Xn. is not an element of

"von Neuman ordinals",

v

= (0. (0). {0.{0))

I

{0.{0).{0.{0))).***1

I. The

Chapter

10

0: Preliminarv Constructions

(which can be used as a model for the natural numbers) is not an

V Q 3.

entity,

V

atoms, s o

#

(We have taken

IN.)

I, as a set of

and thus

(R.

However, because we have some flexibility in

how we choose to code objects from analysis and because we may take

Xo

(3 , E , =)

justified to call The

IR, we feel that i t is

to be any set of atoms containing

rest

of

this

a model of classical analysis.

section

locates

the

pieces

construction of section (0.0) in the superstructure take

* transforms

of

the

We will

3.

of these constructions in the next section.

Arithmetic operations can be viewed as functions, hence are elements of

3.

Partially defined functions also belong to

Define the functions

: IRxR + IR

Define the function

by

fj

f 1 ( r , s ) = r+s , f3: IRx[IR\{O}]

+

=

= h!.

f5(h,k)

IR

by

.:

R or RxIN by k = h , f6(h,k) =

Define the functions on f4(h)

= r-s.

f2(r.s)

f3(r.s)

T-.

k].

binomial coefficient,

as well as.

(0.1.2)

EXERCISE: Find

terms o f

Simple

p

B(R)

f l , .. . ,f7 E X .

that

P

f7

Express

set-valued section

functions (0.0).

are

For

also

involved

example.

F1

:

defined by

F1(n)

in

f2, f5 and f 6'

constructions of 4

so

= T = {t E IR : (3 k

E

R)[1

i k 5 n!

k

& t = -]}

n!

in

the

[IN\{O.l}]

Section

0.1:

+ B(LN)

F2: [IN\{O}]

defined by = W = IN[l.m]

F2(m)

(0.1.3)

11

SuDerstructures

<

= {k E IN I 1

k

<

m}

EXERCISE: f l . * - - . f7, F1. F2 E: Xp

Find p s o that

We also need some restricted forms of set functions which do not belong to each

>

p

as elements in their unrestricted forms.

9L

0,

define

GY

functions

:

Xp

+

For

Xp+l

and

H Y : X x X + X by the following restrictions P P P+2

GY(Y) = B(Y),

Y,

the power set of all subsets of

and

HY(Y.2) = 2 Y , the set of all functions from

(0.1.4)

If

q

p'

s o that

>

extends

For each

p

The

X .

that

4

GI;' extends

GY(Y) = GY(Y)

if

Y

E:

GY

X . P

as

a

HY'

ALSO,

2 0, the set of all finite subsets of

f: IN + 9L finite

E

as a function.

satisfies

function 9L.

HY

GY, HY

p , show

function. that is.

P'

Z.

into

EXERCISE: Find

Fin

Y

Finp

E Xp+l.

given by cardinality

so

Finp

f(p) = Finp function

E

Xp+2.

XPU Xo.

However, the

is not an element of

counting

the

number

of

elements of a set can be restricted to any of these sets in order to make i t an entity.

For each

G;(Y)

p > 0 . define = # [Y].

G Z : Fin

P

+

IN

by

Chauter

12

The

summation

function

restricted entity. functions

whose

can

Let

also

Fct

9

domain

0: Preliminary Constructions

be

defined

in

terms

a

of

denote the set of all real-valued

satisfies

X .

D

Summation

9

can

be

defined by the function

H;(f,Y)

H;

where and

Y

= Z[f(y)

Y E Y].

(f,Y)

is defined on the set of pairs

a finite subset of the domain of

with

f

f.

(0.1.5) EXERCISE: F o r each

p.q

>

0

find

r

SO

that

Gg.

Hl E

xr.

€

Fct

4

13

(0.2) Results from Logic:

Leibniz' Principle, the Internal Definition Principle

This section gives the main results from logic needed to apply Robinson's Theory of Infinitesimals to analysis along with some examples directed Further

our basic

at

random

examples are given in Appendix

treatment

can

appendices. applying

be

found

The main

in

the

sampling scheme.

A more complete

1.

references

thing one must

given

consider

in

the

carefully

in

+-transforms is the bounds on the quantifiers of the

statement being transformed.

(0.2.1) DEFINITION:

A statement in classical analysis is said t o have a standard bounded terms

of

reLations

formalization i f

indiuiduals E

and

and

=

c a n be expressed

it

entities

in a way

so

X

of that

using

in the

the quantifiers

occur only in the bounded f o r m s : ( V x)[x

where

A and B

E

A 3 . * * ] or

are e n t i t i e s ,

(3 y)[y

E

B

A.B E X.

The bounds on the quantifiers, "for all exists

y

E

&***].

x

E

B" are sometimes abbreviated (V x E A)[-**]

or

(3 y E B)[..*].

The statement that a particular entity in cardinality has a standard bounded formalization. any fixed

A" or "there

X

has finite In fact, for

p 2 0. we may characterize the elements of the set

Fin (from the last section) as follows: P

14

Chapter

X

)([Y E Fin ] P+1 P (3 f E Fct )(3 n E lN)["f maps Y 1-1 and onto W[l.n]"] P ( V f E Fct )["(f maps Y onto Y ) a (f maps Y 1-1 to Y)"] P

(V Y E

a a

0: Preliminarv Constructions

}

In other words, an entity of

X

property that functions from

Y to Y are surjective if and only

is finite i f i t can be mapped P bijectively onto an initial segment of IN or if i t has the

i f they are injective.

The statements in quotation marks are

abbreviations for easily formalized statements. bounds, Fin

P

and Fct

P

The quantifier

play an important role in the *-transform

of this statement. We

shall

(Y, E. =)

assume

that

over a ground set

* :

there

Yo

is

another

superstructure

along with an injection

1 + Y

satisfying Leibniz' Principle (0.2.3) below.

The principle is

stated in terms of *-transforms of statements about

Z,

so

first

we give the definition of *-transform and an example.

(0.2.2) DEFINITION:

If

Q

is

a

statement

with

formalization, the *-transform of obtained

by

indiuidual in

The is:

applying

the

*

Q

map

a

standard

bounded

i s the statement to

each

entity

*Q and

Q.

*-transform of being a finite entity at the

pth level

Section

(V

Y E*Xp+ ) { [ Y €*Finp]

a

(3 f E*Fct

a

( V f E*Fct

We

15

0.2: Results from Logic

shall

P P

)(3 n E*H)["f

that

the

on

*'s

and onto

1-1

a

maps Y onto Y)

)["(f

see

Y

maps

*IN[l.n]"]

(f maps. Y

1-1

to

Y)"]

the quantifier bounds

restrictions on the informal meaning of

* finiteness.

* sets can be finite because this property is only *Fct . On the other hand, * finite functions from

}

cause

Infinite

tested with

sets can be P treated as if they were finite as long as the operations are

"internal."

We

give

the

transfer principle

first and

then

discuss the meaning of "internal . "

(0.2.3) LEIBNIZ' TRANSFER PRINCIPLE: The

*

mapping

superstructure

%

over

over the individuaLs Any

if

(91

and

.

E

is

card+(%)

an

injection

of

the

into the superstructure

* Yo = Xo, (%

about

bounded only

*

Xo

is

Yo, w h e r e

sentence

standard

*

: 9: + 91

satisfying:

. . =) E

formalization if

its

91

holds

*-transform

with

a

in

3

holds

in

*9:

is

=).

polyenlarging.

in

particular.

saturated.

Part b of the transfer principle will be explained further

in section

(0.4).

A special case of part (b) is that the

natural extension of every infinite set is strictly larger than the

set

of

(extended)

standard

elements

of

the

set.

In

Chapter

16

particular, cl

IR =

* { r

the standard

elements of

real numbers, embedded

form a proper

I r E IR},

*IR

0: Preliminarv Constructions

*IR

as

We

call

the

*IR.

subset of

hyperreaL numbers.

in

The Dedekind numbers,

IR.

are a maximal archimedean ordered field (in a given set theory), so

we

see

that

Leibniz'

infinitesimal numbers

Principle asserts

6 E *IR

the existence

*0 <

such that

6

< *r

of

for every

r E IR.

positive The

nonstandard

extension

map

acts at

many

levels

(or

"types") of set theory and this can be quite confusing at first. At

the ground

*1

identity, so

level we have

*0

*IR

in

as a multiplicative identity,

on, by part (a) of Leibniz' Principle.

as an additive

* 1+ * 1

=

*2.

and

Since the extensions

of the fixed numbers behave the same way as the originals, we usually do not write

*area of *IR and

the

1

*unit

*'s

on them.

*circle,

0

Hence,

T

E

*IR

is the

is the additive identity of

is the multiplicative identity.

Writing no

*'s

on

standard numbers may be thought of as an identification

Hence, we may drop the sigma on the in this.

The set

statement about and no more.

(0.2.4)

IR

OIR

IR.

but there is a danger

is an external

set: no

transferred

can refer to all the standard numbers in

This is explained fully below.

DEFINITION: I f

X E 9\Xo

is an entity. the discrete

standard

*IR

X

part o f

is the set U

X E XI

If = {*x

XI,

: x E

this

17

0.2: Results from Lovic

Section

is

*

x

= { x : x E

X}

we usually drop the

u

for example,

quite

reasonable

superstructure, but

IN

* c IN

means

the

at

c

u~

first

uX

E

*IN. In fact,

level

is useful above that level.

u

X

and identify

of

the

Whenever we

feel there might be confusion, we will pedantically include all the

and

*'s

U'S.

DEFINITION:

(0.2.5)

Y E 91

An element

a)

entity o r

Y =

indiuidual if there is an

Y E 91

such that

is called an internal entity ( o r

indiuidual) if there is an

Y E 91

An element

c)

X E 5

*x. An eLement

b)

is called (an extended) standard

X

E I

such that

Y E *X. Y

is called an external entity i f

is not internal.

Every hyperreal number belongs = it

*X o ,

*IR.

to

*IR

Similarly, every

is internal.

In fact,

is

u~

1 that

internal

individual

The extended set

is shown in Appendix

external!

r E

IR

*IR

of

because

y E Yo

91,

is "standard". while

viewed

as

uIR

C

for some

p.

then

E

*IR

is

is external f o r euery infinite entity

X E I.Extended standard sets are internal because i f

*X

it

*X

P

by Leibniz Principle.

X E X

P

Chapter

18

0: Preliminarv Constructions

(0.2.6) EXERCISE: Show set

f o r any

that

standard

of

I\Xo. uD(A)

E

*D(A)

A,

subsets of

is

D(*A)\*D(A)

A , and

internal subsets of

A

entity,

the

*natural

numbers, A

B E

of

O(IN)

*IN.

If

*B ,

A =

such that

For example,

* prime numbers. *IN I (Va.bE*IN)[l

which are

dy

"D(N).

E:

an extended standard entity. the set of

set

t s t h e set o f

Let us consider a special case of entities of

then there is a

the

A.

external subsets o f

subsets of the

is

that is.

A

A

is

might be equal to

In that case, by Leibniz Principle,

<

A = {p E

a

<

b

<

p 3 a - b # p]}

Part (b) of Leibniz' Principle asserts that nonstandard prime numbers, is nonempty.

*B\'B,

=

*B .

the set of

There are unlimited (or

infinite) primes!

*IN

Next we consider an internal subset of the

F2

extension : IN +

B(IN)

of

any

standard

F2(m)

given by

set.

We

= {k E IN I 1

which is not

use

<

k

<

the

function

m}.

We know

that (Vm so

by

E

(Vk

E D(#)]

N)[[F2(m)

E

w)[k

E

1 < . k < mll

F2(m)

*-transform (Vm

E

*

N)"*F2(m)

(Vk E *IN)[k

E

For an infinite integer standard, that is, The fact that

m E

*F2(m) uIN

*F2(m)

E

E

*

D(")1

a 1

<

&

k

*NUN, *F2(m) *D(IN)\'B(N).

<

m]]. is internal but not

is external means that E

D("IN)\*[D(IN)].

19

0.2: Results from Logic

Section

Hence we have the strict inclusions

' *[O(~)l ' uCa(wl

O(*W (0.2.7) EXERCISE: h E *H\H

Let

*natural

be a n unlimited

* f4, * f3

extended functions

*F1

and

number.

Use the

(0.1) t o

o f section

show that the set

T =

{t

*IR :

E

*[O(lR)]

is a n element o f

Also, u s e 1

<

j

<

m}

E

fl,

k

8 1

<

k 5 h!]}

and hence is a n internal set.

f5

*[O(IN)]

=

(3k E *lN)[t

F2

and

where

m = n

Informally, we might denote

n

T

W =

show that

to

{j E

*H

n = h!

and

by

T = (6t,26t.**-,l}. where

1 6t = -, h!

and

W = {1,2.***,m}. The set of all internal sample sequences taken f r o m urns with elements along the

in

T-time axis is the internal set

1 R = W Y = * Hl(T,W), where

Hi

taken from internal, n = h!). an

the previous but

*finite

exercise. and

m-element

P

E

Fin )[Gi(HT(X.Y

P

W

are

is not

only

and

set

R

mn

elements

"There are

or E Fin ) ( V Y

The

contains

Here is what we mean.

n-element set into an

(VX

T

is the function of section (0.1) and

mn

(where

functions from

Chauter

20

0: Preliminarv Constructions

(0.2.8) EXERCISE:

H,W E

Verify that is a n i n t e r n a l

{k

E

*IN

:

1

<

*Finl.

so t h a t

R

function mapping

k

<

mn}.

( S e e the

#

[R]

n

= m ,

bijectively

and there

onto

t h e set

+ - t r a n s f o r m o f " f i n i t e " at t h e

b e g i n n i n g o f the s e c t i o n . )

officially an entity.

*B(H)

may write

internal

subsets

of

is not

However, because of exercise (0.1.4). * 1

to mean

any internal entity,

13 D(X)

X

The power set function

Y

E

G1(H)

*X

Y

P' is

and, in general, i f

Y

we is

then the internal set of all denoted

*B(Y).

but

it

is

officially,

*O(Y) the extended function

*GY

but certainly internal set

= *GY(Y).

applied at a , possibly nonstandard. Y.

(0.2.9) EXERCISE: S h o w t h a t t h e i n t e r n a l set o f i n t e r n a l e u e n t s . & =

is a

*f i n i t e

*a(n)

*c a r d i n a l i t y ,

set a n d c o m p u t e i t s

#

[&I.

ANDERSON'S INFINITESIMAL RANDOW WALK Finally, use of the

* summation

Hi,

function,

of section

(0.1) makes i t clear that the extended formula,

(0.2.10)

B(t.o)

= X[P(w,)fi

s E

H 8 6t

<

s

<

t].

defines an internal function called Anderson random walk once we

have given property

an

internal #

that

function

[/3-'({-1})]

/3 :

W

+ {-l,+l}

=

= #[/3-'({+1})]

done explicitly in several ways.

5.

with This

the

can be

Here are two independent ways.

EXERCISE:

(0.2.11) a)

21

0.2: Results from Logic

Section

Show that the definitions

and

both

1

<

define

w

For 9

w

<

m/2

+1,

if

w

>

m/2

functions

W

from

= {w E

*

IN

:

with the same number o f +l's and -1's. n * m = n ,, n = h! and h i s an unlimited natural

number,

-1

if

m}

where

b)

internal

-1.

h

€ *IN\IN.

(i.j)

= (1.2)

and

(2.1)

and for

k = +1

and

show that

There is a very useful logical principle that one may use in situations like the previous exercises.

We used explicit

functions for the exercises to help the beginner.

more general rule.

Here is the

22

Chapter

0:

Preliminarv Constructions

THE INTERNAL DEFINITION PRINCIPLE:

(0.2.12)

A fo r m u la

infinitesimal analysis is said to be a

of

bounded internal formula if i t c a n be expressed in terms of

E,

=

and internal entities o f

quantifiers.

A = {y E B @(y)

:

An entity @(y)},

A E

where

using bounded

?4

4,

is internal i f and only if

B

is an internal entity and

y

is a bounded internal formula w i t h

as the only

@(y).

f r e e uariable o f

For example, W={kE*lNIl
*lN.

<,

1

and

p2

above are internal, s o the exercise

and the internal constant

this principle. using

The formulas for

(0.2.9) follows from

appropriately

defined

standard

applied at the nonstandard point

moral of this functional approach is that if f o r nonstandard

a,

confusing, because making

b

p,

However, the reader may solve the exercise by

extensions of

*K1, *K2

m.

internal.)

then f

b

C DxR.

is internal. so

*f E

m.

b =

functions

The general

* f(a),

even

(This is trivial, if

*Dx*R.

and

b E *R.

23

Basic Infinitesimals

(0.3)

times

At

cumbersome.

the notation

of

infinitesimal

ON.

such as

we

have

=

*f(*a)

is

* [f(a)].

Hence,

*'S

valued

h!, m ,

function

a

f.

function

on the n

as well as

Because of

etc.

the natural

extension,

that

*

this we drop the

is.

on

f.

may mean either the standard sine at Euler's

sin(=) T

number over pi or nonstandard,

-.

+, real

a

*f

ex tension

on familiar sets in

Of course, we also drop the

arithmetic operations Whenever

**S

We remarked in section (0.2) that we drop the

on standard individuals and drop the

X1

analysis can be

its natural extension, whereas, if

sin(6)

really means

If we fear a

(*sin)(6).

possibility of confusion, we will u s e all the

is

6

*IS

and

O".

(0.3.1) DEFINITION:

A

1.1

<

number m

r E

*IR

some

for

is

Limited

standard

"finite") i f

(or

Otherwise. i t is unLimited (or "infinite"). limited scalars by

0

A number

*IR

6

E

infinitesimal

(small

E

write

a

a

<<

b

5

if

scalars

if

a-b

b

if

a

b

but

a

<

Oil.

We denote the

161

is infinitesimal if m

a.b

E

(big oh).

every standard natural. number

*IR.

m

natural. number

by

o

€ ON.

We denote oh).

is infinitesimal we write

<

b a

or

*

a

Z

b.

1 < ; for

Also. a

%

b.

the for We

whereas, we write

b.

In Appendix 1 we show that

*IN

"looks like"

IN

followed

Chapter

24

0: Preliminarv Constructions

by infinitely many new numbers, that is, i f n

>

m

for

m

every

*natural

nonstandard

IN

€

= ON.

In

n

other

then

IN\OIN,

words,

numbers are "infinite".

may cause confusion with

Y

€

all

the

Since that term

"*finiteness" we also to call these

numbers "unlimited. "

It

follows

IR

extension of just

says,

from

algebra

are

1 3 (3 m E a)[€ < --]I.

no

axiom

*Ei.

in

holds!)

for

m E IN,

standard

-.-> 6>

> n

*#\IN.

E

>

6

Principle

standard

Of

* - a .

such as numbers

around each one.

asserts

that

time,

It also asserts

that

there are

E.

>

0

62

>

6m

>

6" > * * *

>

0

the

IR,

in

x, 1

with

a

x # 0 there

the

x+r # r ,

implies are

unlimited

of

inf initesimals

w i t h maximal order ideal to

Ei

a.

a r e a totally o r d e r e d ring

a n d the homomorphism

c a l l e d the standard part.

01%

is

-& IR

is

T h e quotient field o C 0

the

clustered

Here is a fancy way to express this idea.

0,

so

* real

limited numbers are just

"monad"

T h e limited numbers,

an

This is because we may transfer

course, but

distinct points

(0.3.2) PROPOSITION:

isomorphic

0

same

x E IR.

the statement that for all

numbers

>

€ R)[E

B

the

(At

infinitely many

infinitesimal distance away.

r + 6 # r+ti2 # r

(V

Leibniz' Principle says that around each

there are

real,

field

Archimedes' axiom

infinitely many distinct infinitesimals. for E

ordered

infinitesimals,"

Thus, Leibniz'

infinitesimals

there are

proper

a

must be non-archimedean.

"There

*archimedean

that

Section

In other words, every s

E IR

25

0.3: Basic Infinitesimals

infinitely nearby,

infinitesimal

limited st(r) =

The fact

L .

r E or

s

that

*Ut

has a standard r =

for some

S+L

the infinitesimals

form a

maximal order ideal means that i f a number has magnitude less than an

infinitesimal, then

is

it

also

infinitesimal, only

infinitesimals have unlimited reciprocals, and a limited number times an infinitesimal is infinitesimal ('moderate size times very small is very small'). found

in one of

The proof of this assertion can be

the references

such as Stroyan & Luxemburg

[1976. (4.4.4)-(4.4.7)].

For our work in measure theory i t is very convenient to define the extended standard part into the standard two point compactification of

IR,

[-m.+m].

(0.3.3) DEFINITION: The map

st :

st(r) -m

*IR

a[-m,m]

-B

,

if

r

,

if

r

,

if

r

The love knot symbols

fm

is g i u e n b y

i s limited.

> <

0

and

r

0

and

r

r E 0 i s unlimited is unlimited

represent standard objects, not

nonstandard numbers. The standard part map extends to the finite coordinate spaces

*Rd

dimensional

by

St((X l.*-..Xd)) = (St(X,),*".St(Xd))

Similarly, we write

x

%

y

for

x.y

E

*IRd

if

xj

-

yj

for

1 i j < d . The notion of

infinitesimal is external.

The interplay

26

0: Preliminary Constructions

Chapter

between

internal

external

notions

following

notions, is

the

is a useful

principle"

that

infinitesimal

says

at

to

which of

"art"

Robinson's

"continuity internal

the boundary

transfer

principle"

sequences between

applies.

and

theory.

The

"permanence

or

cannot

stop

limited and

being

unlimited

We will use this frequently in stochastic analysis in

indices.

conjunction with the saturation principles of section ( 0 . 4 ) .

(0.3.4)

ROBINSON'S SEQUENTIAL LEMMA: If

an

is an internal sequence and

a m

m E ON. then there is an unLimited

standard

Z

f o r all

0

n E

*I

such

that ak 2 0

f o r all

k E *IN

such that

0

<

k

<

n.

PROOF : The set

I = {n

E

*IN

I (V k E

<

*IN)[k

n 3 k E dom(a) & lak]

is internal by the Internal Definition Principle. the

external

nonstandard lakl

1

< r;

set n

E

ON

by

hypothesis

k

When

is

and

<

1

i;]}

It contains

thus

contains

a

<

n,

infinite and

k

z 0.

POISSON PROCESSES: Now we will construct two independent approximate Poisson processes.

Let

a1.a2 E IR

be

standard

6t = h! and W has m = h! h! an internal function a1 : W 4 (0.1) by Recall that

1 ,

,

if

w

<

[ma16t]

otherwise,

positive elements.

numbers. Define

Section

[-I

where that

27

0.3: Basic Infinitesimals

mast-1

bl z a l .

<

-1

#

<

[ a l (l)]

mast.

integer function. We see # -1 [a, (111 = b16t, with

so

Define an internal function

=

a2(w)

*greatest

denotes the

{

<

1 ,

if

w

1 ,

if

[malat]

0 ,

otherwise.

[mala26t

<

#W W + (0.1) by

:

2

3

<

w

a2

[ma16tl+~(m-ma16t)a2~t]

(0.3.5) EXERCISE: #

-1 [a2 (111

Show that

a)

For

b)

#

= b26t

where

b2Z a2.

CWl

(i,j) = (1-2) and

(2,1) and f o r

k = 1

and

0

show that

Now suppose that j = 2.

(a.a)=(a

raj)

for either

We define an internal stochastic process

J

:

j = 1

or

HxR

*IN

+

by (0.3.6)

J(t.o)

= Y . [ a ( o s ) : s E H. 6t .( s
This process increases by unit jumps as close

to a Poisson process with rate

distribution, we have the following with

increases.

t

a. b

Z

It is

In terms of a.

the

Chapter

28

0: Preliminary Constructions

k

(0.3.7)

P[{o

: J(t.o)

<

CtK -

1 [$]ph(l-p)

k}] =

hl

.

p = b6t

h=O

[$]

=

h=O

t k = (1-b6t)6t h=O

1

1

t -

h (b6t) (1-b6t)

t (1-b6t) 6t

(1-b6t)

6t

h- 1 TI (t-j6t) (b)h j = O h! ( 1-b6t)h

1k 5h h=O

Z e-at h=O One may verify that

t -

(1-b6t)6t

by

using

*binomial hand-side.

the

series

for

Z

expansion

for

b

Z

X

e

a,

and

applying

the

theorem and Robinson's Sequential Lemma to the leftSee Stroyan & Luxemburg [1976] for this and other

elementary properties of

X

e

using infinitesimals.

Next, we want to calculate the distribution of Anderson's infinitesimal random walk (0.2.9). another elementary property of

eX .

For this computation we need

Section

29

0.3: Basic Infinitesimals

(0.3.8) STIRLING'S FORMULA:

For

n

E

*IN. 1 12n+l

<

-

1 n! en < e12n &Ginn

The proof of this formula can be found in Feller [1968, vol. 1. p. 541. We also need a very simple form of

the integral on the

The treatment of Lebesgue integrals in Chapter 2 below

line.

includes this case, s o we present the special case without much explanation.

(0.3.9) "CAUCHY" INTEGRALS: The

integral

f : [a.b] 4

of

a

continuous

standard

<

b, step at]

satisfies:

f(x)dx

Z )[f(t)dt

:

a

f o r any positiue infinitesimal

t

<

6t

We compute the distribution of sums.

function

B(t.w)

* finite

using the

This fact about integrals simply tells us that we have

computed the classical normal distribution at the end.

(0.3.10) EXERCISE: -

Suppose

that

f : [a.b]

--f

IR

F

and

are standard functions such that wheneuer

*IR

and

F(x+6x)-F(x)

6x

is

a

= [f(x)+c(x,6x)]6x

positiue where

a

:

<

[a.b] x

<

inftnitestmal L

Z 0.

b

*R in then

Proue that

ChaDter

30

[

f(x)dx

using

I[f(x)6x

Z

a

<

x

* linearity

the

* triangle

:

0 : Preliminarv Constructions

<

b. step 6x1

* finite

of

Z

F(b)-F(a)

by

summation,

the

inequality, and the estimate

I~[L(x,~x)-~x

x

b. step 6x11

a

<

max[IL(x.dx)l

Obserue that the

Of course

<

:

* finite

:

a

<

x

<

b , step 6x]*(b-a).

maximum i s attained by transfer.

this exercise is one half

of

the Fundamental

Theorem of Integral Calculus. on

f(x)

The usual continuity hypothesis 6F is contained in the assumption that g Z f(x) at

nonstandard

between

X'S

and

a

appear in the summation from

a

to

b. b

Such

x's

in steps of

certainly 6 x * - - . see

Barwise [1977. Ch. A.61.

For convenience in our discussion of

some parameters.

Let

6t =

'

B(t,w),

and whenever

we define in

t

H

is an

2n 1 even

multiple

probability

vt

of

z.

2nt = t

We

first

of "exactly half heads" after

or equivalently, of

where

6t.

{w : B(t.w)

estimate

n = nt

the

tosses,

= 0).

n = nt. Using Stirling's formula for

t

such that

n

t

Section

31

0.3: Basic Infinitesimals

is infinite, we obtain:

1 4 q (1+~~).

= -

Notice

that

even

for

> t d K

L

t

infinitesimal

is infinite.

defined),

n

make

infinitesimal.

L~

where

Z

0

t

> fi

(with

nt

This is all that is needed to

Changing back to the time variables,

we have

Tt =

(0.3.11)

The

only

-fi (l+rt).

values

B(t,w)

negative integer multiples of

We

shall only be

limited (in

can assume are positive 2 c .

consider the probabilities of = 2kfi.

{w

:

we let

so

6x

= x}

B(t.o)

!!- Z

0

and

1x1 = k6x

for

(since

and

= 2&?

x

interested in these when

in which case

0)

z 0.

Lt

G

is

A/(2nl)

is

“1

d

limited,

q

is

infinitesimal

infiniteeimal is infinitesimal). {w

and

:

B(t.w)

because an so

t

have

Notice that

k

times

{a : B(t.o)

= x}

*number

the same

is in one by having

limited

of

elements

more than half “heads.“

interchanging “heads“ and “tails“ internally maps one on the

other. n

w

= -x}

and

Let

k

be as above and

n = nt ,

where

t

*

0,

so

32

Chapter

P[B(t.o)

=

X]

=

0: Preliminarv Constructions

-

2n ] n [ ' = .22n

k-1 TI [n-h] h=O k TI [n+h] h= 1

k-1

2h

2h

h

by formulas extended from algebra.

since

log

is differentiable at

by the formula for

Lh

1,

and estimates like (0.3.10) for

8.

Section

0.3:

= qte

2 Notice that

33

Basic Infinitesimals

nt =

k

'

where

(1+rt).

[Tkfi e

3

L

t

=: 0.

is finite and

nt

with

e

L~

2 f i = 6x

Finally we substitute (0.3.11).

=: 0.

(n=n,)

so

=: 0.

2 2 k /n = x /(2t)

and

to obtain:

DE MOIVRE'S LIMIT THEOREH:

(0.3.12)

If

1x1

is finite ( o r Limited in

0)

and

t

(as

0

and

aboue) is not infinitesimaL. then

Hence, i f

a , b E 0.

P[a

B(t)

is nearly

uariance

<

and

B(t.w)

*

t

<

0,

b] z

then

G

[

2

e-k

df,

normally distributed uith mean

t.

PROOF : Aside from the substitution mentioned before the statement,

Chapter

34

0: Preliminarv Constructions

all that remains is to associate the sum below with the integral.

The sum over possible values of

B(t),

(0.3.10). infinitely close

is, by (0.3.9) and an estimate like to the integral

This proves the result. The

reader

may

wish

to

continuity of the paths of Appendix

1

give

the

B(t).

S-continuous.

smaller

The sets

*probability,

ahead

R:

the

formulation of

off a set

o

to

proof

of

The preliminary results in

infinitesimal

while (5.2.3) shows that for is

look

A =

continuity R",.

B(*,o)

are internal and of smaller and

but the set

A

is external.

The measure

of chapter 1 allows us to conveniently discuss such "Loeb sets" and their probabilities. Results of section (5.3) show that we can take "standard parts" of the paths of sense.

J(t.w)

from (0.3.6) in the appropriate

What is required there is that two or more jumps do not

occur in infinitesimal time.

Again, this happens on an external

set which can be approximated by bigger and bigger internal sets of probability finitely less than

1.

It is convenient to have

a probability defined on the external set s o that

J

has jumps of size

1

with probability

that we can say 1.

35

(0.4)

Saturation & Comprehension Part (b) of Leibniz' Transfer Principle (0.2.3) requires

that the

*

mapping is "polyenlarging."

All things considered, we feel that a

what that term means. polyenlargement analysis.

is

the

There are

simplest

Loeb

for

measure

models, but we

questions to the specialists.

The bounded

framework

interesting

various special nonstandard

axioms for set theory.

This section explains

infinitesimal questions

shall

in

leave such

(Some of these require additional

For example, see Ross [1984].)

forms of

Nelson's

axioms

for

Internal

Set

Theory hold in a polyenlargement. but not in an enlargement. Moreover, polyenlargements have a stronger "saturation property" than Nelson's Idealization Principle.

(0.4.1) DEFINITION: A

superstructure

poLyenLarging

if

it

given

as

successive enlargements, where

K.

satisfying

The

K

image

polyenlargement. detail below.

>

is

*

extension a

:

direct

z-91 Limit

is of

K

is a reguLar cardinal.

card(Z).

of

polyenlarging

a

extension

is

called

a

Successive enlargements are described in more First we give the main properties which make

polyenlargements useful. We say that a family of sets intersection property nonempty intersection.

if

every

S

has the finite

finite subfamily

of

S

has

Chapter

36

THE SATURATION PRINCIPLE:

(0.4.2)

*

Let Suppose

: L + 91

P

that

internal

Y

0

Y

F

E

3

this.

internal

sets and

F E 0 is an internal subset

Y

E

*Xm+l

m

! m E

standard parts,

to

st P[Bm]

=

*X

m

The

* U Xo.

is a countable sequence has

the finite intersection m Bm = fl A is a nonempty

3 E

j=O

The sequence of

probabilities

for some m.

*X

that we will use saturation is

that

Then for each

internal set. internal

{Am

% =

intersection

external; only its elements,

91, then

the primary ways

an

n[F : F E % ] # 0.

Then

If every

E

extension.

subsets o f

finite

are subsets of the standard set

Suppose that

property.

internal the

card(%).

need be internal.

One of

of

<

has

is usually

of the internal set sets

polyenlarging

a

which

card(%)

The family

~

be

i s a family o f

set

property and

F E %.

0: Preliminary Constructions

Bm's

these

may assign i t the probability

the

decreases. fl[Bm

inf[bm

If we assign

decreases.

sets, then

= bm,

that the external intersection,

j

:

m

sequence

Saturation says f 0,

€

of

hence we

This is the

: m E

whole secret of Loeb's measure extension. Another way to view saturation is as a kind of "completeness." put

the "biggest

because

[am.;]

Consider the gap in

1

5+5 > 5). , for

infinitesimal"

*IR

where we might like to

(no such number

[

exists

No countable nested sequence of intervals

6,,,% 0. will have empty intersection; there will

always be an infinitesimal

11

>

6m

for all

m E

An extremely useful consequence of saturation is a function extension property called "comprehension."

The model can

0.4:

Section

37

Saturation 81 ComDrehension

comprehend "small" infinite sets by extending them (we can even

* finite).

make the extension

(0.4.3) THE COMPREHENSION PRINCIPLE: fE 9

Let

f,

the d o m a i n o f

D

R

and

:

<

card(dom(f))

satisfies

be internal entities such that

R 2 rng(f).

F

be a n external f u n c t i o n and suppose that

f

Then

D + R.

has

an

card(%).

Let

D 2 dom(f)

and

internal

x E dom(f).

that is. f o r each

extension = f(x).

F(x)

The main way we will use comprehension is in the countable case.

{Am : m

For example, i f

internal subsets of an

D =

*IN.

R

such that Many integers n

=

*D(V) =

A(m)

E

Am

for standard first

V,

set

is

E

*IR :

to

treat

T

is

The initial segments

k = ;&

(3 k E *IN)[t

S-dense in

shows that the cardinality of least the continuum.

infinite

Another temptation

n

1

[O.l].

T

so

*IN[l.n]

is infinite.

I k I

The

is

n]}

internal one-to-one image of the initial segment of The set

*D(V)

A:*H

sums as some sort of countable series.

are all externally uncountable sets when

T = {t

take

m.

inclination

These are misleading analogies.

set

then we may

like countable ordinals.

* finite

is t o consider

internal

and find an internal sequence

people's

*IN\'I

is a countable family of

E

*IN

an

above.

the external map

(and hence

*IN[l.n])

st

is at

(In certain nonenlargement nonstandard

models the cardinalilty can be exactly the continuum.) Another

important

property

of

polyenlargements

is

the

"internal homogeneity" property of Henson's Lemma given below.

Chapter

38

0: Preliminary Constructions

One consequence of this property is the following result which we will find useful in construction o f nonmeasurable sets.

The

result says internal sets are homogeneous in size.

(0.4.4) PROPOSITION:

*I

Let sets,

A l l infinite internal

be a p o l y e n l a r g e m e n t .

including

*f i n i t e

unlimited

sets, have

the

same

external cardinality.

The

common

gigantic

by

cardinality

of

the

non-set-theorists'

internal

will

be

In t e rna 1

standards.

* finite

cardinalities, in particular,

sets

ones, "seem smaller to

the model" because those cardinalities can only be tested with

For example, there is no internal functior

internal functions. from

onto

*lN[l.m]

while there is an external

*I[l.m+l].

bijection.

(0.4.5) EXERCISE: a) euery

Use

saturation

infinite

x E X).

b)

Why i s Prove

X E

(HINT:

L.

*X\{*x} that

to prove

S =

Let

# 0

for

* * { X\{ x}

I

internaL?)

there are positive

considering the f a m i l y

*X\aX

that

S =

* { (

i n f i n i t e s i m a l s by

0 , ~ :) a E

R+}.

Now we give the detailed definition o f a direct limit of successive enlargements. Let [L'

Z1

We begin with a single enlargement.

be a superstructure over a set of individuals

= U Xm. 1 where

Xi+l = O(U m Xk). l as in section 0.1.1 k=O

Let

I2

Section

39

Saturation & ComDrehension

0.4:

be a superstructure over a set of individuals an injection

2 il(Xo) = Xo 2 1

Principle

x2

is a superstructure extension i f

satisfies part (a) of Leibniz' Transfer

is an elementary

(i?

and =

E

X1

extension of

for

with a constant for each element of

the 3').

is a superstructure extension, then an entity Y E X2 2 i -internal provided there exists X E L1 such that 1 2 2 1 2 il(X). An internal entity Y E il(Xm) is called il-finite

If

19

is

Y

+

2 il

and

language of

L'

14 :

We say that

Xg.

E

if every i:-internal

(0.4.6)

injection of

Y

Y

into

is onto.

DEFINITION:

A superstructure extension enlargement o f

2

il-finite

Y

L

1

X

i f for every entity

2 such that

E L

Enlargements

2 1 i l : L -+L

may

be

2 {il(x)

I x E

constructed

by

E L

1

is

an

there is an

2 X} E Y C il(X).

forming

"adequate"

ultrapowers as defined in Appendix 2

(0.4.7)

EXERCISE:

-,

rf i2 : z1 L is enlarging for 1 3 2 i 2 : L + L3 is enlarging f o r L2, then 1 enlarging for L .

L1 i20i

and

4

is

SUCCESSIVE ENLARGEMENTS: Let

K

be an infinite cardinal.

successive enlargements of Let

L

A direct l i m i t of

is given inductively as follows.

L0 = L . be our original superstructure.

be an enlargement.

If

X

K

Let

io1 : L0

-+

L1

is a cardinal less than or equal to

ChaDter

40

K,

Preliminarv Constructions

0:

i7 = i ~ o i ~ .whenever

satisfies

a

01

is defined and

4 Z7

: %a

:i

and the family of enlargements

< P <

we

7 .

proceed

inductively in two cases.

1:

Case

h = P + 1,

If

enlargement and define Case 2:

If

i

h

a =

let

'PP+lOiPa'

P+l: ZP

'I3 for a

+

be an

< p.

is a limit ordinal note that the usual

algebraic definition cannot be used because the limit mappings go into a superstructure over a set of individuals.

Xt

a set of individuals under

x E Xg

and

U[Xg

equivalent to the union

x

the usual identification, y E XE.

Then

= {ip(x)

superstructure levels, ia(A)

y

Px) i,(

if h ia

define h

h

I

First take :

a

<

h]

= y. where

by

induction

on

I x E

This completes the detailed definition of direct limit of

A set

successive enlargements.

A

=

i:(B).

element

B E %"

for some of

some

internal. refer to

2 card+(%), K

= io.

the

set.

0-standard and

successive enlargements K

and

a-standard

A polyenlargement

*

A E

*

:

9L + 9

where

K

is

a-internaL

The

if

a-standard

A

if

terms

is an and

standard

0-internal. is the direct limit of is

regular

first cardinal greater

and

than

Notice

K

satisfies

card(%),

Successor cardinals are always regular.

in Chang-Keisler[1977,A.25.p.505].) are

xK

and

(See the proof

that

* finite

sets

iK-finite in the sense above. 0

PROOF OF THE SATURATION PRINCIPLE: Let card(%)

%

<

be a family of internal subsets of

card(%)

*X

satisfying

and having the finite intersection property.

Since successor cardinals are regular, there is an

a

<

card+(%)

Section

E X z such that

and a family %a Since

B

is enlarging,

iK a

41

Saturation & ComDrehension

0.4:

3 C d C i;(B").

The

last

5 = {iK(F a a ) I Fa

is contained in a set

has

* finite

the

Xu

property by transfer of that property from

n

Sa} g i;(B").

E

* finite

set

d.

intersection

and

therefore

z 0.

5 a n d

PROOF OF THE COMPREHENSION PRINCIPLE: Use the notation of ( 0 . 4 . 3 ) . the set

Ax

F E n[Ax

:

=

{F : F

:

D

x E dom(f)]

For each

R is internal

+

x E dom(f)

&

define

= f(x)}.

F(x)

Any

satisfies the assertion.

PROOF OF ( 0 . 4 . 4 ) :

A

Let

B = iE(Ba).

is an

E

f

injection

F~+' E A~+'.

asising by transfer. an injection.

Let

F

+

Ba."

ga+2

fa+l

G

The

* finite

a F of

:

transfer

this

Fa+1 +

is embedded in a be an injection

oia+l :

fa+l a

A" + ga+l

is a finite subset of

fitl

of

Aa

set has such a map.

Aa

says that

The composition

Next, i f

* finite

an injection

:

iaa+1

is an injection extending

in a

Let

says that every

The enlarging property of

Ba+'

Xu.

We know that."for any finite subset

statement to

* finite

A = i:(A"), Aa+m = ia+m(Aa),

be infinite internal sets

Aa,Ba

for

Ba+m = iz+m(Ba). there

B

and

is

Ba+'

there

G.

Embedd

and also defined on

subset and transfer this property to obtain

. .

Ga+2

~

Aa+2 extending

ia+20f-1 a+l

a+l*

The map

a+2 injects B ~ +into ~ Aa+2. Continue this procedure ga+2O ia+l back and forth thru a countable number of steps s o that the injection

fa+m+l

extends the injection

l i m i t mapping is a bijection of

Aa+O

onto

ia+m+l -1 a+m oga+m. Ba+~

The

42

ChaDter

0:

Preliminary Constructions

BACK i

Ba+ 1

fa+1

I

; & : i

Ba+2 ia+3

a+2

ga+2

1

Aa+3

-

Ba+3

€r3 1

a+3

HENSON'S LEMMA:

(0.4.8)

For

each

card+(%)

first order

L

language

constants and relations.

with

A

if

eLementariLy equiualent structures f o r

L

less

than

B

are

and

whose domains

a n d r e L a t i o n s a r e internal. e n t i t i e s o f a p o l y e n l a r g e m e n t

*L .

A

then

and

B

are isomorphic.

PROOF : Let

A = (A.Rk)

and

B = (B.Sk)

denote

domains and relations of the L-structures. regular, such that La.

there exist

A = iz(Aa).

the

a

<

etc.

L-structures

card+(L), Since

internal

card+(%)

is

Aa, Rak, Ba, Sak in

Xa

*L

(Aa,Rak)

Since

the

is an enlargement of

(Ba.Sak)

and

are

elementarily equivalent. We may use the enlargement property of construct

an

elementary

(i:+l(Ba).iz+l(Sak)) o f (0.4.4) above.

L-monomorphism

ia+l, %a a

from

~

(Aa.Rak)

$a+l

to into

and continue back and forth as in the proof See Henson[1974]

for details.

43

CHAPTER

-FINITE

1:

&

HYPERFINITE UEASURES

[1975]

This chapter is an elementary treatment of Loeb's

construction of measures and related work as described in the

*Finite

foreword.

* finite

sets were defined in section (0.2) and the

summation

extension of illustrated

operation

may

*H;

the function many

(0.3.9-12).

uses

be

defined

the

sums

The standard part function,

natural

(0.1.5).

from exercise

*finite

of

by

in

We

(0.2.9)

st.

and

is defined in

section (0.3). According

to

"Littlewood's

Principles."

integration just amounts to three basic facts:

2)

sets are almost intervals. continuous.

3)

Lebesgue

1)

Lebesgue

Lebesgue functions are almost

Convergence is almost uniform.

The principles

analogous to (1) and (2) for hyperfinite measure theory allow us to replace measurable

sets and

functions by

formally

ones.

Saturation corresponds t o the third principle.

(1.1)

Limited Hyperfinite Measures

A

* finite

6 p : W + *[O.m),

weight

where the domain

*f i n i t e (positive) * function p : B(W) of

W,

A E

*B(W).

measure

by

p

Is

W

an

function

6p

is

The

the set

defined on all internal subsets

* summation = 2[6p(a)

internal

is an internal set.

associated with

+*[~,m)

p[A]

We say that

function

finite

of the weights,

:

a

E

A].

is a Z i m i t e d ( p o s i t i u e )

*f t n t t e

measure

if

p[V]

E

0.

function

The :

B(V) * [ O , m )

:

-

inner

O(V)

Hvperfinite Measures

The o u t e r m e a s u r e associated with

-p

-p[U]

E

1:

Chapter

44

= inf[st

[O,m)

defined by

p[A]

U C A

:

associated with

measure

is the set

p

*B(V)].

E

is

p

the

set

function

defined by

~ [ u ]=

sup[st

p[B]

*O(V)

:

B C U]

The inner and outer measures are defined on both internal and

V,

external subsets of

V.

subsets of

I

Since

may be defined from

while

p

is

by

is only defined on internal

p

* finite,

the weight function

6p(v) = p[{v}].

our treatment to situations where

p

One may generalize

is not given by a weight

function (see EXERCISE (1.1.8)).

DEFINITIONS:

(1.1.1)

T h e s i g m a a l g e b r a g e n e r a t e d by

*f i n i t e

of a

set

V

A set

M

r[M]

= ;[MI.

V

is c a l l e d The

the internal subsets

is c a l l e d t h e L o e b a l g e b r a ,

I(*O(V))

= Loeb(V).

p-measurable prouided

collection

of

p-measurable

sets

is

to

the

d e n 0 te d

Meas (p) . The

ltmtted

limited

hyperfinite measure

*f i n i t e

measure

p

6~

associated

ts

the

set

function

Section

p

:

1.1

Limited Hvperfinite Measures

-

Meas(p)

Theorem

tR

45

giuen by

PCMl =

SCMI

below

justifies

(1.1.6)

= ECMI.

this

terminology.

The

intermediate lemmas simply prove various parts of the result.

* finite

The

measure

can be manipulated with formal

p

combinatorics, but there are two things “wrong” with

*IR

takes values in

p

can

solve

this

problem

st p : *O(W) + IR.

IR.

instead of by

taking

This brings up

class of

internal

algebra.

Suppose that

(Am

:

intersection property

and

is not a

Am C n[A,

If

V.

(Am\Am

:

:

Am = n[A,

intersections

of

internal

total

Therefore we

internal and s o

sets are not

is

has the

cannot have an empty

m E “IN]

:

the sigma

m E “IN]

m E “IN)

intersection by the saturation property ( 0 . 4 . 2 ) . cannot have

parts,

is a properly decreasing

internal. then the countable sequence finite

standard

*O(W).

m E “IN)

sequence of internal subsets of

is limited we

p

the second problem:

V.

subsets of

Since

First,

p.

countable

internal unless

they

reduce to finite intersections.

(1.1.2) REMARK: Since U1

C_

and

U2.

N

C

p

and

~ [ U l li G[U,l

M.

N

st

are monotone, and

is measurable.

~Cu,l i

rCU,l.

E[U]

<

; [ U ]

Thus if

and i f P(M)

= 0

Chapter

46

1:

Hvperfinite Measures

(1.1.3) LEMMA:

M C V positiue

is

measurable

standard

and

there

B.

B E M E A

s u c h that

if

and

only

exist

<

p[A\B]

if

for

internal

euery

A,B

sets

B.

PROOF : M

If internal =

sets

>

F(M)

is measurable, A Z M a B

-

p[A]

5;

then

for each standard

B E M G A

for

and

p[A\B]

such

p[A\B]

> 0 < a;

B

every

<

such

V[B1

that

+

exist

5>

E(M)

Conversely, assume that

a.

A.B

there are internal

F(M) - r(M) <

then

there

B

such that

a.

(1.1.4) LEMMA:

If

M

N

and

M fl N.

M\N.

a r e measurable, so a r e

M U N.

PROOF : M = V

showing that

0

internal

W e begin with

Fix a standard and

<

p[A\B]

>

B

a.

the

so

intersection,

p[A\B]

F = B n D

satisfy

The

<

5,

<

C 1 N 2 D.

<

p[BC\AC]

a

let

number and choose internal sets

+ p[C\D]

A.B

a

>

NC

be

A,B,C.D

< 5. E 1 M n N a F p[C\D]

B E N E A

AC C NC

and

0

is measurable.

so that

Now the complements satisfy

BC\AC = A n BC = A\B. For

and

V\N

a

BC

and

is measurable. given

so that

standard

A 2 M 2 B,

Then

E = A fl C

and

p[E\F]

and

5 p[A\B]

a. rest

M\N = M n.'N

of

M

u

the

N =

proof

(aC n ~

follows

~

1

~

from

.

set

algebra,

Section

47

Limited Hyperfinite Measures

1.1

(1.1.5) LEMMA:

If

(Mk)

UMk

then

is m e a s u r a b l e , a n d i f

U Mk] =

p[

is a c o u n t a b l e s e q u e n c e o f m e a s u r a b e s e t s ,

Z

k= 1

p[Mk].

the

are d sjoint,

Mk

the conuergent s e r i e s .

k= 1

PROOF : Without any loss of generality, we can always assume that

Mk

the

are

disjoint

to prove

that

their union

M

is

measurable. Given

<

p[Ak\Bkl

B

E OR+.

Zk+l' €

choose internal sets

Extend the sequence

Ak 2 Mk 2 Bk

(Ak.Bk)

with

to an internal

sequence by the Countable Comprehension Principle (0.4.3). Use the Internal Definition Principle to pick an infinite

n1

such

that

For each infinite

n

€

*1.

n

<

nl,

n hence

;[MI

<

6

+ Z p[Bk]. 1

Again the Internal Definition Principle tells us that there m m is a finite m € such that c[M] < E + Z p[Bk]. But Z p[Bk] 1 1

Chapter

48

m

5

= p[U Bk]

E[M],

that

so

;[MI

1 standard positive number.

so

<

E

1:

Hyperfinite Measures

+ E[M],

and

was any

E

Moreover,

that

re, this proves tha

Since the terms m

and additivity of

this

completes

the proof

of

countable

p

(1.1.6) THEOREM:

If

W

is a

*f i n i t e

6p : W +

set a n d

internal w e i g h t f u n c t i o n that sums t o a Limited measure,

p[A]

= 2[6p(a)

: a E

A],

*

finite

A.

then

(V,Meas(p),p)

is a

for internal

the limited h y p e r f i n i t e measure space

is a n

*[O.m)

complete countably additive finite positive measure space. Moreouer,

p

is t h e u n i q u e c o u n t a b l y a d d i t i v e e x t e n s i o n

st p[*]

of

p-compLetion o f

to

Loeb(W)

Meas(p)

and

ts

the

Loeb(V).

PROOF : The remarks and lemmas preceding the theorem show that is a complete countably additive measure.

internal

sets are

p-measurable.

so

JI

It is trivial that

Loeb(W)

E Meas(p).

The

Section

1.1

49

Limited HvDerfinite Measures

uniqueness and completion remarks follow from the approximation lemma since we can find internal

increasing and decreasing chains of Bk C M E Ak with p[Ak\Bk]< r;1 forcing

sets

= S-lim p[Bk]. k

= S-lim p[Ak]

p[M]

k

Because of saturation, we can approximate measurable sets by internal sets up to an error of

p-measure zero.

The error

is in the sense of the s y m m e t r i c s e t d i f f e r e n c e .

M v N = (M\N) U (N\M)

This

result

is a special case of Lemma (1.2.13) below.

cannot assert that there is an internal set p[A]

1 p[M].

The set

A 2 M

We

such that

in EXERCISE (1.1.13) has measure

a(r)

zero, but every internal superset has noninfinitesimal measure.

(1.1.7) LEMMA:

A set

(Sets are almost finite.)

M

is m e a s u r a b l e

hyperfinite measure

p

w i t h respect

to a limited

i f a n d o n l y i f it d i f f e r s f r o m a n

i n t e r n a l s e t by a s e t o f m e a s u r e z e r o , t h a t is. a n internal

A

p[M

s u c h that

The proof of (1.1.7)

v A]

t h e r e is

= 0.

is left as an exercise in case our

reader is only interested in probability measures and plans to skip

section

observing

that

rephrasing.

(1.2).

In

the proof

that of

case,

solve

(1.2.13) works

the here

exercise with

by

minor

50

Chapter

1:

HvDerfinite Measures

(1.1.8) EXERCISE:

V

Let

a)

V.

arbitrary algebra o f internal subsets o f

a monotone

finitely additiue

w i t h limited

p[V]

V

measures o n

function

is given.

E 0

be an

d

be an internal set and let

Suppose that :

p

d

* *[O.m)

Define inner and outer

by

= inf[st

L[U]

p[A]

E A

: U

E d]

and = sup[st

r[U]

and

p[B]

p-measurability by agreement o f the inner and outer

measures.

Show

that Theorem (1.1.6) still holds where d

is the smallest sigma algebra containing

Loeb(d)

Suppose in addition to part (a) that the algebra

b)

and set function

: d + *[O,m)

p

LEMMA (1.1.7) hoLds, where

d

algebra

V

are internal.

Then

A E d.

Suppose i n addition to parts (a) and ( b ) that the

c)

on

2 B E d],

: U

* finite.

is

by

v

u

if wheneuer

V

= ( W E )

u E A E d.

* finite

-

T h e set

Define an equiualence relation

is.

v E A.

then

and the uniform hyper-

V

finite measure ouer part (b). it t s a

(Note:

* finite

Since

(Ak.Bk) However.

p

in

[v] = n[A:

v E A]

belongs to

d

of

since

intersection o f an internal algebra.)

may the

is isomorphic to the extension

be

proof

external, of

the

one

must

analogue

extend to

Lemma

more

than

(1.1.5).

the sequences of numbers have additive extensions out

Section

1.1

51

Limited HyDerfinite Measures

to some infinite

nl.

The next proposition is the famous Caratheodory trick that we could have used in the next section.

(1.1.9) PROPOSITION: p[W]

Let

M C

U'

is

a limited

be

=

A

measure.

i f and only i f f o r each

p-measurable

F[U]

*f t n i t e

set

U C U',

L[U n M] + F[U\M].

PROOF : Fix a standard

B

Now take an arbitrary that

<

p[C]

U

+

F[U]

Thus, measurability

0. Suppose

A.B

we may pick internal

so

>

g.

with

E V

M

is measurable, s o that

B C M C A

and let

C 1 U

and

We see that

implies the condition, since the opposite

If the equation holds for + ]#\rO[;

= ;[MI

(1.1.10)

= ;[MI

We say that a

every

M

+ F[V]

U,

and all

- E[M]

and

take

;[MI

-p.

U = V, = &[MI.

DEFINITION:

p

ertenston

v

E

V.

5.

be an internal set

inequality always follows from the monotone property of

F[V]

<

p[A\B]

are

* ftntte

measure

non-atomic

p

a n d its hyperfinite

provtded

6p(v)

Z 0

for

so

Chapter

52

1:

HvDerfinite Measures

(1.1.11) EXAMPLE: One of the most important types of example is a uniform

* finite

*cardinality

#

1. t = k6t. k

E *IN},

[V] = n and 1 6p(v) = n for all v in V. As long as n is infinite p is non-atomic. One example of this is the discrete time axis from measure.

0

to 1 6t = n

This is when the

U = {t

1.

*IR

E

<

0

:

for some infinite

n

<

t

*1.

in

where

6p(t) = 6t

In this case

Another uniform probability is the space R = W # n 1 of Chapter 0. In this case [R] = m and 6 p ( o ) = 6P(o) = -.n for all

t.

U

m

(1.1.12) EXAMPLE (A nonmeasurable set): Let

n

external

cardinality

{k E *IN

of

1

:

<

k

<

[Recall that in polyenlargement models all the

same

1 < k < * finite

external

log2(n)}

cardinality.]

V.

for any

A

the cardinality of

68

choose

B

P+ 1

let and

<

xP

= BP

Ba A\U

=

since

7 .

#

yP B

E

Let

: 6

7

P

<

) C

a}

*l 3 ( V ) ,

E[U]

U = {xa : a

= E[V\U]

minimal.

V={kE

*

IN:

d =

inductively. card(A

P

(A a : a <

Let

) =

7 )

Bo = 0. we may

7 ,

For successor ordinals let

<

A

7 )

For limit

in

and observe that

i[V]

a

U fl A

si.

If

= 0.

probability measure then. since

= card(V)

card($)

be the first ordinal with

and

7

A \B p .

are nonempty for each

so

that the

finite sets have

Let

The only internal subsets of either finite,

is

for such a pair of points.

Let

p
n}

*

and index the elements,

both from

u {XP.YP) U

€ sl.

card(B

so

be the collection of all infinite

Since d

Ba = {x6,y6

We define sets

p

d

and let

subsets of

= card(A)

For

*IN

be an infinite natural number from

= 1.

U p

F[U]

or

V\U

is

a

#

must be non-atomic

r[U].

1.1

Section

53

Limited Hyperfinite Measures

T h i s s h o w s that t h e s e t

U

is n o n - p - m e a s u r a b l e

non-atomic h y p e r f i n i t e probability measure

f o r every

p.

(1.1.13) EXAMPLE:

V = U = {t

In this example we let

<

5

E

*IR

= k6t, k E *IN,

: t

6t = 1 n. for some infinite

*

IN.

We may

think of this as an infinitesimally discrete time axis.

We let

1

k

p(t)

n}

where

= 6t

for all

0

number,

<

r

<

1.

If

U.

in

t

n

IR

in

r

a(r) = {t E U : t

let

st(t) = r .

"instant" of times such that

in

is a standard

r}

Z

denote

the

a(r)

is

Prove that

6t-measurable.

If

r

[r] = max[t

in E

T

external set o f

*IR

satisfies

: t

<

r]

standard

that

measure

of

{t

E

m

p[(I(r.m)]

H

is

then the internal map

1.

<

0

<

r-[r]

6t.

The

<

t}

I > _ H.

is internal and

0

satisfying

<

5 1.

r

For any

show that

such that

= ;1;;. 1,

st ( t )

:

I

in

E T : [r]

{t

U

IR

in

r

there is a finite

Show

<

satisfies

Suppose

is n o n - m e a s u r a b l e .

1 2

r

' p o s i t i v e half i n s t a n t s '

H =

fixed

<

0

<

<

t

Use for

= I(r,m).

[r] + ;}1

that

to prove

example, by

a

that

sort

the outer

of

covering

argument . Suppose

J

is internal and

the complement of

J.

J C H.

Replace

I

above by

reverse the inequalities and show that

ChaDter

54

v[J]

Z

0.

Hvperfinite Measures

1:

H

so that the inner measure of

i s zero.

(1.1.14) EXAMPLE: The example of a nonmeasurable set worked out in the last example for a specific space

(U,Meas(p).p)

can be generalized

to any non-atomic hyperfinite probability space

(V,Meas(a),a)

as follows:

V

Ignore those points of and

for

a

fixed

V = {vl,---,vn}, n

Such a function an

Define

of

the

elements

V.

in

internally and bijectively onto

* [O.l].

of

g(V)

"isomorphic copy" of S-dense in

V

carries

= 6p(vi),

6a(g(vi))

* [O.l]

takes the value zero,

a

infinite, consider the function

g

S-dense subset

where

* enumeration

a

i = 1 , - * * .n.

and you have an

U

(U,a), where

in the form

(V.p)

and

(Verify this.)

is non-atomic. with

a[U]

is

= 1.

Now we use the same argument as in (1.1.13) to prove that under these wider conditions the set

H =

is

not

p-measurable,

a-measurable.

{t

E

and

T

: st(t)

conclude

<

t}

that

g

-1 (H)

is

not

Section

1.1

55

Limited HvDerfinite Measures

(1.1.15) REMARK:

T

Panetta [1978] has shown that n o well ordering of be measurable

with

respect

to

the uniform

* finite

can

measure

on

T2.

(1.1.16) EXERCISE: S h o o t h a t t h e C a r a t h e o d o r y t r i c k o f (1.1.9) a l s o w o r k s f o r the t n n e r m e a s u r e , so t h a t m e a s u r a b i l i t y c o u L d be d e f i n e d w i t h e t t h e r t h e o u t e r m e a s u r e or t h e i n n e r m e a s u r e by i t s e l f .

M

is

Nottce when

p-measurable

that

U1

and

F

if and only if for euery

is s u p e r a d d i t i u e .

U2

are dtsjotnt.

~ [ u ,U

A set

U C V,

U2]

>

JL[U,]

+ r[U2].

56

(1.2)

Unlimited Hyperfinite Heasures

01

Let internal

* finite

be a

function.

Every

* finite

measure

associated

internal

= 2[6p(a)

p[A]

In this section we assume that

* finite

measure, that is

6p : V

set and

€

*O(V)

has

an

: a E A].

p :

*O(V)

Q 0.

p[V]

A

set

be an

*[O,m)

+ *IR

is an unlimited

This may seem like a small

change. since infinite measures like Lebesgue's are easy to work with.

Unfortunately,

hyperfinite finite.

this is a

extension

of

an

false

first

impression.

*measure

unlimited

This section is much more complicated

is

The

not

sigma

than the

last.

For purposes of basic probability, including the later chapters of

this

book,

you

may

skip

this

section

to

avoid

these

technicalities. Let

so

st :

*IR

=

+m.

st p[V]

+

denote the extended standard part, The outer measure

associated

with

p

is

given by

= inf{st p[A]

i[U]

It

is

understood

whenever

an

internal

associated with

k[U]

p

=

F[U]

that

A

I

U E A

+m

€

*D(V)}.

p[A]

if

contains

U.

The

is given by

= sup{st p[B]

I

U 2 B

€

*D(V)}.

is

unlimited

inner

measure

1.2:

Section

I t is understood that

= +a

E[U]

i f either the set of values is

unbounded o r i f there is an internal p[B]

57

Unlimited HvDerfinite Measures

B

U

contained in

with

unlimited.

(1.2.1) DEFINITIONS: An internal set

L E *D(V)

p[L]

E 0.

I C V

r[I]

=

Any

< -.

;[I]

W e denote

M ll I

is

_C

v

is a

called

the

class

p-measurable

r[U]

= E[U].

(when

p

unpleasant

countably

Meas(v).

not

Extensions of

They

are

if

p-integrable p-measurable

I.

integrable

the internal

st p

A set

Meas(p).

E[UI =

We

I;[u].

p-measurable by the condition

sets do

additive

p-integrable

form a st p

measures

on

not

same

the

sigma algebra

by

and by

the complete and

each

feature (caused by non-sigma-finiteness).

the extension of by

these

is unlimited).

both yield algebra

because

if

of

sets by

p-unique extension set i f

We can not.define sets to be

p-limited

i s called

f o r euery

integrable

denote the class o f

u

is

M C V

A set

Intg(p).

sets by

if

set

i s called

sigma

has

an

However,

is unique on the Loeb algebra generated

subsets of

chosen the larger extension,

01.

-p ,

Loeb(V) = H(*B(V)).

We have

for the hyperfinite measure so

that integrable sets are contained in limited ones.

1:

Chapter

58

Hyperfinite Measures

( 1 . 2 . 2 ) DEFINITION: Let

* finite

be an unlimited

p

unlimited hyperfinite measure the

outer

measure

-

restricted

p

;[MI,

=

p[M]

p-measurable sets,

for

restricted

to

beyond

st p

to

st p

Meas(p) Loeb(V).

Loeb(V),

and

but

there can be measurable

E[U]

;[HI

=

H

sets

the

to

<

F[U]

is the

p

show that

is a different measure

Of course,

is

M € Meas(p).

Now we shall justify this terminology, show that unique extension of

The

associated with

p

function

W.

measure on

extension of for all sets,

E[H]

with

= 0

and

m.

( 1 . 2 . 3 ) REMARKS:

The

inner

<

r[U1]

implies

and

outer

E[U~]

measure is superadditive, U2

;[Ul]

and

L I U l U U2]

is subadditive.

and

measures

U U2]

U 1 E U2

monotone,

.( ;[U2].

< ;Cull

i[Ul

are

The outer measure

+ ii[U21,

while the inner

2 ;[Ul]

when

+ ;[U2],

u1

are disjoint.

( 1 . 2 . 4 ) LEMMA:

The

Meas(p)

sets

and

Intg(p)

are closed under

ftnite intersectton.

PROOF : Let

F.G

with finite F,G

are

p-measure

E

Intg(p).

p-measure measurable

and

re

can

Then there is an internal p[V']

in

the

apply

such that new the

V' 2 F U G.

space,

V'.

results

in

with the

V' C I

Thus, limited

preceding

(1.1.4).

paragraph:

M.N E Meas(p),

Suppose

N fl F

€

59

1.2: Unlimited HvDerfinite Measures

Section

Intg(p),

so

M

fl

F E Intg(v).

and take any

(N n F )

E

Then

Intg(p).

The pattern of the argument in the last proof, referring to the preceding paragraph applied to some limited superset, will be used systematically in the sequel.

The basic fact is that

U

the outer and inner measures of a set

suffer no change if we

consider only the internal subsets of a fixed internal set that contains

U.

(1.2.5) PROPOSITION: All

internal sets of

ftntte measure a r e integrable.

a n d a11 t n t e r n a l s e t s a r e m e a s u r a b l e .

PROOF :

It is obvious that i f so that

A

A

A fl B

and

F

obtained by restricting

= E(A),

is limited.

F

€

Intg(p).

E ( A fl F)

B 2 A

fl

F

p[B]

is internal and

that there is an internal

The sets

-p ( A )

is internal, then p[A]

is integrable if

Now, if so

A

with

<

+-,

limited.

are measurable for the limited measure p

to

B.

hence

A

n B

fl

F = A fl F

measurable.

(1.2.6) PROPOSITION: (i)

A set is t n t e g r a b l e

tf

and

only

tf

tt

ts

tf

and

only

tf tt

ts

measurable a n d has ftntte measure.

(ii)

A set ts t n t e g r a b l e

m e a s u r a b l e a n d ts t n c l u d e d t n s o m e l t m i t e d s e t .

is

ChaDter

60

A

(iii)

set

is

measurable

1:

Hyperfinite Measures

if

and

only

its

if

i n t e r s e c t i o n w i t h a n y i n t e g r a b l e s e t is m e a s u r a b l e .

PROOF :

Lemma

(1.2.4).

<

p[M]

F

If

(i)

p[A]

-

n

A

E

<

p[M]

A

finite,

M = M

F

then

Conversely, assume

Since

+a.

Intg(p)

E

By

Meas(p)

and

A 2 M

with

of

Meas(p)

from

Intg(p)

Intg(p).

In

(iii)

C Meas(p).

one

direction,

Conversely, if

E Meas(p).

E

definition

-

follows from finiteness of

(ii)

M

that

according to

there is an internal

+m,

Intg(p).

E

Meas(p)

E

p.

follows

it

for each

applying (ii) we have

M

fl

F.

integrable

F

E

M n F

Intg(p).

(1.2.7) LEMMA: Meas(p)

Intg(p)

and

are

closed under

countable

intersections.

PROOF :

(Fn)n P M 5. Intg(p)

Let

n E

Then if

OM.

F1,

containing

V' nFn

and define

F' n

=

F1

n

Fn

for each

is an internal set of finite measure =

nF;I

can be regarded as a countable

intersection of measurable sets in the space with finite measure

V'.

Apply (1.1.4) and (1.1.5). Suppose now

(Mn)

Meas(p)

and take any

F

E

Intg(p).

nEum Then

nM,

first part.

n

F

=

n(Mn n F).

and

the result follows from

the

Section

1.2:

61

Unlimited HvDerfinite Measures

( 1.2.8) LEMMA:

And

Meas(p)

and

Intg(p)

are closed under d i f f e r e n c e s .

Meas(p)

is closed under complementation.

PROOF :

W

Since

is measurable,

consequence of

Meas(p)

from

measurable,

the

(#\N) ll F = (M Thus, implies

for

each ll F)

f l F)\(N

only

it

Meas(p)

same property

then

remains

in section (1.1):

embed

measure

(1.1.4)

apply

is closed under differences of

if

Intg(p):

integrable

M,N

F.

set

are

the

set

i s integrable.

to be

proved

that

F,G E Intg(p)

As usual, this follows from the results

F\G E Intg(p).

and

is an immediate

being closed under differences.

Also, the fact that follows

the second part

F

V'

in an internal to

prove

that

of

finite

F\G = F fl (V'\G)

E Intg(p).

(1.2.9) LEMHA: (i) (ii)

Meas(p)

is closed under countable unions.

Intg(p)

is closed under dominated countable

unions. PROOF :

(i) €

Meas(p)

Let

(an)m'En

E

Meas(p).

Then

according to the last two lemmas.

(UMn )" = flME

Again by (1.2.8),

UMn E Meas(p). (ii)

Suppose

(Fn)

E

Intg(p)

and

UFn C F E Intg(p).

nd"' then by (I) and Proposition (1.2.6)(ii),

UFn E Intg(p).

Chapter

62

1:

Hvperfinite Measures

(1.2.10) LEMMA:

(Mn

If of

:

n E

is a disjoint countable sequence

measurable sets, then m

and m

ECUMnl =

b)

t4CMnI. n=O

PROOF :

Mn

In case some

is not integrable,

(by Proposition (1.2.6)(ii)), are

UMn

is not either

hence both sides of the equation

+@ for the outer measure.

Mn

If all

M = UMn

and

are

integrable, the property is proved using (1.1.5) by embedding in some limited superset.

This case is the same for both the inner

and outer measure. To complete and

Mn

E

(a)

for all

Intg(v)

M

we only need to show that i f n E

then

z

C

p[Mn]

Intg(p) = +@.

Observe that since i t is a series of nonnegative terms, i t is enough to prove that for some infinite sum is infinite.

An 2 Mn internal

such that

.

(A,)

For each n E 1 p[An] < - + v[Mn]. 2" Let Sn be the

n,

the

nth

partial

there is an internal Extend

(A,)

to an

nP# *extension of the

nE*# n externally defined but standard series

Z

p[Mk]

= Sn. The set

k=O

{n

* E

IN

is internal and contains

" : L p[Ak]

<

1 + Sn}

1

so

that there is an infinite

n

Section

1.2:

63

Unlimited Hyperfinite Measures

n For that

in it.

Y Ak

n.

the set U Mn. n

n p[U Ak] 1

therefore


Sn

hence case

Any

for

is internal and contains n

is infinite.

P[u Akl

But

1

n 2 PEAkl 1

<

is infinite. the

inner

measure

where

each

Bn E Mn

int egrab e may be treated by choosing internal sets with

> E[M,]

p[Bn]

m

2 st

Z

>

p[Bn]

n=O

-

L. 2”

When

m

is

m

is

Mn

E[UM~]

finite.

m -2e =

E[M,]

Z

n=O

Z

-

F[Mn]

2e.

Since we know

2 F[UMn]

in this case.

n=O

m Z

F[Mn]

F[UMn],

we see that

E[UM~]

n=O The final case for the inner measure is when some not integrable. question is: We

claim

Of course

there

are

-+

Bk

two of

cases.

UMn

measure up to?

Either

limited sets,

or there is one point

E[UM~]

unlimited.

only

is

is also not integrable, but the

What do internal subsets of

increasing sequence

tf[Bk]

UMn

Mn

there

Bk

a E UMn

UM,.

is

an

with

with

6p2(a)

We prove this claim after we show why i t suffices.

This condition establishes the countable additivity of the inner measure

trivially

unlimited weight.

so

the

series

is

superadditivity of

in the second case of a point with

In the first case,

infinite.

For

shows that

each

standard

m.

the

ChaDter

64

Hyperfinite Measures

1:

Now we establish our claim about the weights for

Bk

sequence

exists.

B.

internal set

{6p(v)

: v E

{61.62."'. q = min{j

as

.

The

J

Z

:

has

B}

6P}

6j

UMn

must

contain an unlimited

yet there is a limited

A E UMn

limited

then

>

>

p[A] an

b.

b E 0

such that no

Order

the

increasing

6 9

weight

I f no

p.

internal

weights sequence

unlimited

is

where

b}.

i=1 This completes all the cases of our proof.

(1.2.11) DEFINITIONS:

A sigma algebra wheneuer a set euery

U

measure

measurabLe

set

is said to be

has the property that

p-integrable

A

54

F,

space of

then

U

E

U fl F E 9

for

9.

caLLed

is

if

p-saturated

infinite measure

semifinite contains

if

each

integrable

subsets o f arbitrariLy Large finite measure.

Clearly, we have defined

Meas(p)

so that i t is saturated

and complete (every subset of a set of measure zero is measurable). An infinite measure space is called sigma finite i f i t is a countable union of sets of finite measure. sigma

finite

hyperfinite

measure

measures

unlimited weights

is (as

semifinite. well

6p(v) e 0 )

inner measure usually is.

as

Hence a (infinite) Non-atomic

hyperfinite

unlimited

measures

with

are not even semifinite. but the

1.2:

Section

65

Unlimited Hvperfinite Measures

( 1 . 2 . 1 2 ) THEOREH:

a)

An

(V,Meas(p),p)

unlimited

hyperfinite

measure

space

is a complete saturated countably additive

t n f i n i t e m e a s u r e space. b) is

T h e inner measure,

aLso

complete

a

E,

countabLy

restricted t o

additive

Meas(p)

infinite measure

space. c)

T h e inner a n d outer m e a s u r e s a g r e e o n the L o e b Loeb(V).

algebra,

so t h e e x t e n s i o n o f

tnternal sets to

f t n t t e , but

E

E

stgma-

6p(v) C 0 ,

t f t h e luetghts a r e aLl L i m i t e d ,

If

the

LnftnttestmaL measurabLe

=

t s never

is semtfinite.

e)

;[HI

is untque.

An u n L i m i t e d i n n e r m e a s u r e

d)

then

Loeb(V)

f r o m the

st p

weight

6p[v]

ualues.

H

set

E

functton

Meas(p)

T h e extension of

m.

Z

for

0.

such st p

then

there

only is

E[H] = 0

that to

has

p

Meas(p)

a and

is n o t

u ntq u e .

f)

Non-atomic u n l t m t t e d h y p e r f t n i t e ( o u t e r ) measures

are not semtftntte. a n d thus are aLso not stgma ftnite.

PROOF : The

proof

of

(a)

and

(b)

is

contained

in

the

previous

lemmas. To

prove

:

{M E Meas(p)

E[M]

(c). =

;[MI}

all the internal sets. is

E[M]

a

monotone

5 u[M]

<

;[MI.

set

observe

that

the

collection

is a sigma algebra which contains

Therefore i t contains function

that

Loeb(V).

extends

Thus any measure on

Loeb(V)

st P.

If

u

then

that agrees

Chapter

66

with

on internal sets must equal

p

Now we prove part (d).

Hvperfinite Measures

1:

on Loeb(V).

p

E

We show that

is never sigma

finite by showing that any countable union of measurable sets of

V.

finite inner measure always omits an infinite amount of

W 2 UIm

Suppose

with each

E[I,]

Im C

sequence is increasing,

We may assume that the

m.

(by taking finite unions).

Bm E Im

Next, choose an increasing sequence of internal sets

-

2 r[I,]

with

p[Bm]

p[V]

is unlimited and

i.

We know that

r[Im] <

p[B,]

>

p[V\BJ Extend

m.

m.

since

Bm

to an

increasing internal sequence using countable comprehension. internal set of indices

n

an infinite

r[V\UIm]

n,

thus

such that

>

p[V\Bn]

2 r[V\Bn]

=

n

must contain

m.

To prove the rest of (d) suppose that a set

ELM]

=

I v

{6p(v)

B E M

and

03

E

B}

corresponding

B1 = {bi I i

is

Order

M

the

weights

{61.62.***. 6P) { b l , ~ ~ ~ , b=pB. ) Define a sequence

min[j

.i

6h

Z

:

>

11)

and

. i

:

Z 6h

>

p[Bk]

+ 11).

We have

-

Bk+l

h= 1 {bi I i 5 min[j

has

in increasing order

to points

<

unlimited.

The

<

k 5 E[B,]

m.

h= 1 The second part of (e) and part (f) follow from the first because i f

H

contains an integrable set

be approximated by internal subsets of

I.

I.

then

Since

E[I]

E[H]

may

= 0, H

cannot contain integrable sets of positive standard measure. Now we complete the proof of the theorem by constructing

H.

Let

limited

9=

V

be the collection of all internal subsets of

p-measure constructed at or before the stage

the direct limit defining our polyenlargement.

Since

9La

V

of in has

Section

1.2:

67

Unlimited Hvperfinite Measures

the finite intersection property.

V fl %a+l

which are not in any

H = {x a

family

A E 9a

card+(%)}

H

Clearly, p-limited

<

I a

has

<

a

for some

finite outer

measure

<

card+(%)

card(H fl I) because

<

I

card+(%)

be a

+

= card ( 9 1 ) .

card+(%).

is contained

by the above.

for

card+(%).

in a

any

because

p-limited

F(H)

This means that

=

I

of

set, s o m,

with

of

internal

<

p[A]

F[H

I]

fl

where

E

= 0

by

is an

E

<

F[H

fl I]

Fix an arbitrary

t

and

Since

p-measurable.

family

card(H fl I)

We know that

We will show that

A 2 H fl I

H

this makes the

<

but

We know that any set

arbitrary standard positive number.

consider

x a E V\f19a.

p-integrable set.

finding an internal

= 0.

Thus we may select a

card(H fl A )

also.

have

= card+(%).

card(H)

Let

I E 9a. with

A.

,

Hence there exist points in

card(H)

internal set

V\I

9a,

unlimited measure the complements of sets of

k[H

fl

: a

{$€(a)

sets

I]

H

E

fl

I},

where

=

d,(a)

: a E A

{A E *D(V)

& p[A]

al.***.am E H fl I.

Given any finite set

<

t}.

the set

{al.***,am}

belongs to each da(ai) since p is non-atomic. m + f l “€(ai) # 0. By card (%)-saturation words, i=1 a E H fl I] # 0. so there is a single A such that and

p[A]

<

B

fl[d,(a)

H

fl

:

I E A

t.

Finally, we show that subset of

In other

H.

then

is finite and

card(B) p[B]

Z 0.

k[H] = 0 .

<

card+(%).

If

B C H

is a limited

Therefore, by ( 0 . 4 . 4 ) .

An unlimited set

B

cannot be a

Chapter

68

H

subset of internal {b.

B : j

E

J

2

and

<

k Z

:

min[k

1 > 5. If B'

p[B']

the

>

11).

then

B'

B'

subset

By

this

=

definition.

i=l

j

>

define

p(bi)

HvDerfinite Measures

B = {bl.b2.*-*.bn} for an

because we may write

sequence

1:

C H,

is finite, s o

p[B']

0.

This completes the proof.

The next result is the "Littlewood Principle" that says, 'sets are almost finite'.

(1.2.13) THE SET LIFTING LEMMA:

A set p-ltmtted

F

p-integrable

ts

A

tnternal

i f a n d o n l y i f t h e r e is a

s u c h that the s y m m e t r t c d i f f e r e n c e

A v F = (A\F) U (F\A)

sattsftes

p[A

v F] =

0.

PROOF :

F

The set there

exist

internal

Bm -C Bm+l r F C Am+l E Am with 1 < p[F] + ; < @ . Extend the sequence

sets

p[Am\Bm]

1 < ;

(Am,Bm)

to an internal sequence and select an infinite

that The

m

<

sets

n

and

p[Am]

implies

An.Bn

= p[F].

p[tlA,]

and

Bm E Bm+l C Am+l C Am

U[Bm

:

m

E

u#] E F

P[A,,,\B~I

and

14F1 = vCUBm1

satisfy

<

n[Am

n

v[Bnl

:

<

F

or

Bn and

from completeness of

so

<

i4An1

m E uLN],

that the symmetric difference has measure zero for either and

m.

is integrable if and only if for standard

1

E.

< so

An

F.

The converse of the second part follows

p

and the measurability o f internal sets.

1.2:

Section

69

Unlimited HvDerfinite Measures

(1.2.14) THEOREM:

A set measurable

M in

v, p ( ~ ) =

T E

is

p-measurable

the

;(T

sense

of

if

and

if

only

Caratheodory:

it

for

is

every

n M ) + ;(T\M).

PROOF :

M E V

Assume that Caratheodory's condition holds for

F

let

be an integrable set.

T E V',i.e.,

for every condition in in

M

-p v , ,

agrees with the outer measure

p

V'.

n F

V'.

V'.

n V')

so that

satisfies

Therefore, by (1.1.9).

fl

(in case

Remark

PCFI = ;[TI.

F

the Caratheodory

M fl V '

is measurable

;[TI

(1.2.3)). But

V.

Hence

is also integrable.

M E Meas(p).

Suppose now

from

M n V'

of

the outer measure

and then i t is integrable as a subset of

= (M

measure

V' 2 F

Take an internal

finite measure: then for each subset of

-

and

For each

T E V

of finite outer

= +-

the Caratheodory property follows

there

is

W

~

an

F is

integrable integrable,

F 2 T hence

with it

is

Caratheodory-measurable (apply Lemma (1.1.9)). so

The next exercise shows why we cannot define a set to be measurable simply i f

E[U]

= ;[U].

The function

st p

has a

unique extension to such sets ( i f the extension is continuous),

70

ChaDter

To show this we need a

but these sets do not form an algebra. non-measurable subset of a

set.

Hvperfinite Measures

1:

Example (1.1.12) yields a non-measurable

p-limited set when

p

is non-atomic.

(1.2.15) EXERCISE:

Let

p

be an unlimited hyperfinite measure w i t h a

E[T n F] <

F].

Show that

a)

b)

c)

F[U

k[U]

n F).

= W\(T

= F[U]

such that

but

<

r[UC]

and

E[F]

F[Uc].

=

F[F].

but

n F].

Show that

Another

u

= F[U],

E[U]

F

and an integrable

Let

Shom that

n F] <

E[U

G[T n

E V

T

nonmeasurabLe set

approach

U

is not measurabLe.

one

might

try

to

extend

an

internal

measure is to take the sigma algebra generated by the integrable sets.

E[W]

These sets form a sigma algebra with the property that = ;[MI

f o r each

M,

however, the internal sets are not

all included.

(1.2.16) EXERCISE:

Let weight Show

p

be an unlimited hyperfinite measure whose

functton takes only

that one c a n divide

internal sets

A U B = 01

Limited values.

V

roughly

mith netther

A

6p(v)

in half

nor

sigma algebra generated by the LntegrabLe sets.

B

by

E 0.

two

i n the

71

(1.3)

Almost and Nearly Sure Events, Measurable and Internal Functions

f.

Recall that a real-valued function, respect to a sigma algebra,

B E

IR.

= {v 1 f(v)

<

set

--,

= {f

<

f-'(B)

if

sufficient

<

r) E I:

to

E

P

show

for every

r

for every Bore1 -1

that

f

in

Moreover,

IR.

(-m,r)

to also take the extended standard values

still

r} E L

is

r) = {f

f

we may allow and

It

1,

is measurable with

is meaurable if and only i f

f

for each

r

in

01 + IR

be

f-'[-m,r)

IR.

(1.3.1) DEFINITION:

f

Let

:

*

f

projection o f

an

internal

function.

The

is t h e e x t e n d e d - r e a l - v a l u e d f u n c t i o n : N

= st(f(v)).

f(v)

(1.3.2) THE FUNCTION PROJECTION LEMMA:

-

Let

T

:

Loeb

v

V

be

[-m,ml

measurable

*f i n i t e

a

set.

The

f

o f a n internal function and

since

Loeb(V)

:

projection

v * *IR

E Meas(p).

p-measurable f o r any hyperfinite measure

p

on

f

V.

PROOF :

{T < {T 2 {T <

r} = u[{f

<

r-l/m}

:

r} = n[{f

> <

r-l/m}

: m E

r+l/m}

:

r} = n[{f

m E

m E u~~

r E IR

is

is

+m

ChaDter

72

Hvuerfinite Measures

1:

(1.3.3) DEFINITION: g: W +

If

[--.+-I

f : 01

i s a function,

+

*IR

is

N

internal and i f

f = g.

f

then

i s caLled a ( u n i f o r m )

g.

lifting o f

v-

II

v

f

g

*R ISt

[-m,+m]

N

f

Naturally,

i s a lifting of its own projection

f.

The

proof of (1.3.2) above shows one implication of our next result.

(1.3.4) THE UNIFORH LIFTING THEOREH:

A function g and

onLy

tf

for

r}

(v : g(v)

:

V +

each

has a uniform lifting i f

[-m.m]

rational

(v

and

:

in

r

g(v)

2

are

r}

the

IR

sets

countable

intersections o f internal sets.

PROOF (of converse): We

shall

construct

a

sequence

of

internal

funct ons by first making partitions with in

Q

g.

approximating

For a rational

r

let (g

2

r} = {v : g(v) 1 r} =

A(r.03)

= nA(r.n)

and

where

A(r.n).B(r.n)

assume

are sequences of internal sets.

A(r,n) 2 A(r,n+l)

and

B(r.n) 2 B(r,n+l)

We also because

n f l C'(r.k)

= C(r.n)

is internal and decreasing when

C'

is

k= 1 just internal.

Let

{rl.rz.---}

enumerate

Q.

For each

m.

Section

1.3:

Measurable & Internal Functions

{s1.**-,sm} = {rl.***,rm}

let

following sets partition

{g

>

sm} =

[n

Bc(sk.m)]

s1

<

s2

<

* * *

<

sm.

The

V:

n [n

k

Notice

with

73

k

that each of

these sets may be written as a

intersection with each set

B(sk.m)

or

A(sk.m)

2m-fold

represented

once with either itself or its complement in the intersection. We may

code this with a function

E

:

{1.***.2m}

blank,^}

so each set above is Q(E)

for an appropriate

n cn

= cn k

function

~~(~~)(s,.m)l

k

E

with

the blank or complement

74

1:

Chapter

values as described. refer

to

the

Hyperfinite Measures

We shall call this partition

sets above

in

terms of

the

and

Q(m)

corresponding

e

function. For each

Our next step is the following claim: n 2 m

exists

so

that for each nonempty

in

Q(E)

there

m Q(m).

the

corresponding set

satisfies

n

P(6.n)

Suppose

Q(E) =

{s

i

<

g

= [A(si,m)\n

<

Q(E)

# 0.

contains

s.}

J

v,

n B(sj,m)\n

B(si,n)] n

and there is an

A(sj,n)] n

A's

B's

Since

A(si.n) 2 A(si.m) contains

v 4 B(si.n)

that

so

Since

P(e,n(i,j))

and

n

so

decrease we may take and

B(sj.n)

v 4 A(sj.n).

and

2 B(s

n(i.j)

= max(m.n).

m)

we see that

j'

A similar argument works for the full

v.

expression of each of the

Q(e)

sets in

be the maximum of all the

n(i.j)'s,

etc.

Q(m)

and we let

This

n

n

fulfills

the claim.

For this sets

n,

R(m)

let

{A(sk.n);B(sk.n)

:

1

written as the collection of

[n A8(2k)(sk.n)] k

be the partition generated by the

k

<

m},

which, by the way, may be

2m-fold intersections

n [n k

B8(2k-1) ('k' n, 1

for

75

1.3: 'Measurable & Internal Functions

Section

the

set

all

of

functions

Define a function by summing over the nonempty

+ (b1ank.c).

9 : {1,-*..2m}

€-functions describing

Q(B)'s: : Q(E) #

fm(v) = ~[g(vp(€,n))Ip(€,n)(~)

01

I

is a is the indicator function and v P(a,n) P(a,n) sample of points taken from P(a,n) f l Q(e) for all the nonempty

where

Each

Q(E)'s.

sequences

fm

{v,}

R(m)

is internal, because are.

j.k

property: "for each

This

<

is and finite

function enjoys

the .following

m.

Ifm <

E B(rj.n) E B(rj.m)

rjl

and

Use comprehension to extend

an internal sequence with decreasing sets and property in quotes above up to an infinite Then for each = g(v)

st[f(v)]

r

<

in

s

9.

to

{(fm,B(r,,m)...A(rm.m))}

<

{r

f

<

s}

satisfying

n.

Let

E {r

<

g

the

f = fn .

<

s}.

as claimed in the theorem.

(1.3.5) DEFINITION: Let We

say

that

surely p[V\W]

= 0.

be a hyperfinite measure

v

the property

(p-a.s.)

(p-n.s.) U 2 V\W

(V,Meas(p),p)

or

We say that

E

W C V

palmost

v

E

W

everywhere

holds

i f there is an internal set

and

p[U]

%

0.

hoLds

p

p

space,

almost

(a.e.)

if

nearly surely

U E V

such that

so

1:

ChaDter

76

v E W

Clearly, i f

n.s., then

surely implies almost surely."

Hvuerfinite Measures

= 0.

p[V\W]

"nearly

so

The converse fails.

(1.3.6) EXERCISE:

T

Let t Q a(1/2)

then

p[U]

and

b e as

p

U

a.s. while i f SJ

(1.1.13).

in

Show

that

0(1/2) E U,

is internal and

0.

(1.3.7) DEFINITION:

(V,Meas(p),p)

Let

f : 8 +

If

[-=.=I

be a h y p e r f i n i t e measure space.

g : I

is a f u n c t i o n and

+

*IR

is

N

internal and

f = g

p-lifting o f

f.

If

f

has a

p-almost

p-lifting,

g

surely, then we call

g,

then

{v

:

f(v)

a

# g(v)}

is

N

p-measurable with measure zero.

f

is

+

g

is Loeb measurable,

p-measurable.

The projection

V

Since

*IR

map

from

the

set

of

to the set of measurable functions

surjective, even ignoring sets of zero

internal

functions

[-=.=I

V

is not

p-measure.

(1.3.8) EXAHPLE:

n

Let lkl

<

n2},

be an infinite natural number, and

p(v)

=

for each

v E V.

function of the set of finite numbers since

0

n V = U{v

E V :

1.1

<

n)

n indicator function

f

has no lifting.

0 fl V

V =

{E

: k E *B,

Then the indicator is measurable,

is a Loeb

set.

This

Section

77

Measurable & Internal Functions

1.3:

N

g : I + *R

Suppose

almost everywhere. measure

<

p[A]

A E I

Then there is an internal set

1.

g = f

is an internal function with

such that for each

v E V\A,

g(v)

Z

of

f(v).

Then, g - 1((5,2)) 1 U A 2 V n 0

Since

the

left hand

I fl [-h,h]

g

For

-1

same

V fl [k.n2]

g-1

Hence,

is

internal and

U

A 2 V

\ 0.

h E ON.

for all

is contained in i t , we can deduce that

((15.2))U A

the

side

g-'((-Z.Z)1) 1

and

2 V fl [-h.h]

reason,

A 2 V

A 2 V

fl [k.h].

g -1 U-f

since

for all infinite

((-f,$)) U

for some infinite

h

E *IN

$1 u A

contains

k E *IN.

fl [k.n 23

k E *IN

for some finite

which is impossible since

<

p[A]

1.

(1.3.9) THE FINITE FUNCTION LIFTING LEMMA: If

f

:

V

+ [-m,m]

p{f # 0)

p-ftnite carrter. the measure g :

V

a.s..

+ *IR

p

is

<

+m

is limited),

s u c h that

g

and

has

( i n p a r t i c u l a r , luheneuer t h e n there t s a n internaL

w

f = g

then we c a n choose

p-measurable

8.5.

Woreouer. i f

s u c h that

lgl <

If I <

b

b.

PROOF : Let and

c

{di 1 2

be an unlimited positive number

do = -c,

<

IR.

i E

IN}

be a dense subset of

let the latter set be dense in

[-b.b].)

(If

For each

d

1 = +c,

If1

<

b.

m E IN

ChaDter

78

By

hypothesis

FYI <

p[U

these

sets

are

Hence for each

a.

1:

HvDerfinite Measures

measurable

m

and

there is an

h

for E

IN

each

m

such that

i p[ U

i >h so

FYI < .;1

AS E FS

Choose internal sets

for

5 i 5 h

0

that h Z

< .;1

p[FY\AY]

i=1 h Use intersections of these sets to define a partition of

{BY

: 0

define

<

5 j an

(For

Bm

so

we may

k choice v

if

example,

FY,

is a subset of some

J

finite

such that

B; E FY.

E FY.)

Each

internal

{do.***.dh}

Bm J

k}.

U A; i=1

take

The internal function

E

gm : U BY j=1

function

Bm

and gm(v) . i the maximal i gm

= di,

then

such

may be extended to

+

that

V

by

k taking the v E

gm(v)

= 0

then

B

.I'

gm(v)

< m

lgm(v)-f(v)l so

BY.

Suppose that we interpret

j=1

I-c

= 0.

U

v C

+

inequality k U j=1

if

< .;1

Igm(v)-f(v)l

to be

true,

that whenever

so

Also, whenever

< .;1

lf(v)I

< ,;1

Thus we see that

This shows that the sequence of projections

A .

.

gm

converges to

f

Section

Measurable & Internal Functions

1.3:

in measure. associated Igm(v)-gn(v)l Ign(V)-f(v)l

Consider another internal 2

2

2

gn

,;

1

.;

either

I

:

-

n E IN

we

79

with

*R.

m

In

have

order

<

m

Now we use countable comprehension to extend internal sequence and select an infinite n m

<

n.

4

I .;

2 ];

p[ (gm-gnl

and the to 1

2 ;

or

n.

{g,}

to an

N

gm

converges to

N

in measure.

have

such that for all

This shows that

N

gn

n

Igm(v)-f(v)l

Therefore we see that for

2

<

We already know that

N

gm

+

f,

hence

gn =

f

(v-a.e. ) . The reader should check the proof just given t o see that we actually have shown the following.

(1.3.10) THE EXTENDED FINITE LIFTING THEOREM:

f

Suppose that Least

f o r each

m

does not haue f i n i t e c a r r i e r , but at in

stiLL exists a n internal.

The example.

extension applies

IN. g

v{ If I >

1 --}

f(x)

=

m.

g = f

such that

to

< N

-X

on

T h e n there

a.e.

the

line, for

Recall that the indicator function of the finite part

of the line has no lifting.

(1.3.11) DEFINITIONS & REHARKS: We

say that an internal scalar

almost surely if

v[l?l

=

-3

= 0.

function

f

is

finite

Chapter

80

HyDerfinite Measures

1:

This is equivalent to

for each infinite

h

*IR+. It

in

is also equivalent to

that is, for each finite positive

k

such that

>

h

say

We

implies

that

infinitesmiat

an

almost

>

p[lfI

there is a finite

e,

<

k]

e.

internal

scalar

if

p[T

surety

h

f

function

z 01

= 0.

is

This

is

equivalent to

for all finite

k

ulN

in

and by Robinson’s sequential lemma

there is an infinite

n

such that the infinitesimal condition

0 5. k 5. n

in

*IN.

holds for

f

the cond tion that

is n e a r l y s u r e l y i n f i n i t e s i m a l . that is.

there is an internal set in

U.

then

f(v)

z e r o a l m o s t sureLy sureLy.

Therefore. this is equivalent to

U

with

p(U)

is infinitesimal. if a n d only i f

f

0

Z

and if

v

In other words,

is not

T

is i n f i n i t e s i m a l nearLy

This is not the case with finiteness.

Now we look at the property of an internal function being n e a r l y

sureLy

internal set

U

f(v)

is

is finite.

with

ftnite.

p[U]

If we let

Z

that 0

is.

and if

such that v

b = *sup[lf(v)l

there is an

is not in :

f

v e U]

U, then then

b

is finite, since the internal set is bounded by every infinite

Section

1.3:

Measurable & Internal Functions

number.

The standard part

st(b)

81

=g is an essential bound

T,

for

p[

IT1 > %]

= 0.

N

so

f

that

the

E Lm(p).

p-measurable functions. by

if

b

5 b

Ig(v)l

If

f

space

of

(Recall that

except on a

essentially

bounded

is essentially bounded

g

p-null set.)

T

is finite n.s., then

all

is in

and when

Lm(p)

is limited the finite lifting theorem says every

in

g

p

Lm(p)

N

has a then

f

p-lifting N

N

f = g

for some n.s.

f 1g

remarks above that

uniformly finite if

f

Unfortunately, integral,

for

and

g

it

E

Lm(p)

follows from

n.s.

Finite a.s. only means weaker than finite n.s.

finite

f

Also, if

that is finite n.s.

that

= m] = 0.

p[lfI

so

it

Certainly finite n.s. is weaker than

there are sets of

infinitesimal measure.

can be finite n.s. and still have an infinite

example, take

uniform counting on point where i t equals

n

the

space

of

elements. and let

Example

f = 0

(1.1.11).

except at one

nn.

(1.3.12) EXTENSION TO ALGEBRAS (con't.):

For is a

p.d,V

as in (l.l.S(c)),

Loeb(d)-measurable

[or

that there is an internat g = f

may choose

a.s.

g

f : V

b.

[-oJ,~]

--f

d-measurable function

Moreover. t f

bounded by

if

p-measurable] function, show

N

that

is

f

is bounded by

g

b.

such we

82

Chauter

M

A topological space

1:

Hyperfinite Measures

is called a Polish space i f the

topology is given by some complete separable metric, that is i f

M

there is a countable dense subset o f metric

p

M

on

and a Cauchy complete

that induces the topology.

(1.3.13) METRIC LIFTING: Suppose ualues

in

a

f

:

1 +M

Polish

space

hyperfinite measure o n g :

w+*

M

is a

s u c h that

1.

M

p-measurable function w i t h where

p

is

a

limited

S h o w that t h e r e is a n i n t e r n a l

p[p(f,g)

*

01 = 0.

This exercise can be solved by rephrasing the proof of the scalar (*R.

1.1,.

lifting

theorem

in

terms

of

(*lN.p)

rather

than

83

(1.4)

Hyperfinite Integration Our next question is the relationship between the sum of an

f.

internal function

always

€

W],

and the integral

s,"

of its projection, We

: v

H[f(v)6p(v) f(v)dp(v).

have

the

inequality

(see ( 1 . 4 . 9 ) ) . but the following examples show that the converse inequality is not always true, even in case the sum is finite. We build our examples on the space of Example (1.3.8) which we may 'visualize' as an infinite discrete line.

W = {v

unlimited, let with

constant

weight

*R :

€

v = n

function

for k € 1 6p(v) = -. n ,

Let

*E

n

€

*

with Ikl (Any

IN

be

<

n2}

unlimited

nonatomic space has similar examples.)

( 1 . 4 . 1 ) EXAMPLE:

For functton

-

f(v)

(1.4.2)

and

as above, the constant infinitesimal

p

1 = n

-

f(v)

= 0.

J-

1 y n

and

= 0.

f(v)dp(v)

SO

= 2 +

B f(v)6p(v)

satisfies

EXAMPLE: For

g(0)

W

= n,

W

and

= 0.

g(v)

unltmited multtple

B g(v)6p(u)

J g"(v)dp(v)

=

as above, the function

p

1,

= 0.

of

z

if

v

the

tndtcator

whtle Nottce

0

(g(v)

thts

= nI{O~(v)~

an

p-a. s . , rematns

restrtcted t o the Ltmtted measure subspace

It is not enough to have the sum of

such that

functton) satisfies

g"(v) = 0 that

g

f

true

* [O.l]

so

when

fl W.

near the integral

Chapter

84

of

7.

be

like

We want the structure of the space of such functions to the

G(T,A) = lATdp.

function of

1 L -space, or

standard

F(f.A) = I[f(v)ap(v)

function

SL1

Hvuerfinite Measures

1:

! v E

equivalently,

want

the

be

near

the

A]

to

We define the appropriate space of

S-integrable functions below in (1.4.4).

1 (1.4.3) DEFINITIONS OF FL , S-AC & S-HC:

Two obvious necessary conditions for

f

to be in

SL1

are: (FL11

I[f(v)ap(v)

:

v

E

V]

H[f(v)6p(v)

:

v

E

A]

i s finite (or limited)

and

(S-AC)

p[A]

Z

%

A

0 wheneuer

is internal. and

0.

L 1 -norm is finite.

The first condition says that the internal

The second condition is "standard absolute continuity."

The

reason the second condition must hold is the absolute continuity N

f

of the projected integral, the integral of

over

A

is zero.

S-absolute continuity fails for Example (1.4.2). For unlimited measures,

a

third

condition

corresponding

to

the

standard

monotone continuity of integrals is also necessary:

(S-MC)

I[lf(v)lap(v) and

:

f

v

E

A]

Z

T

is internal

A.

i s infinitesimal o n

This must hold since the integral of of zero.

A

0 whenever

on

A

is the integral

This condition fails for Example (1.4.1).

show that the three conditions imply that all senses of integration and (S-MC)

f

We shall

is close to

is not needed when

N

f

p[V]

in

1.4: HvDerfinite Inteeration

Section

85

is limited.

(1.4.4) DEFINITION: An internal if

f

function

:

W

above.

W e denote the space o f

SL'(~).

IF

f

is i n t e r n a l .

is c a l l e d t h e i n t e r n a l

*IR

is

S-integrable

(FL1), ( S - A C )

satisfies the conditions

it

+

and

(S-MC)

S - i n t e g r a b l e f u n c t i o n s by

Ilfll = x[lf(v)(6p(v)

1-norm o f

f

:

v E VI

w i t h r e s p e c t to

p.

In the case of probability spaces, we have the following characterizations of

SL1(p).

(1.4.5) PROPOSITION: Let

be

p[V]

limited and

f

:

V

+

*IR

internal.

T h e following are equivalent: ( a ) f E SL~(~).

1 s a t i s f i e s (FL ) a n d (S-AC),

(b)

f

(c)

For e v e r y i n f i n i t e p o s i t i v e n u m b e r

H[(f(v)(Gp(v)

:

(f(v)(

>

k]

Z

k,

0.

PROOF : (a) 3 (b) by definition. (b) 3 (c):

k E *IR+. measure.

{v

:

Since lf(v)I

>

f k}

satisfies (FL1), for any infinite is an internal set of infinitesimal

Then (c) follows from (S-AC).

(c) 3 (a):

Let

k

be any infinite positive number.

Then

Chapter

86

Since

p[W]

Hvperfinite Measures

is finite and the internal inequality is valid for

k,

all infinite positive

If

1:

A

(FL1) holds.

is an internal set of infinitesimal measure, for

each infinite positive

k, reasoning as above,

--1

',

k = p[A]

Taking

we get (S-AC).

A

Finally, if

is internal and

then for all standard positive

Since

for all

v

E

A.

8,

EXERCISE: Suppose

Show

0

%

is finite, (S-MC) follows.

p[W]

(1.4.6)

f(v)

f

If

p

is

a nonatomic

1 (S-AC) i m p l i e s (FL ) ,

that

only if

that

so

f

limited

measure.

SL1(p)

if and

E

s a t t s f t e s (S-AC).

6p(vo)

@

0.

f o r some

vO'

t h e n (S-AC) d o e s n o t

1

i m p l y (FL ) .

An

indicator

p-measure is

function

S-integrable and

of

an

p[A]

internal =: p(A)

set

of

finite

by our construction

Section

of

1.4:

87

Hvuerfinite Integration

We now build on this fact by taking limits just as i f we

p.

were constructing the standard integral. The

space

ex terna 1 Ilfll =

SL1(p)

space

of

with

I[ (f(v) 16p(v)

v

:

S-integrable

an

E Or].

functions

int erna 1

is

seminorm

The quotient o f

an for

obtained

SL1(p)

by identification of the functions with infinitesimal integral,

f

o

E

Ilfll z 0 , is isometric to

iff

Suppose values.

:

b

For

is internal, but only takes limited

is internal and every infinite

for all

bounded. functions

V + *IR

f

Since

<

If(v)l

f

~'(p).

f

v,

the

rest

satisfying

b

satisfies

must actually be uniformly finitely of

the

section

the hypotheses of

we

shall

refer

(1.4.7) as

to

p-finite

functions.

(1.4.7) PROPOSITION: Suppose values and

f

that

internal.

# O}]

p[{f

that

is

only

is l i m i t e d .

I[f(v)6p(v)

S-integrable and

takes

:

v E V]

Z

limited

f

Then

is

J," fdp.

PROOF :

The first part is easy by all the finitesness assumptions. We may assume that from p[{f

<

y,

and

f = f+-f-). # O}],

in

Let

is positive (and finish the general case

M

be a finite real number greater than 0 = yo

and choose a finite sequence OR.

with = 0

p(?-'(yi))

because for

f

y

>

0,

ym

>

max(f),

for all -f 1 (y) C

i =

{T

such that .m

1.a..

# 0) C

can only be a countable number of such

<

yl

yi-yi-l

<

<

... e/3M

(this can be done

{f # 0). and then there y

with

p(?-l(y))

>

0).

Chapter

88

1:

Hvverfinite Measures

Then, i f we define the four sums

i = 2:-.,m,

For any

-- 1 f

Then,

(Yi-l.Yi)

S(p)

r

-1 f

(Yi-1.Yi) E

I ~ ( J A ) . and hence

-1 f

cYi-l.Yil c

12 f(v)ap(v)-J

-- 1 f

Tdpl

CYi-l.YiI

<

B.

This

completes the proof.

When "viewed" with standard tolerances, the space "looks like"

L1(p).

SL1( p )

The next lemma begins to make this remark

Section

1.4:

precise.

Hvperfinite Integration

{fm I m E “IN}

We say that a sequence

functions is an standard

>

e

there exists a standard

<

j.k 2 m.

Ilf -f II . i k internal function

of internal

1-norm if for every

sequence in the

S-Cauchy

0

89

such that for all

m

8.

We say that the sequence

fm

f

as its

fm = f ,

1-norm i f for every standard such that for all standard

>

E.

>

j

S-lim

S-limit.

in

there is a standard

0,

Ilf -fll . i

m.

has the

<

m

E .

(1.4.8) LEMMA:

of

f E SL1(p)

Euery

(a)

p-finite

functions.

SL1(p)

sequence in

L1(b)

(b) measurable # 0)

is the

Moreouer,

has a n

is

S-limit

S-limit

o f a sequence

euery

S-Cauchy

SL1(p).

in

t h e c o m p l e t i o n o f t h e set o f b o u n d e d

functions.

with

g.

p-fini te

carrier

< -.

PROOF : (a)

functions in exists

an

I m E

{fm

Let

SL1(p), m

P

the internal set

{n E

*IN

I ( V j,k E*1)(m

contains an infinite

“IN

in

: v E {fm

an internal sequence,

“1.

S-Cauchy

that is, for each

2 p

E[lfj(v)-fk(v)16p(v)

be an

n

intersection property.

< -P1 . m E *IN}.

I j.k I

P

natural numbers from

Or]

such

P

.

m

P

that

p for

sequence of

in

“IN,

all

j.k 2 m

Extend the sequence

For each finite

n 3 ~[lfj(v)-fk(v)16p(v)]

there P’

fm

to

p

in

<

k)}

The countable family of intervals of to

n

P Saturation

therefore has the finite lets us pick an infinite

90

Chapter

Hvperfinite Measures

1:

n I n

f = f n'

for all p E aIN (see ( 0 . 4 . 2 ) ) . Then if P S-limllf-f II = 0. Next we show that f E SL1(p). m The

(FL1) is

property

continuity (S-AC). take any

so that i f

A

Therefore,

Z[lf(v)lap(v) f(v)

: %

0

= f(v)

obvious: v E A,

Set

gm(v)

if

lfm(v)I

Z

0

and

%

gm

p[A]

v

and let if

If(v)l.

E

absolute m.

0,

%

A.

where

A

is a

p.m

be as in the last

Ifm(v)

I <

If(v)l.

and

The following facts are are

If(v)-gm(v)l is

the

0.

for all

v E W,

for each gm(v)

v E A]

= fm(v)

>

prove

Then for some finite

E

V,

certain internal subset of

gm(v)

p

is an internal set with

Now assume

paragraph.

To

clear.

,(

If(v)-f,(v)l;

for

S-monotone continuous.

Hence,

by inequalities as above,

Thus,

To

I[lf(v)lGp(v) finish

f E SL1(p) n E

*IN.

the

:

v E A] =: 0.

proof

of

is the limit of a

define

(a)

we

are

p-finite

to

show

sequence.

that

any

For each

Section

91

HvDerfinite Integration

1.4:

It is easy to see that for finite Moreover, for each infinite

IIf-fnll

Therefore, (b) (a):

+

fn

n.

is

n,

n + m.

0 as

The proof is basically the same as the last part of

pick

f E L1(p)

and for. each

n E

Convergence Theorem guarantee that

Ilf-fnll

fn

define

The Finite Lifting Theorem (1.3.9) and

above.

p-finite.

+

as

the Dominated

0.

The next lemma shows that the projection is a contraction.

( 1 . 4 . 9 ) LEMMA:

If

5 B

f

:

V

+

If(v)l6p(v)

*R

t s an tnternal functton,

*[O.+m]

(in

lotth

03

Z

p

tf

Jvl?ldp

p

ts

V postttue tnftntte). PROOF :

If

2[lf(v)l6p(v)

relation is trivial),

:

v

€

for each

V]

is

n

€ ulN

finite define

(otherwise

fn

the

as in the

Chapter

92

proof of

lfnl 5 If

carrier.

fn

Then each

(1.4.8)(a)

1:

Hvperfinite Measures

is finite, has finite

(lTnI) 1 IT1

and

pointwise.

By

the

Monotone Convergence Theorem and the Proposition (1.4.7).

and

JW (1.4.10) DEFINITION: g

Giuen

L1(p),

€

f E SL1(p)

a function

such that

N

f = g p-a.s.. i s called an

S-integrable surely,

the

liftings only

error

S-integrable

are

to

sums

integrable function has an

more

than

an

g.

p-lifting o f

just

close

almost

infinitesimal.

Every

S-integrable lifting.

(1.4.11) THE INTEGRABLE LIFTING LEMMA: The maps then

projection

~~'(1.1) the

norm

onto

of

internal

~'(p).

in

SL1

zw

f(v)61.r(v).

S-integrable

satisfies

functions

f,g

Moreover, i f

E

SL1 (PI,

Ilf-gll 1 ll?-ill

and

n N

f(v)dp(v)

JW PROOF :

We know that the mapping

f

-

T

is continuous by (1.4.9).

By the Dominated Convergence Theorem, each g L 1 -limit of its truncation g,(v) = g(v)I

in

lg I Sm)

<,{

indicated in (1.4.8)(b).

lf,l

<

m.

Each

gm

by (1.3.9). and by (1.4.7)

L1(p)

has a lifting fm

N

fm

(v). fm

is the as with

is isometric.

Section

1.4:

Hvperfinite Intearation

Jlgm-Tm1dp = 0.

Since S-limit

f.

We know

fm

is an

say, by (1.4.8)(a).

s

T,dp

Z

S-Cauchy

sequence, with

By continuity,

2[fm6p]

and

Jw

= 0. hence

S-lim 2[lf-fm16p]

93

-f = g

T,dp

s

Tdp

N

fdp z H[f(v)Bp(v)

p-a.s.

:

while

v E V].

INFINITESIMAL HULLS: The Lebesgue space of the hyperfinite measure,

L1(p),

can

be identified with part of the norm infinitesimal hull of the * l L (p) (Stroyan & Luxemburg [1976. chapter internal space 103).

The norm

infinitesimal hull of

finite mod infinitesimal internal

L

(p)

consists of

*L1-functions; the continuity

L1

conditions locate a classical

* l

space inside the norm hull.

We shall denote the finite elements by

FL 1 (p)

= {f

:

Blf(6p

is a finite scalar}.

We denote the infinitesimal elements by

IL1 ( p ) = {f

:

The hull is the quotient space Henson

Moore

and

characterization

of

2lf)6p z 0 ) .

FL1(p)/IL1(p),

[ 19741

abstract

show

L-spaces

that implies

Kakutani's there

A

measure

p

(not hyperfinite

p)

such that

Hence w e have the following embeddings and projections:

is

a

Chapter

94

HvDerfinite Measures

1:

U

II

(1.4.12) PROPOSITION:

(a)

I f

f,g

A

1

SL ( p ) .

E

then

A

N

f = g

N

f = g

i f f A

(b) N

f

I f

SL1(p).

E

g E FL1(p)

a.e.

A

f = g.

and

then

N

f = g

a.e. and

g

E

SL1(p).

PROOF : Part (a) is just another way of saying that the projection

is an

S-isometry, and part (b) follows from (a) and the obvious

facts that

E SL1(p)

IL1(p)

direct proof.

By definition of

the map

Or]

then

H[lf(v)-g(v)l6p(v)

:

v

E N

JOrl?-gIdp = 0: thus,

A E V

If

H[lf(v)l6p(v)

:

is a vector

1 0;

v

where

internal sets the p E

f = g

IN

f = g

means

(1.4.10).

Lemma

a.e.

E

A]

A

Z

0, so that

be an internal subset of

N C A.

(Ap(p E

p(N) with

S-absolute continuity of U

by

is a

A

N

only takes infinitesimal values. A\N.

".

Here

A

is an internal set of infinitesimal measure,

Finally, let

in

space.

such that

f.

Then,

f(v)

10

= 0.

Choose a

N E A

E A,

P for each

B

p[Ap] E

V

where

g

for all

v

sequence of

<

l/p.

By

there is a

1.4: Hvperfinite Integration

Section

H[lf(v)lap(v)

f

Since

: v

H[lf(v)(Gp(v) the

same

FL1(p)

: v

-

know

-

6

A]

E

e.

inequalities

I[lg(v)lp(v) We

A\Ap

S-MC, on

has

E

A]

so

used

is

As

P

]<

e.

its sum is infinitesimal, hence

Now.

that i t is infinitesimal. in

(1.4.9).

(p)

SL1(p)/IL1(p).

v E A

(S-AC)

proving

give

us

Z 0.

after

L1

:

95

a

(1.4.10) that

contraction

a matter

of

and

is

fact, the

completely characterizes the space of

the

projection

isometric

on

isometry property

S-integrable functions

SL1b).

(1.4.13)THEOREM:

If

f

:

U'

-

*IR

is

internaL.

the

following

are

equivalent: (a)

f E

(c)

7

SL'(~).

E L1(p)

lTldp

and

1

H[lf(v)l6p(v)

:

v E U'].

JU'

PROOF : (a) implies (b) is part of the claim of Theorem (1.4.11).

On the other hand, Lemma (1.4.9) proves that (b) and (c) are equivalent. Assume

7

E

L'(p)

is

the

projection

of

an

internal

. d

function f E FLf(p).

f Q SL'(~),

and

Ilfll =: Ilfll.

hence either (S-AC) o r

Then

obviously

(S-MC) are false for

f.

But this is impossible: Let

(1.4.9).

A E V

be

an

internal

set.

According

to Lemma

Chapter

96

Suppose that

I[ If(v)lap(v)

f(v)

v

%

0

for

in

A.

:

v

E

A]

1:

*

Hvperfinite Measures

and

0.

PEA1

%

0

or

In either case,

hence

IlTll Z Ilfll.

contrary to our hypothesis

(1.4.14) PROPOSITION:

If

f

<

lg(v)I

is

If(v)l

S-integrable, v,

for aLl

then

g

g

internal

is is

and

S-integrable.

PROOF : Exercise.

<

Ig(v)l

If(v)l

Also, show that i t is not sufficien, to assume a.s. or n.s.

(Make

g

big on a sma 1 set.)

(1.4.15) ASIDE: Using the hull completeness theorem (Stroyan & Luxemburg,

[1976].

(10.1.20)).

is

it

immediate

that

SL1(p)/IL(p)

complete, thus shortening the proof of (1.4.8)(a). hand, by (1.4.12)(b), of

L1(i).

SL1(p)/IL1(p)

What is more:

P

On the other

is a true normed subspace

using the Integrable Lifting Theorem,

Lemma (1.4.9) and (1.4.12)(a), projection

is

one easily shows that there is a of norm one from L l ( Ap ) onto SL'(~)/IL'(~), so

Section

that

1.4:

the

Finally

latter

f

is

(1.4.13)

location of is not

97

Hvperfinite Integration

complemented

a

tells

subspace

of

the

former.

us

something about the geometrical 1 * whenever within L ( p ) : IIP;II < ll;ll

SL1(p)/IL1(p) S-integrable.

(1.4.16) EXTENSION TO ALGEBRAS (con’t): For

that

a

p-integrable

&measurable

The

etc. as in (1.1.8)(c)

p,d.

and (1.3.12).

function

has

an

and

Perkins’

show

internal

S-integrable lifting.

following

lemma

is

Hoover

[1980]

formulation of an old uniform integrability criterion (due to de

la Valee-Poussin?)

similar

to Burkholder, Davis

and

Gundy’s

[1972. Lemma 5.11. We shall need i t in chapter 7.

(1.4.17)LEMMA: Let

f

:

V

p

be a n internal limited positiue measure.

Let

f

is

* + R

be

an

internal

function.

Then

S-integrable i f and only i f there i s a convex. increasing, internal

function

?J : * [ o . w )

+ *[O..D)

cP(0)

with

= 0

such that

(a)

sup[L

: x

@(XI

<

1 nl z

(b)

?J(2u)

4@(u)

(c)

I[*( If(v) 1)6p(v)

o

for all :

for infinite

n.

u 2 0.

v E V]

is finite.

PROOF :

Sufficiency of this condition is quite easy. exists and that

n

is infinite then by (a) there is a

If such a a(n)

“u

0

?J so

Chapter

98

I

If(v)

1:

Hyperfinite Measures

o(n)@(

Summing, we obtain

This shows that

from

(1.4.5)(c)

f

holds and

is

S-integrable.

(Notice that we did not require (b) or convexity, though they are important in chapter 7.) Conversely, suppose that construct

f

is

b

in

{aj

*A}.

The conditions are:

Let

= 0

. a

internal

order

and choose

inf

a2 = 1 + inf[b

to

al = 1 + inf[b

E

is

greater

than

*IR:

(c.l)(b,l)

&

finite by

S-integrability

(c.2)(a2.1)

both hold.

may

use

construct

is finite by (1.4.5)(c)

a

because

of

.j)

.j+l

and

an

E

*IR

and

the

f

internal

:

and

sequence

(c.2)(b,O)].

(c.2)(a1,0)

tnf.

(c.2)(b.l)].

Next, Again, (c.l)(a2,1)

The holds select a2

is and

This is an internal process which we

to select an internal sequence

(c.l)(a

We must

We will use two conditions, (c.1) and (c.2). on a

@.

number : j E

S-integrable.

such that {aj : j € *A} ( ~ . 2 ) ( a ~ + ~ , j ) hold for all j in *A.

1.4:

Section

(The

inf

is always nonempty.)

finite and when Define

99

Hvuerfinite Integration

is infinite

j

= j

q(t)

for

When aj

j

is finite

a

j

is

is infinite.

t E [aj-l,aj)

and define

@

by the

2cp(t).

since

internal formula

Condition (c.1) implies that for all if ,(

then

t E [aj-l.aj)

2cp(t).

2t

This fact about

cp

<

t,

aj+l.

<

cp(2t) so

cp(2t)

<

(+tw

gives us conclusion (b) of the

lemma because

Conclusion Since

cp

convex. of last

@(O) = 0. Since

Clearly.

j

with

[O,x] aj

infinite whenever

Finally,

the preliminary

is positive and increasing,

over

cp

(a) and

@(x)

<

< x

kx

conditions are easy. @

@(x)/x

is increasing and is an average value

and more than half of that average is the x,

we see that

is infinite.

when

x

<

ak,

@(x)/x

is increasing and

This forces

so. by

(c.2)(aj,j),

100

Chapter

m

< [ 1 +

z k=2

T h i s shows that

the sum of

the proof of our lemma.

@(If

1:

k]

HvDerfinite Measures

H[ If(v) Isp(v):v

E

V].

pk

) 6 ~ is finite a n d completes

101

(1.5) Absolutely Continuous Measures 8 Conditional Expectation Until now we have dealt only with positive real measures, but

now

we

want

to

consider

general

measures (or even complex measures). and let of

so

%

finite measure countably

V

Let

finite

real

* finite

be a

set

be a sigma algebra containing the internal subsets

’%

V,

(signed)

2 Loeb(V),

the Loeb algebra.

is a function

additive.

that

u

is,

: ’% + IR

Recall that a

such that

F = UF

if

is

a

u

is

disjoint

rn

countable union. then

u[F]

=

1 u[Fk]. k= 1

the finitely absolutely

convergent real series.

(1.5.1) EXAMPLE (Keisler): As the reader has no doubt guessed, in section (2.3) we

IRd

will show that every Bore1 measure on

representation in terms of a weight function

* finite

set.

has a hyperfinite on an

6p

S-dense

However. not every probability measure on Loeb(V)

is hyperfinite, that is, equal to the extension of an internal

measure Let

*B(V).

p :

V

*B(v) + * Lo.=). be

* finite

and let

91

be a free ultrafilter in

the algebra of internal subsets of

function of

91

[u(U)

= 1

if

U E 91.

u[W)

V.

The indicator

= 0

a finitely additive measure on the algebra of

if

W e 911

internal sets,

u(0) = 0 and m

in

u( U Ak) = Z u(Ak). k= 1 k=1

for finitely many disjoint internal sets

Ak,

1

is

<

k

<

m E

Chauter

102

By

(0.4.2)

we know

1:

Hvuerfinite Measures

that no disjoint

strictly

countable

union of internal sets is internal, s o that the hypotheses of Caratheodory’s

extension

theorem

&

(Wheeden

section 11.51. for example) are fulfilled.

-

countably additive extension Since and

= 0

for every

then

6~(v) =: 0

P[Bp(v)

:

1

<

k

<

n

Or]

E

for

=: 1.

The number extension

v

If the extension

V.

E

*D(V)

Since

91

-

.

u = p

+

*[O,~),

u(V)

V

by an internal map

m

u({v

:

Z[bp(v(h))

:

1

h

<

= 1,

v(k),

internally, so

v = v(k).

<

p[{v

1 :

k

<

m})

v = v(k),

N

is measurable

with

h(N) = 0

and

M C V\N.

hyperfinite measure

p

it

exists.

The

takes only the values

<

1 5 k

= 112.

m}]

This

over the same sigma

o.h

o 1 A.

algebra are said to be mutually singular.

singular over

1/21.

as claimed.

# p

measurable

<

k]

is defined

Recall that two finite measures

Let

:

p

v E V.

every

Enumerate

zero and one while

u

has a

and define

m = max[k:

proves

u

Loeb (V).

to

the hyperfinite extension of some internal

v

Thus

C1977.

is a free ultrafilter. no finite sets are in

91

u({v})

u

Zygmund

p,

a(M) = 0

such that We

the measures

shall

-u

i f there is a

see

and

that p

whenever for

M

every

are mutually

Loeb(V).

be any (positive) limited hyperfinite measure on

(induced from an internal weight function

6 ~ ) . Recall that a

V

Section

1.5:

finite

measure

v[N]

= 0.

the condition

<

IUCMll

is

denoted

p.

Since

that

9 >O.

standard

IR

u : Loeb(I)

c o n t i n u o u s w i t h respect to

implies

103

Conditional Expectation

<

p[V]

for every

<<

standard

M

such that for

<<

u

u

m,

called

>

Loeb(V),

E

p[N] = 0

if

p.

is equivalent to

p B

absolutely

there

0.

p[M]

<

9

is a

implies

E.

If

u

<<

on

p

Loeb(V).

then

extends

u

to

the

p-measurable sets, because

= {M

Meas(p)

the

E V

: 3

p-completion = 0,

u[W\U]

we

U.W

E

E M C W

Loeb(V).U

of

Loeb(I).

Since

may

extend

u[M]

=

&

= 0

p[W\U]

u[W]

= 0).

p[W\U]

implies

= u[u].

using

finiteness.

(1.5.2) PROPOSITION:

If for some

t s a finite measure o n

u

limtted hyperfinite measure f

there is a n tnternal

u[U]

unique; if

g

u

SL1(p)

U

E V.

h

in

u

<<

p.

I, t h e n

s u c h that

U]

: u E

Moreover.

SL1(p)

and on

p

f

is a l m o s t

h a s the property aboue. t h e n

and any

represents

in

z H[f(v)6p(v)

for every internal

SL1(p)

Loeb(V)

with

Ilf-gll

IIf-hll

%

Z

0

0

in

also

o n tnternal sets.

PROOF :

W e shall base our proof on a well-known classical result,

ChaDter

104

the Lebesgue

Decomposition

Theorem, stated

from

the

Radon-Nikodym

Integrable

part

of

for

the

reader's

The proposition (1.5.2) follows

convenience as (1.5.3) below. easily

Hvperfinite Measures

1:

Lifting

Lemma

A

(1.5.3).

(1.4.11)

complete

and

the

of

the

proof

proposition can be fashioned on a partition argument like the one that Wheeden and Zygmund [1977, section 10.31 use to prove Lebesgue Decomposition. justify an

We feel that the slight changes do not

independent proof.

we appeal

so

to the classical

fact. Proposition

(1.5.2)

shows

that

hyperfinite measure with "signed weight

If we

already

knew

u = a

that

is

U

function"

for

a

6p(v)

= 0

limited

f (v)6p(v).

positive

hyperfinite measure given by a weight function have the weaker hypothesis that

the

6n(v)

implies

limited

and if we = 0,

6a(v)

then we may define

=

aa(v) (apAv)

E

SL1(p),

,

f(v)

In order to have

f

,

condition that whenever I[f(v)6p(v) 6a

# 0

6p(v)

= 0.

we need to know the external z 0, then

p[A]

v E A] = .[A]

a[A]

0, because

Z

(see (1.4.5.b)).

In general, i f

is not positive, we can work with the separate

internal max[O.6a(v)] and

:

6p(v)

a-[V]

measures and

4-

a

-

and

a

max[O.-6a(v)].

with

weight

f

p[A]

=: 0

a+[V]

FL1.

to satisfy

We also need the infinitesimal absolute continuity.

(1.4.6).

functions

We need to know that

are both limited in order for

nonatomic. then we only need

positive

implies

a[A]

If

p

is

=: 0, by

Section

1.5:

105

Conditional Expectation

(1.5.3) LEBESGUE DECOMPOSITION THEOREM: (X.3.A)

Let let

be a f i n i t e positiue measure space a n d

be a f i n i t e measure o n

u

3.

T h e n t h e r e is a u n i q u e

decomposition

where

Moreover.

N,

u

and

a

+ u[F].

= a[F]

u[F]

are measures,

<<

a

f

t h e r e is a n a.s. u n i q u e

E

E

3

A

u 1 A.

and

L1(A)

a n d a null

= 0. s u c h t h a t

h[N]

= JFfdA

a[F]

deriuatiue o f

a

f

= u[F

u[F]

and

F E 3. T h e f u n c t i o n

for

F

for

fl N],

is c a l l e d t h e R a d o n - N i k o d y m A.

w i t h respect to

We conclude the discussion of example (1.5.1) by showing

u

that

for every hyperfinite measure.

1 ~l

Let

theorem.

Either

u[N]

Let

= 1.

a =

fdp.

measure

If

g

Jl.

u[N]

induced by

= 0,

be a

u = 0.

reasoning in (1.5.1).

-u 1

N E Loeb(V)

the weights Thus

which

p-S-integrable

then

-u[N]

be a

and

u = u

-

be as in the decomposition in

-u =

p = A

3 = Loeb(V)

nonzero bounded hyperfinite measure, from (1.5.1).

Let

and

a

case

lifting of a

g(v)gp(v).

-

= u[V]

u = 0,

= 1

f

or

where

is the hyperfinite contrary to the and

p[N]

= 0, so

1:

Chapter

106

HvDerfinite Measures

( 1 . 5 . 4 ) NOTATION:

V

When 2[6P(v)

:

measure.

* finite

is

v E V]

Z

1

p =

then

P

The letter

6 p = 6P : V

and

P

j

satisfies

*[O,m)

is a hyperfinite probability

looks more like "probability."

There

If

is also a custom of calling integrals "expected values."

f

:

v

*IR is internal,

is the internal expected value of

f.

Let

g

:

V

.-, IR

be

P-integrable. then

One

important is

(1.5.3)]

use

in

Radon-Nikodym

of

showing

that

derivatives

conventional

[see

conditional

expectations exist.

(1.5.5) DEFINITION: Let space, let g : V

IR

$

be

expectation o f

is

be

(W.Meas(P).P)

the a.s.

such that

C Meas(P)

P-tntegrable. g

given

unique

J$dP

let

The (external) condittonaL

0,

= E[g(v)lF(v)l =

probabtltty

a sigma algebra and

%-measurable

E[h(v)lF(v)l or

be

a hyperfinite

JFgdP

functton

}

for

h

F

:

V tn

IR 8.

1.5:

Section

= 0

(IF(v)

107

Conditional ExDectation

= 1

v Q F.

if

v

if

E

F.

for

F

is the indicator

function.)

If we let p

5,

to

u[F]

then

=

h

s,

g(v)dp(v)

5

and restrict

from the definition is the Radon-Nikodym

u = 0 because

derivative and

in

<<

u

p.

(1.5.6) EXAMPLE:

V = R

Let

A time

(0.3.12). on

R

t

as follows. Two

restriction. t

at = u ,

p =

and in

P

lI

as in (0.2.7-10).

determines an equivalence relation

wt = ol[O,t]

Let

sample points

=

Cut] = { u

E

R

:

( W ~ . W ~ ~ , * = * . O ~ the ).

R

w,u E

that is. if they agree up t o

denote the class by

(0.3.7) and

are equivalent

the time t

wt = u } .

t.

if

We may

We shall want to

know internal conditional expectations like

(1.5.7) DEFINITION:

V

Let

2[6P(v) Let the

:

p

v

be E

* finite.

V] =: 1

Let

and Let

6P

:

f

V

:

V

+

*R

be

satisfy internaL.

V

be a n internal equiuaLence relation o n

equiuaLence

class

of

v

v

+

E[flp(v)],

ECf IP(V)l

p(v).

denoted

(internal) conditionaL expectation o f function

+ *[0,1]

f

giuen

p

with The is the

where = ZCf(u)6P(u)

An internal equivalence relation

p

UPVI/P~P(V)l.

:

on

V

induces a sigma

Chauter

108

subalgebra of the Loeb algebra,

1:

Loeb(p)

Hvuerfinite Measures

C Loeb(V).

defined to

be the smallest sigma algebra containing the internal sets with the property that if

upv

and

Any equivalence relation

v

= {u : upw for some w

W(p)

the set of

p-closed

E

U,

then

u

E

U.

(external or not) induces a

p

sigma subalgebra of a sigma algebra let

E

U

W}.

W C V,

For any set

8.

The sigma algebra is

%-sets.

(1.5.8) EXERCISE: Proue

that

is a sigma algebra.

%(p)

i n t e r n a l a n d p r o u e that

Loeb(p)

Let

p

be

= Loeb(V)(p).

An easy proof of this exercise can be given by using the separation theorem (2.2.3).

( 1.5.9) REMARK:

P

Let

* finite

be a

probability

internal equtvaLence reLation.

:

M

is in

number such

Meas(P)

k. that

and

M = M(p),

E M

C

W

and

be

an

We know that if

then for every finite natural

there exist internal sets U

p

= M(p)}

can also be thought of as a sort of completion.

M

let

The sigma algebra

= {H E Meas(P)

Meas(P)(p)

and

P[W\U]

<

U.W 1 i;.

contained in The

internal

I

set

1.5: Conditional Exuectation

Section

= {v : 3 u

U(p) U

U(p)

C_

then

M.

C_

Meas(P)(p)

contains

N

Also, if

there exist internal

5 W.

M = M(p)

and since =

N(p)

N

Wc C N ,

and

S

Therefore, a set

S = S(p)

i f and only if

U

V\M = MC.

=

E M C [Wc(p)lC

U(p)

U

U & upv}

E

109

is in

and for every finite

W = W(p)

U = U(p),

p-closed sets

k, such

that U

E SC W

and

<

P[W\U]

k.

(1.5.10) PROPOSITION: (V.Meas(P),P)

Let space and

Let

p

S-integrable

an

Lifting

of

S-integrable Lifting o f

a

hyperfinite

probability

internaL equiuaLence

= Meas(P)(p)

Meas(P.p)

Let

be

be

If

f r o m (1.5.9).

f

E[flp(v)]

then

g.

relation. is a n is

an

E[g(Meas(P,p)].

PROOF : We know that i f F(u)

= F(v).

F(v)

= E[fIp(v)],

Therefore the sets

Loeb, hence in

Meas(P,p).

S-integrable. e.g.. use

then

<

{i(v)

upv

are

r}

p-closed and

It is easy to see that

H[F(u)6P(u)

:

>

F(u)

k].

implies

F(v)

k

is

infinite.

n

We know that

If

internal. H[f(v)GP(v) where each

U’ p

U :

v

gdP % P[f(v)6P(v) JU is also p-closed.

U(p)

U] = B[B[f(v)6P(v)

:

E

:

v E U] = U.

v E p(u)]

whenever

U

is

then :

u

E

U’],

is an internal selection of one representative from

equivalence class in

by definition of

the

U.

The second sum equals

internal conditional

expectation.

The

110

1:

Chapter

Hvperfinite Measures

latter sum equals

B[E[fIp(u)]6P(u)

Hence

F(v)

: u E U].

has the property that

JugdP =

for

all

internal

p-closed

J UFdP U.

sets

The

(1.5.9)

remark

completes the proof.

(1.5.11) EXERCISE: Let

g

S-integrable between

be lifting

E[glLoeb(V)]

P-integrable of and

g.

f?

and

What

is

let

the

f

be

an

relationship

111

(1.6) Weak Compactness and

SL1

We believe that the following result can be a useful lemma in probability.

(1.6.1) THE DUNFORD-PETTIS CRITERION: (V.Meas(P),P)

Let

If

space.

functions,

H

H

is an

SL1(P), 1

N

H E L1(P).

be

a hyperfinite

internal

set

S-integrable

then the set of its projections, m

o(L (P),L (P))-weakly

is

of

probability

relatiuely compact.

that is, has compact closure in the weak topology.

The reason for this is the following.

(1.6.2) LEMMA:

H

If H E SL

1

is an internal set of

(v).

S-integrable functions

then 1

N

H

(a)

The set of projections

(b)

For euery standard positive

positive

such that if

9,

p[A]

<

A

is bounded in

there is a standard

E .

E W

L norm;

is internal. then

8 t m p l t e s H[lh(v)l6p(v)

: v E A]

<

for all

B

h

E

H.

PROOF :

(a)

The set

{H[lh(v)lS~~(v)

:

v

E V]

:

h E H)

is internal

and only contains finite numbers, so its supremum exists and has to be finite. (b)

(S-AC):

Follows given

B

from

E aR+.

the

S-absolute continuity condition

the set

ChaDter

112

<

A E *l3(V))[v[A]

( 5 E *IR+:(V

8 2

(V h E

is internal and

H) B[lh(v)l6p(v)

contains all positive

Eberlein-Smulian

:

v E A]

< €1)

infinitesimals, s o

it

5.

contains a standard positive

The

Hvperfinite Measures

1:

Theorem

& Schwartz

(Dunford

[1958,

p. 4301) renders the following consequence about weak sequential compactness:

( 1 . 6 . 3 ) COROLLARY:

Let

(Or,Meas(P),P)

H

space, and

be

a hyperfinite probability

an internal set o f

S-integrable functions.

T h e n for every

sequernce

S-integrable

and a subsequence

g

hn

from h

H,

there is an

so

that for every

"k finite internal

f,

S-lim Z[h (v)f(v k "k

bP(v)

:

v E V ] = B[g(v)f(v)bP(v)

:

v E Or]

The projected set need not always be weakly closed.

( 1 . 6 . 4 ) PROPOSITION: N

If

H

closed in

is an internal subset o f 1

L (P)

SL1(P),

for the norm topology.

then

H

is

I f . i n addition,

N

H

is

S-convex. then

H

is also weakly compact.

PROOF : Since in any Banach space the weak and the norm closure agree on convex sets, the second part follows from the first and from the last theorem and its corollary.

Section

1:

113

Weak Compactness

We leave the proof of norm closure as an exercise. The

following

an

is

example

of

an

internal

set

of

S-integrable functions on a probability space whose projection is not weakly closed:

(1.6.5) EXAMPLE: Let

V = {vl,**-,vs},

s

an

infinite

factorial, be a

probability space with uniformly distributed weights. n E

*IN.

define an internal function

hn(vi)

= +1

if

i

hn(vi)

= -1

if

i

Not only are all the is

internal

hn

too.

(incidentally.

E

hn

V + *R

as follows:

O.-**.n-l (mod 2n) n.-..,2n-l

(mod 2n).

internal, but the set

H

= {hnln E

*IN}

for all

n

lhnl = 1

Moreover, since

hn

:

F o r each

equals the constant function

n =

from

1

s

N

on).

the sum of

lhnl

and the integral of

1;

both infinitely close to

hence,

Observe that all the subsets of

with

p

finite,

r

lhnl

( = lhnl)

are

H C SL1(P). H

of the form

infinite, are also internal; and their

u

projections

contain all but a finite number of terms of

H(p.r) N

the sequence

(hm Im E a # ) . N

Assume that subsequence of

.,

H

is weakly

(hmlm E a l N )

sequentially compact.

Then a N

converges weakly to some

hn.

We

prove next that this subsequence cannot have a subsequence with

114

ChaDter

1:

HvDerfinite Measures

so

that

N

a weak

limit within

projections compact.

of

each

internal

H(p.r), sets

are

not

some

weakly

of

these

sequentially

Thus i t is neither weakly compact nor weakly closed. N

If

n

is finite, the set

H(2n.s)

is not sequentially .-

compact, since no function in i t is almost equal to the weak h

9

E

topology

H(2n.s).

let

is Hausdorff). k = [q/n]

Let

us

prove

hn

(and

if

this:

2 2; then the set

[(k-l)n.kn)

2 [n,2n) U C3n.411) U

U (C(k-2)n. (k-l)n)

depending on whether k is even or odd. Then p[AO] 1 1 5 p[{vili = l.**-,q}]; and a similar argument can be carried over

to

every

interval

{vili = jq+l,-**,(j+l)q].

For

j = O.-**.s/q-l.

A. = {v E: Vli = jq+l.-**.(j+l)q J i thus PCAjl 2 V[{V

E

Vlhq(v)

If

# hn(v)}l

n

is

= J4AJ

infinite,

+

the

&

h (v,) # hn(vi)} q . Therefore,

5:

'.*

PrAs/q3

+

set

>L!lS=l

7 .

3 s q

G(l,[n/2])

is

not

sequentially compact, since no function in i t is almost equal to N

hn

(to prove

this, take

before, inverting the roles of

hq q

€

H(l,[n/2]) and

n).

(1.6.6) QUESTION: What is the closure of the sequence

{h,}?

and

proceed

as

115

CHAPTER 2:

MEASURES AND THE STANDARD PART MAP

T h i s chapter makes several connections between hyperfinite measures and classical constructions in Euclidean spaces. of

the

results

noted below.

have generalizations to

topological

Many

spaces as

T h e idea of the chapter is to measure all points

with a n appropriate hyperfinite

measure

that

being measured by the classical measure. the inverse of the standard part map.

lie near

points

T h i s means that w e use We begin with Lebesgue

measure because a complete measure is easier to treat than the naked Bore1 algebra which follows. Let

0d = {r

€

*Rd : r = ( r l . - * * . r d ) with each r

. i

denote the Cartesian product of the limited scalars.

limited}

Recall the

definition of the standard part map f r o m section (0.3).

(2.1) Lebesgue Measure Let d-fold

Ud

be a

* finite

Cartesian product.)

every standard point of

Rd

subset of We

*Rd.

say that

Hd

( I t need not be a

is

S-dense

is close to some point of

W e say that a n internal weight function approximates volumes o f standard rectangLes in bounded standard rectangle, for example.

if

Ud,

6a : Ud + *[O,-)

Rd

i f for every

1 = {r

the

2:

ChaDter

116

E IRd

d-dimensional volume o f

: aj

I

<

Measures 8 Standard Parts

r

<

j

b.}, J

is approximated by

a,

d

TI (b -a ) = d-vol(L)

j=l

j

j

Here is a good exercise for beginners at infinitesimal analysis: Show that for every weight

function

S-dense

*finite

that approximates

Td

set

*Rd,

there is a

standard rectangles.

(See

Appendix 1.) An nice example of a weight function and let

h

E

*I

be an infinite integer,

n = h!

S-dense set is to and take

and

Notice that Fd contains all standard rational points, because E,P * T I [ k : l < k < h & k # q l is an internal expression. It 9 n follows that for rectangles I = {r : aj r j < bj} with

<

rational edges.

aj.bj

E

Q,

we have

d-vol(1)

= a[

*I

fl

Fd 3 .

The infinitesimal volume approximation for all standard bounded rectangles follows by finite rational approximation. Notice

that the internal measure

d-dimenstional

Lebesgue measure d h ( S ) = a(*S fl T ) for all sets S.

a

x

does not represent by

the

For example,

formula Qd

has

Section

2.1:

Lebesgue measure zero, yet given.

117

Lebesaue Measure

*Qd 3 Ud

Also, we cannot generally say that

the hyperfinite measure, because

d

T -example just

in the

Td

X(S) = aCUSnTd] for need

not

contain any

standard points. Throughout conventions.

this chapter we use

= (st(r,),*.-.st(r,)).

rjEO.

or

r = (rl.***.rd).

r E *Rd,

If

provided

r E O .

If

S

the following notational

C *Rd,

each then

of standard parts of limited vectors from

We usually discuss

st

restricted

to

then

component st(S)

is

st(r) limited,

denotes the set

S,

Td

because we are

comparing hyperfinite measures and classical ones. In this -1 context we write st ( E ) f o r the inverse of the restriction,

In

particular,

our

formula

for

Lebesgue

measure

h(L)

= a[st-'(L)]

with this convention in effect.

is

The next result plays an important role in our proofs.

(2.1.1) PROPOSITION: T h e standard part o f

*IRd

a n internal subset o f

is

closed.

PROOF : Let

r E closure(st[B])

for each finite natural number

for an internal m.

st[B]

B

_C

fi { s E IRd I

*tRd

.

Ir-sI

Then --} 1

118

f

B n {s

hence

0

Bm

decreasing chain and i f to

b

nB,.

E

arbitrary

I

*Rd

E

Measures & Standard Parts

2:

ChaDter

2

<

Ir-sI

-} m

has nonempty

then

st(b) = r .

topological

spaces

Bm

=

T h e countable

# 0.

intersection by

saturation

(The generalization of this can

be

found

in

Stroyan

&

Luxemburg [1976]. section 8.3.) Note: measure

Perhaps

preserving

i t would be better

in our

next

to say that

result,

but

we

have

a

*f i n

st-'

is

used

the

customary terminology.

(2.1.2) THEOREM:

ud

6a

Suppose

+ *[o,m)

ts

te

weight

f u n c t i o n that approxtmates volumes o f standard r e c t a n g l e s

Rd

in from

as above.

the

Then

Lebesgue

measure

Lebesgue

is a measure-preservtng map (R

space

d

(T .Meas(a).a).

hyperftntte space ts

st

measurable

if

,Leb,X)

tnto

spectftcally.

and

only

h(L) = a[st

a-measurable, and then

d

-1

tf

the

L C Rd

st-l(L)

is

(L)].

PROOF :

Td

First, the limited points of

form a n

a-measurable set

because they a r e a countable union of internal sets,

0

d

ll

Td = U{t E Td I I t 1

<

m} E Loeb(Td).

m

Next, we establish three general claims which we u s e in the proof of each implication of the theorem.

Claim 1)

If st-l(I)

I

ts

E Meas(a)

a

and

bounded

open

rectangle.

h ( 1 ) = a[st-'(I)].

then

Section

2.1:

I

Since

119

Lebesaue Measure

is open, whenever

contains a finite neighborhood of

c * I n ud

st-'(^)] <

and

I

We may write rectangles, and

I

r,

and t E

*

since I -1 Thus, st (I)

t

%

* I.

r,

n u d 1) = ~(1).

st(a[*I

as a countable increasing union of compact

= UI,.

Since

2 st(a[*Im

~x[st-'(I~)]

aI

r E

*I

Im is closed. d n U 1) = h(1,).

fl Ud Z ! st-l(Im)

Moreover, we have

the inclusions

u

* I~ n ud

E

m

u

st-l(Im) = St -1 (I) E *I n ud.

m

Now we use rectangle approximations to see that

Hence.

st-l(I)

claimed.

is

a-measurable with

(It is easy to see that

a-measure X(1) -1 st (I) E Loeb(Ud).

as see

(2.2.8).)

Claim 2)

T h e family of

z1 =

sets

{L E IRd

I st-l(L)

Meas(a)} d Borel(IR ) .

i s a s i g m a a l g e b r a c o n t a i n i n g the B o r e L a l g e b r a

This claim is easy "commute" with algebra.

to verify, because

set algebra

Claim

1)

inverse functions

I1

operations, s o

implies

that

I1

€

is a

contains

sigma

all

open

rectangles, thus the whole Bore1 algebra. Next, suppose that zero.

X(L) = 0.

countable cover of than

a.

L E Rd

is a Lebesgue set of measure

Then for every standard

L

By Claim

B

>

0.

there is a

by open rectangles of total measure less

1)

it

follows that

a[st-'(L)]

= 0.

so

120

ChaDter

st-l(L)

is an

P2

The collection

I1

since

P2 = Leb,

L

a-null set and

contains

P1

=

Borel(R

that is, i f

L

L1

E

fl Leb

d

)

in particular. is a sigma algebra, but

and all Lebesgue null sets,

Leb.

E

Measures & Standard Parts

2:

then

st-'(L)

E

Meas(a).

The family of sets where two defined measures agree is a sigma algebra.

Thus,

sigma subalgebra of

Z3 = {L

Leb

Leb : a[~t-~(L)] = h(L)}

E

is a

which contains all Lebesgue null sets

and contains all open rectangles.

Z3 = Leb

Hence again

and

this proves the next claim.

Claim 3 ) Leb = {L C IRd

The

L

third

E

Leb,st-l(L)

claim

trivial because i f

makes

L

E

Meas(a)

one

= h(L)}.

& a[st-'(L)]

implication

of

the

L

is a Lebesgue set. then

theorem

belongs to

the right-hand-side of claim 3 . hence satisfies the conclusion. Now we prove the reverse implication of the theorem. suffices to prove that whenever st-l(L)

E

Meas(a).

particular.

then

L

Rd

is a bounded set with

= X(L)

a[st-'(L)]

It

and

L

E

Leb

in

The reason this suffices is because we may treat

the general case as a countable union

L = U(L n Im)

where the

m are bounded rectangles. We shall establish Lebesgue Im measurability of bounded a-measurable sets by classical inner and outer approximation.

This is the step where Proposition

(2.1.1) is used.

L

Let st-l(L)

f

be a set contained in a bounded rectangle

Meas(a).

Let

a

I

with

be a standard positive tolerance.

We know that there is an internal

B E st-l(L)

with

a[st-'(L)]

Section

<

2.1:

a[B]+a.

<

a[B]

The set

st(B)

L

is at least

1 a[st-'(L)]-e

h[st(B)]

is compact (by 2.1.1).

= h[st(B)].

B)]

a[st-'(st

measure of

121

Lebesgue Measure

Therefore, the Lebesgue inner

a[st-'(L)],

with

because

>

a

1 a[st-'(I\L)]-e.

h[st(A)]

satisfies measure of

The set

I\L

is at most

L

and an

st(A) 5 I\L

I\st(A)

5 a[st-'(L)]+a.

h[I\st(A)]

and

arbitrary.

B

obtaining a compact set

0.

C L

st(B)

Now we apply a similar argument to the set arbitrary

by claim 3,

so

with

L

covers

and

Thus, the Lebesgue outer

u[st-'(L)],

since

a

is arbitrary.

But this means that the inner and outer Lebesgue measures of coincide,

L

making

a[st-'(L)].

Lebesgue

measurable

with

L

measure

This completes the proof.

We shall be most interested in the next definition in the

case where

is internal and the set vectors, M = U d n ud . If

limited

g

S-continuous

od n ud.

on

then

A

+ IR

g : st(U)

is well-defined by

a)

Hd

Let

g

function

and

ud +*w

:

a-measurable

set

a-measurable

U

E

g(s)

U

is

g

a-almost

real-valued

function

= st g(t).

:Hd

g(t)

and

+

Let

*

IR

a

be as above.

is

a-almost

M E Hd

E M

satisfy

b) g

equals the set of

DEFINITION:

(2.1.3)

s.t

the

.. g(st(t))

M

with

We say

S-continuous

provided

a[M\U]

s =: t ,

then

= 0

that a

there

on an is

an

such that whenever

g(s) Z g(t)

and

both

Suppose

that

are Limited.

f

d + IR

: IR

is internal.

S-conttnuous lifting o f

be a function. We say that

f

g

provided that

i s an

a-almost

2: Measures 81 Standard Parts

ChaDter

122

= f(st(t))

st g(t)

Suppose

there

n Ud)\U]

a[(Od Then

g

= 0

is

is

S-continuous and and

= st[g(s)].

= f(st(s))

t € U.

takes

other

such

limited values

g

U

on

= f(st(t))

st[g(t)]

words,

that

= f(st(t)).

st[g(t)]

then

s X t,

In

pd.

U E Od n Ud

set

and whenever

s,t E U

because if

a

ud n

a.e. o n

is

a-almost

S-continuous on the set of limited vectors.

Fussy readers will say, "This isn't a special case of the liftings in chapter

They're right, this lifting is two

1."

legged, s o the following diagram 'almost commutes':

udn rd L *IRn u st

I

1

St

Note the standard part on both sides instead of equality on the left.

There are many kinds of 'liftings' in this book-11

can be

sure

they have

in common

is

that

they are

you

internal

objects with special properties that correspond to measurable objects under various kinds of projections built from standard parts.

We strive for simple properties of

* finite

objects.

(2.1.4) ANDERSON'S LUSIN THEOREM:

Td a n d

Let a)

f

f

:

IRd

+

a

IR

has a n (tnternal)

be a s a b o u e . t s L e b e s g u e m e a s u r a b l e t f a n d only t f a-almost

S-continuous

ltfttng

g.

2.1:

Section

f

IRd

-

R

is Lebesgue

integrable

h a s a n almost

S-continuous

lifting

f

b)

if

123

Lebesgue Measure

:

i f a n d only

SL1(a)

in

g

such that

d

L C IR ,

I n this c a s e , f o r euery Lebesgue measurable

P

r

N

=

g(t)da(t)

f(x)dX(x).

JL PROOF :

I f such a

(a)

g

U n st-'{r

difference,

Ud fl Od 2 U

exists then for

N

<

: f(r)

<

a} = {t E U : g(t)

with null a}

and

f

<

a}

is Lebesgue measurable by theorem (2.1.2). Conversely,

<

= st-l{f(r) Thus, by

a}

we

see

that

k.

gk(t) = f(st(t)) sequence implies

gk

a.e. to be

on

progressive

gh(t) = gk(t)

on

It1

select an infinite

such that

It 1

f(st(t))

<

n

m. on

Let

We

may

extensions,

that

<

gk

g = gn.

od n

h,

by

using

such that choose

the

is,

<

the

h

k

internal

to an internal sequence and

<

m

n

implies

g(t),

The function

T~

a.e.

then

the set where

cannot be internal unless the

k}.

gk

= gn(t)

g,(t)

N

= f(st(s))= Note:

<

{It1

Extend

since

a-measurable.

there is an internal function

definition principle.

g(s)

: f(st(t))

is

f(st(*))

N

N

rd n od

the Finite Function Lifting Theorem, for each finite

natural number

when

E

{t

since

infinitesimal

g(s) g

f hull

Now "N

g(t).

is

g(t)

if

agrees with and

S Z t

so this proves (a).

S-continuous and

lifts

f

is actually uniformly continuous, of

an

internal

function

in

an

Measures & Standard Parts

2:

ChaDter

124

(See Appendix 1 for basic

internal set is uniformly continuous. S-continuity.) T o prove

(b)

first

consider

the

case

f+

Lebesgue integrable then apply this case to

I

[-m,ml [-m.m].

(x)

denote

indicator

function

of

= gm(t)

liftings

for

It1

<

f(x)IC-m,ml(x)

the

S-integrable)

almost

C-m.ml

fm(x)dX(x)

m

(t)]

S-continuous

(XI11

and

because

(2.1.2)

SL 1 (a)-sequence

and

such

gm(t) = 0

and

m

g in ( t ) = min[m,g

truncations

min[m,(f(x)I

of

they

dominated

interval of almost

that

gm+'(t)

It1

>

are

bounded

liftings

of

s

convergence

says

The

m.

(hence

f (x) = m

an

says

Let

gm

for

form

is

f-.

and

Using part (a), we may choose a sequence

S-continuous

s

the

f 2 0

where

S-Cauchy im(t)da(t)

=

is

a

fm

L' (dh(x) )-sequence.

convergent Extend

gm( t )

= gm(t)

for

formula

gn(t)

of

SL'((r)

gn(t)

= g(t)

g E SL'(a)

n

>

to a n internal m 2 It1

is and

there

g

for infinite is

an

SL1(a)

the

gn(t)

and maintain the internal truncation

= min[n.gn(t)]

says

sequence satisfying

S-completeness

infinite

limit

is a n almost

n.

of

n

such

g,(t).

S-continuous

that

Certainly lifting of

f.

N

I t remains to show that w

JTr and

N

gm + g

in

g(t)da(t)

L1(a)

\o

= 0.

but we k n o w that

N

each

gm

satisfies

this

integral

^. .formula so the integral formula follows for

g.

T h e rest is a

consequence of (2.1.2).

slf(x)

Conversely, suppose such a

(2*1*2)

= [odlg(t)

g

Ida(t)

exists.

<

OD.

By part

(a) and

125

(2.2) Borel and Loeb Sets In

this

section we

use

some basic

theory

of

abstract

analytic sets or Souslin sets to give a relationship between Some of this machinery will also be

Borel sets and Loeb sets. important

later

in

the

study

of

hyperfinite

stochastic

processes. Let

Seq

denote the set of all finite sequences of natural

numbers.

Let

be a family of subsets of a set

%

F

that a mapping

:

Seq

4

%

is a Souslin

sequence,

Fs

We may think of the sets

s.

scheme.

%-set. F s ,

words. a Souslin scheme attaches an

X.

We say

In other

to each finite

as attached to the

nodes of a tree which branches infinitely many times as each ( s l , * ' * * , s mis ) increased to

sequence

If

F

Seq + %

:

(~~,*-.,s~,s~+~).

is a Souslin scheme, then the kernel of

F

is the set S = U[n(F : m E IN) u m ulm

:

IN u E IN 1,

where the union ranges over all infinite sequences of natural numbers.

u

E

ININ ,

(ul.**..um) = aim. S

the set

and

denotes

ulm

the

finite

sequence

In terms of the tree interpretation of

is the union "along the top" of the intersections

"up each branch. "

( 2 . 2 . 1 ) DEFINITION:

If set

S

F.

%

t s any f a m i l y o f

i s said

operation i f

to

subsets o f a set

be derived

from

5

by

X,

then a

the SousLin

126

the

is

kernel

coLlection

Here

of

are

operation.

of

a

Chapter

2:

Souslin

scheme

these sets is denoted

some

basic

%.

from

The

Sous(%).

observations

about

the

Souslin

Countable unions and intersections are special cases

of the Souslin operation. Hence

Ueasures & Standard Parts

sous ( % )

is

Moreover.

closed

= Sous(%).

Sous(Sous(%))

under

countable

unions

and

There is no loss in generality when studying

Sous(%)

to

intersections.

assume that

is closed under finite unions and intersections,

%

that is, i f

T

is the closure of

intersections, then

= Sous(%).

Sous(y)

closed under finite intersections and The mapping,

s +

=

m

k=l

if

s =

s =

tlm. then

Gs 2 Gt.

=

U

u m

every

F

is

%

is a Souslin scheme.

ns Ik.~

(s1,---,sm).is a decreasing Souslin scheme. that is,

S

so

Suppose that

Gs, given by

G

where

under finite unions and

%

set

S E Sous($)

decreasing Souslin scheme.

~

We have the same kernel,

G

elm

=

may

u n F

~

~

~

.

from

%

u m

be

derived

by

a

127

2.2: Borel & Loeb Sets

Section

(2.2.2) DEFINITION:

A family

0

of

subsets

X

if

9

is n o n e m p t y a n d c l o s e d u n d e r f i n i t e

pauing

of

unions and

of

A pauing

intersections.

X

a set

0

is

is 0

semicompact if euery countable subset o f

caLLed a

said

to

be

w h i c h h a s the

finite intersection property has nonempty intersection.

There are two basic pavings in this book:

the family of

compact subsets and the family of internal sets. denotes 0

the

= Kpt(lRd)

family

of

compact

all

is a semicompact paving.

subsets

of

If

Kpt(lRd)

Eld,

then

The sets derived from the

compact sets by the Souslin operation are called the a n a l y t i c sets,

Sous(Kpt(Rd))

= Anal(El d ) .

Analytic sets may also be characterized as the continuous images

of Borel

sets or

continuous

images of

the

irrationals, see

Dellacherie 8 Meyer [I9781 or Kuratowski [l966]. Let subsets

V

*a(V)

be an internal set. = 0

The family of all internal

is a semicompact paving by the saturation

property of section ( 0 . 4 ) .

We refer to the sets derived from

the internal sets by the Souslin operation as Henson sets,

Sous (*D( I) ) = Hens (V) .

Each of these two pavings has the property that complements

of

%-sets are

Sous(%)-sets

(open sets are countable unions of

ChaDter

128

compacts). by

Therefore, they contain the sigma algebras generated

3,

d

Anal(R

) 2 Borel(R

Hens(V)

If of

Measures & Standard Parts

2:

P

d

)

2 Loeb(V).

flu(%) denote the closure

is a family of sets, let

under countable union and countable intersection.

%

The

next result is an abstract form of a classical theorem.

(2.2.3) LUSIN'S SEPARATION THEOREM: Suppose

A,B E Sous(%)

3

is a semicompact paving

are disjoint.

C.D E nu(%) such that

A

C

of

X.

If

then there exist disjoint

D

and

I,

B.

PROOF : See Dellacherie and Meyer [1978. 111.141.

Two immediate applications are as follows. If A C IRd and Rd\A are both analytic, then A E Borel(R d ) . This is the classical Lusin result. sets, then

H E Loeb(V).

If

H C V

and

V\H

are both Henson

We shall see other applications in

later chapters. Now we begin the specific study of Loeb sets on an d d internal set Td E *Rd, with st(H ) = R .

S-dense

(2.2.4) LEMMA:

If

A1 2 A2 2

internal subsets o f

- - a

Ud,

is

a

then

decreasing

sequence

of

st(n Am) = fl st(A,). m m

PROOF : It is sufficient to show that

n st(A,) m

E st(n Am). m

since

Section

the other

129

Bore1 & Loeb Sets

2.2:

inclusion

is

trivial.

Let

r E

n st(A,)

be an

m

For each standard

arbitrary point. such that to

: n

E

*I :

{n E

E

IN}

to

be

an

*lN)[m

E

n 3 (tm

E

an

internal

Am

sequence

infinite

n.

This

< --)I} 1

Am 8 Itm-.]

is internal and contains all finite indices contains

tm E

The set

*IN}.

(Vm

there is a

Use the comprehension principle (0.4.3)

: m

{(tm.Am)

extend

{(tn,An)

= r.

st(t,)

m.

n E

satisfies

hence i t

tn E

n Am

and

m = r

st(tn)

(2.2.5) PROPOSITION: a) A E Anal(lRd) b) H

E

3 st-l(A)

Hens(Td) 3 st(H)

E

Hens(T d ) ,

E Anal(lRd).

PROOF : If

K

is a compact ,kt.

then

st-l(K)

is a countable

intersection of the internal sets

Im =

Therefore, i f

Ks,

A

{t E

Td

I dist(t.

*K) <

--}. 1

I s the kernel of a compact Souslin scheme,

then

is a Henson set because

part a.

Sous(Hens(H d ) ) = Hens(Y d ) .

This proves

Chapter

130

Measures & Standard Parts

2:

H

As noted above, every Henson set

may be derived from a

I u l m 1 Ial(m+l).

decreasing Souslin scheme,

T h e n Lemma (2.2.4)

shows that

st(^) = n

u

1.

st(x

u m

ulm

) a r e closed. Im is analytic.

Proposition (2.1.1) proves that the sets Since

Sous(Anal(IR

d

) ) = Anal(lR

d

),

st(1

st(H)

(2.2.6) THEOREM:

Td

Let

B E IRd

set

be a n ts

Q

S-dense

IRd

Borel. subset o f

st

inuerse standard part

-1

'*IRd .

internaL subset o f

(B)

A

i f and only i f its

is a Loeb subset o f

Td .

PROOF : B C IRd

Let are

analytic.

st-l(IRd\B) sets.

B

b e a Borel s e t , so that both By

Proposition

n Od

= [Td\st-'(B)]

(2.2.5), the

and

sets

Hd\st-'(B)

st

-1

(B).

are Henson

(2.2.3) shows that

The Separation Theorem

IRd\B

and

st-l(B)

is

Loeb. Conversely, i f st

-1

(B)

and

B E IRd

d

T \st-'(B)

and

st-l(B)

E Loeb(Td),

then both

a r e Henson sets and so, by Proposition

(2.2.5).

B = st(st-l(B))

analytic.

Again, the Separation Theorem (2.2.3)

IR d \B = st[Ud\st-'(B)]

and

are

shows that

B

not

a

i s Borel.

The

reader

should

note

that

we

have

correspondence between Loeb sets and Borel sets. inverse standard parts

of

infinitesimal relation

t

sets.

standard

=: s.

sets a r e

given

S e t s which are

closed under

the

so they a r e not arbitrary Loeb

131

2.2: Borel & Loeb Sets

Section

In the following discussion i t is convenient to introduce If

some modern set-theoretical notation.

V

is an internal

set, let 0 rr0(V)

If

X

0

=

= rnO(V)

*Nor).

is a topological space, let

lTy(X) = {C

: C

is closed in X)

Zy(X) = {U i U i s open and a countable u n t o n

of

closed sets}.

In both cases, continue inductively with countable operations:

These sets generate the Loeb and Borel sigma algebras,

Loeb(T

d

) =

U

O d na(T ) =

where

o1

Z,(OH d )

U a<wl

a
is the first uncountable ordinal.

f o r some of these families are

O d) = n2(R

Gs

and

Two older names

O

Z2(R

d ) = Fo.

We have already seen that standard parts of Henson sets are analytic.

Theorem

(2.2.8)

shows that every analytic set is

Chapter

132

actually

2:

Measures & Standard Parts

the standard part of an internal

levels up the sigma algebra hierarchy.

a6-set.

just two

We need two technical

results to prove that theorem.

(2.2.7)LEMMA:

A

If

S1 = st(A)

nonempty closed subset o f infinite f o r

x E S1,

each

A1.A2

internal

Ud,

i s a n internal subset o f

x

and for each

contained in E

S

j'

S2

and

A

with

is a

fl st-'(x)

then

there

disjoint

A

st(A.) = S J j' i s infinite (j=l,2).

exist

such that

A. fl st-l(x) J

PROOF : Let

Si.Sa

respectively. in

A

st(D.) = S' and for each x J i. countable number of elements in

a

(j=1,2). Notice that

di

<

we have the internal condition

m

m)[dist(Dz,*S2)

A2 = {dk1

there are -1 D . fl st (x) J

i.

D.

< ; 1 &

hence this holds for some infinite and

S'

D1.D2 are countable.

these sets

For each finite

(Vk

E

contained

and extend the = {di ! m E IN} J to internal sequences with values in A (j=1,2).

Enumerate sequences

D1.D2

Choose (external) disjoint sets

such that

exactly

S1.S2.

be denumerable dense subsets of

:

k

<

1 dk 2 C {dl.---.dm}]. 1

n.

Let

n}.

This lemma has the following extension.

A1

1

= {dk

:

k

<

n}

Section

133

Bore1 & Loeb Sets

2.2:

(2.2.8) LEMMA:

If

B

{Cm : m

= C,

st(B)

C,

x E

of

E IN}

is

an

is

a

B

where

sequence

n

st-'(x)

= Cm

Bm

and

B

n

closed

of

is

subsets

of

euery

{Bm I m

s u c h t h a t f o r all is

and

infinite f o r

sequence

st-'(x)

Hd

of

subset

t h e n t h e r e is a d i s j o i n t

internaL subsets o f

st(Bm)

internal

E

tN}

in.

i n f i n i t e f o r euery

x E cm. PROOF : We apply the previous lemma inductively.

S2 = Cl.

B1 = A2.

A1

Choose

A1

Then

lemma again to

S1 = st(A1)

= C2

from

= C

A2

and

as in Lemma (2.2.7) and let

B1

is disjoint from

into two disjoint sets st(B2)

A2

and

S1 = C

Let

C2 = S2,

and

B2

and

and we may apply the

and each infinite-to-one.

splitting

with

st(AZ)

Also,

B2

= C

A1 and

is disjoint

B1'

For have

the

been

st(Am) = C.

induction assume

chosen st(B

i.

along ) = Cj

that

with

a

m

into two disjoint internal sets

Since

Am

= Cm+l.

disjoint

sets Am

B1.***.Bm such

and each is infinite-to-one.

Lemma (2.2.7) to the set ' A = A

st(Bm+l)

disjoint

s ~ ( A ~ + ~= )C.

is disjoint from

and

that Apply

S2 = Cm+l. splitting i t

Bm+l

and

Am+l

such that

and each is infinite-to-one.

B1.*--.Bm.

1

is disjoint from

them.

Nor we are ready to characterize analytic sets.

Chapter

134

Measures & Standard Parts

2:

(2.2.9) THEOREM:

Td

Let

be

S-dense

A E Anal(IRd)

Euery analytic

O d B E n2(T ),

an

internal

subset

*Rd.

of

some

is the standard part o f

A = st(B).

PROOF :

A

Let

be given as the kernel of a decreasing Souslin

scheme of closed sets,

A = u n F

~

~

~

.

a m

We will

apply

Lemma

(2.2.8) to

Bs,

scheme of internal sets a)

{Bm : m E

b)

st(Bs)

c)

Bs 2 Bsk.

d)

=

s E

define a decreasing Seq.

Souslin

satisfying

I}is a disjoint for sequences of length 1.

Fs, for all f o r all

Bsh fl Bsk = 0.

s E

s E

Seq.

k E IN.

Seq.

for all

s E

Seq.

h.k E IN

with

h # k.

To start, apply Lemma (2.2.8) with This gives us the internal sets Once a and

Bs

Bs

C = IRd

and

for sequences of length 1.

has been chosen, apply the lemma again with

Cm = Fsm. Since our internal scheme is decreasing, we have

by (2.2.4). Thus we need to show that

u n B~~~ = n o m

Cm = Fm.

u :A

m n

B = B

Section

2.2:

for internal sets

.:A

Conditions (a),

imply that whenever

Bs

length, then

135

Borel & Loeb Sets

and

s

are sequences of

t

Bt are disjoint.

and

(d)

(c) and

Bs

on

the same

This implies that

and completes the proof. In light of this result one might ask how far through the 0 0 family of sets Za and ITa one must go in order to generate the Loeb algebra.

In the case of the real numbers i t is known

that one must take the union all the way to generate

the Borel

algebra.

Kunen

has

w1

shown

in order to the following

result.

(2.2.10) THEOREH:

T

Let

*I

=

* [O,l].

be

For

an

S-dense

A C I and

internaL

subset

of

a 2 1

a)

A

€

TIz(I)

i f and only i f

st-l(A)

E TIz(U),

b)

A

€

B:(I)

i f and onLy i f

st-'(A)

E

and

Z:(U).

The following is a sketch of the proof kindly sent to us by

K.

K.

Kunen.

Kunen

and

A.

Miller

plan

to

publish

the

topological fact (2.2.11) in a paper tentatively titled, "Borel and projective sets from the point of view of compact sets." The difficult part of the theorem reduces to showing that if

st-l(A)

E

B:(P).

then

A

E

B:(I).

First. we may consider

2:

Chapter

136

St

-1

(A)

Measures & Standard Parts

= q(Ko.K1.K2:**)

0

Za-combination of a sequence of internal sets

to be a

Next, we topologize the problem by

X

letting

be the Stone

*l 3 ( T ) .

space of the Boolean algebra of internal sets, gives

us

a

-

commutative

i : T

inclusion

X

diagram

consisting

and a lifting

f

of

{K,}.

of

the

This natural

st:

We take

Nm

where

is

the clopen

f-'(A)

Saturation implies that there is a

E

Y

A.

Now

X

in = C,

since if

st(t)

by

f-l(A)

Km.

u E f-l(A)\C.

= f(u)

The contradiction shows that

The following

determined

and t E Km -1 t e q(KO,K1.*-.) = st (A), so

such that

u E Nm.

only if st(t)

t E

set

i f and f(u)

= C.

theorem of Miller and Kunen completes

proof.

(2.2.11) THEOREM:

-

Let f : X

X Y

A E B:(Y)

and

Y

be

compact H a u s d o r f f s p a c e s , let

be continuous and onto and let i f and only i f

f-l(A)

€

=

A

_C

Iz(X).

Y.

Then

the

137

2.2: Borel & Loeb S e t s

Section

(2.2.12) REMARK: Let

a set

2

be a sigma algebra o n a set

U C X

is universally

is a finite measure o n p-completion of

2.

2,

Z

then

X.

We say that

measurable i f whenever

U

p

is in the

In other words, there is a n

S

€

2

such that the symmetric difference has zero outer measure. F[S

v U]

= 0.

A n argument based o n the Souslin operation

c a n be used to show that sets in 2

measurable.

Sous(2)

a r e universally

T h i s a l s o follows from C h o q u e t ’ s theorem,

see Dellacherie & Meyer [1978. 111.33(a)].

In particular,

analytic sets a r e universally Borel measurable and Henson sets a r e universally Loeb measurable.

138

(2.3)

Borel Heasures Suppose that

and

Ud

is an

6a : Ud + * [ O , m )

* finite

S-dense

subset of

is a limited internal weight function.

B E IRd,

Since Theorem (2.2.6) says that for every Borel set st-l(B)

is a Loeb set, the hyperfinite extension

Borel measure

= a[st-'(B)].

The mapping from internal measures

For example, i f

to Borel measure

a

is

Pa

is the indicator function of

6p

is the indicator of a different

then both

pa

and

s 10,

point

0,

because

6a

while

one point t

defines a

a

by

Pa

pa[B]

not unique.

*IRd

st-'{O}

contains both

s

are unit mass at zero

pp

and

t.

The purpose of this

section is t o show how to start with a Borel measure

* finite

find an internal

measure

a

such that

P = a

0

~.r

and

st-1

.

(2.3.1) LEMMA: Let Let there

Ud is

LnternaL wheneuer

P

p

be a an

IRd.

be a finite posittue BoreL measure o n

*f i n i t e

S-dense

infinite natural

weight

function

k l . - - * . k dE

i=l

*Z

subset

number

6a : Ud P make

-P

*Rd.

of

h

E

*[O.m)

*IN

and such

k,***,k] i=1

Then an that

Limited, then

Section

2.3:

Bore1 Measures

139

PROOF : Consider the internal property of a natural number

kil

<

m:

(m!)2

k

1

2 1 m!

is a singleton .

This simply says that there is a finite sample from one of each of a finite number of boxes. hold at an infinite define

h.

Since

@

is internal i t must also

Denote the corresponding set by

S

and

by

6a P

d 6a (t) = P

{0

ki-1 ki

d

[r. m]]. i f ,

t E

k -1

s n i=1 n2[+-.

k.

~;i]

otherwise.

This internally defined function satisfies the lemma because

where

is the single point in the box.

s

The reason that we chose

h!

a factorial in the lemma is

because now all standard rationals and

this gives

a P

p

E uQ

may be expressed as

the right measure on all standard

rational-edge rectangles.

(2.3.2) LEMMA: For

edges,

every

-1 st (R)

open bounded

rectangle

i s Loeb and

a (st-'(R))

u

R

with rational

= p(R).

Chapter

140

2:

Measures & Standard Parts

PROOF : The p.q

first

a d

Q

E

.

part

is

in

Theorem

(2.2.6).

If

let

1Cp.q) = { x E alRd

(we will

included

also

use

obvious meaning).

I

<

pi 5 x i

the notations

qi, i = l.---.d)

I[p,q].I(p.q).

with

the

* finite

I[p,q)

can be decomposed in a d ki-1 k. TI (recall that all disjoint union of rectangles i=l

s]

[r,

rational numbers are of the form

b): hence

On the other hand. i t is clear that

therefore

Now, let edges,

hence

R

R

= I(p,q).

be an open bounded rectangle with rational Then,

Section

2.3:

Borel Measures

141

(2.3.3) THEOREM: If

Ud

is a n

is a B o r e L

measure

Rd

on

= av[st-l(B)]

p[B]

*f i n i t e

S-dense

and

a

subset o f

* IRd ,

is a s above,

P

f o r every Borel set

p

then

B.

PROOF : The family

L = {B

is a

E

Borel(IRd)

: p[B]

sigma algebra containing all

= a[st-'(B)]}

rational

edge

rectangles.

L = Borel(IR d ) .

Hence,

(2.3.4) THEOREM: Let

a

Ud

be a n

*f i n i t e

measure

p[B]

= a[st-'(B)]

that

a)

st-1 (M) b) only t f

A set is

S-dense Let

a.

for

M C Rd

is

a-measurable with

A functton f(st(t))

is

f

:

*f i n i t e

*IRd

with

be a BoreL measure s u c h

p

B

subset o f

E

Borel(IR d ) .

v-measurable

a[st-'(M)]

IRd + IR

is

a-measurable.

i f and only if =

v[M].

v-measurabLe i f and

2:

ChaDter

142

Measures & Standard Parts

PROOF :

M

If

is

p-measurable.

then there a r e Borel sets

M v B y 5 B2

such that the symmetric difference satisfies

= 0.

p[B2]

T h e n we have

a-measurable with measure

= 0,

a[st-l(B2)]

B1.B2

st

so

-1

and

[MI

is

p[B2].

T h e remainder of the proof c a n be finished in the same way

If

a s the end of Theorem (2.1.2). then f o r each standard with

a[I]

contained

<

>

a[st-'(M)]-~.

in

M,

a[st-'(M)],

equals

(M)]

a-measurable,

I 5 st-'(M)

there is a n internal The

set

st(1)

is

closed

~ C I I< a~st-'(st(1))1

the

inner

a[st-'(IRd\M)]

p-measurable

a[ s t-l

0

is

p-measure

A similar argument shows that the

IR d \M

to b e

> so

forcing

a[st-'(M)]. of

E

st-'(M)

= v~st(111

M

of

the

p-outer

to

be

p-inner measure

and finiteness of

because

and

p

measure

forces

M

M

is

of

.

T h e reader should notice that i t is easier to show that a completed Borel algebra lifts via

st-'

by a n argument like Theorem (2.1.2) result (2.2.6).

to

a-measurable sets

than i t i s to g i v e Henson's

W e could have done section (2.3) without (2.2)

i f we used only complete measures. The

function

exercise.

version

of

this

theorem

is

left

as

an

T h e following is a n integrated version.

(2.3.5) A CHANGE OF VARIABLES THEOREM: Let

p = a

0

p - i n t e g r a b l e , then

st-' JBf(')dV(')

a s above.

If

f

: R

d

R

is

= Jst-l(B) f(st(t))da(t).

PROOF :

If

f

is a n indicator function of a

p-measurable

set,

Section

2.3:

143

Borel Measures

then the result follows from (2.3.4)(a).

Otherwise break

f+,

into positive and negative parts

f-.

sequence of simple functions such that

f-).

(resp.

gn

Let

gn

f

be a

increases to

f+

We know that

and

Since

a[{t

:

-+ f(r)}].

gn(st(t))

= a[st-l{r

+ f(st(t)))l

:

gn(r)

we may apply the Monotone Convergence Theorem to

complete the proof.

When

f(r)

is a Borel or

the function

f(st(t))

Ud,

need not be

but

it

replacement

g

for

p-measurable function on

is a Loeb or

f.

IRd,

a-measurable function on

If we seek an internal

internal.

we want to make the diagram

udnud A *Rno

I

St

f

IRd

a-almost

commute.

See

(2.1.3)

a-almost

S-continuous lifting.

B I R

Ist

for

the

definition

of

an

144

2:

ChaDter

Measures & Standard Parts

(2.3.6) THEOREM: Ud

Let

* finite

a

be an measure

A function

a)

Let -1

a.

v

measure such that

* finite

S-dense

= a

f

0

st

. is

only if there is an internal function a[{t

E

ud :

st(g(t))

f

A function

b)

f : IR

only if there is an

r

for euery

IR

is

a-S-integrable = 0.

# f(st(t))]

a[st(g(t))

+

p-measurable i f and g : Ud + *IR

with

= 0.

f(st(t))}] d

with

be a completed Borel

p

IRd + IR

:

*IRd

subset o f

p-integrable if and g : Ild + *IR

with

In this case

r

N

I.

p-measurable set

PROOF : Left as an exercise with lifting to the function

the hint

to apply a chapter 1

f(st(t)).

(2.3.7) THEOREM: Let

~.r=

a

a-S-integrable measurable

r

for all Borel

a[A

v st-l(B)]

d

*R

r =

a s aboue.

If

f

:

Ud

+

*

IR

is an

internal function, then there is a Borel

g : IR

JBg+

st-l

0

such that

N

fda =: E[f(t)a(t)

: t E

A],

s t-'(B)

sets = 0.

B

and

internal

A

such

that

2.3: Borel Measures

Section

145

PROOF : N

u(B)

Let measure.

=

f(t)da(t) define a standard Borel ‘s t-l (B) u << p we know that there is a p-a.s. unique

Since

Borel measurable Radon-Nikodym

s

g =

derivative

d’ dw

such that

gdp = u(B).

(2.3.8) EXAMPLE:

Rapidly oscillating internal functions cannot be liftings of Borel functions. n E

*IN

(k-6t t E U.

and

: 0< k

let

<

For example, let

U

be

n , k E *IN}

The function

f

:

the

6t =

uniform

with weights

II

-+

0 , 1 ,

(0,l)

if if

1 n

t E t

for some infinite hyperfinite

6a(t)

= 6t

Hence.

JB

-1 dr

=

:

f.

n * [a.b)]

Jst-l(B) f(t)da(t).

some sense, the constant be to

t € T

p1

for all

given by

is odd is even

has the property that for any interval of finite length,

8[f(t)6t

space

[a.b),

1 =: p(b-a)

for all Borel sets

B.

In

is as close as a Borel function can

146

(2.4)

Weak Standard Parts of Measures This section is related to probabilistic "convergence in

distribution" be low.

( 2 . 4 . 1 ) NOTATION:

BC(IRd)

Let continuous

denote

real-ualued ilq-+ii

u n i f o r m norm

the

Banach

U

+

cp(x)

i f and only i f

A

function

:

Z

+(x)

IRd.

*BC(IRd)

The

:

is

A

x E *Rd.

for all

* + IR

bounded

d x E IR ] induces * [BC(IR d )] giuen by

I

the u n i f o r m infinitesimal relation o n cp Z

of

functions d e f i n e d o n

= sup[ lq(x)-+(x)

U

space

(uniform-norm)

U

S-continuous i f

Ud' C *Kid

Now let

implies

cp Z J,

A(q)

* finite

be a

set and let

be an internal signed weight function. a

as acting on

*BC(IRd)

A(+).

Z

6a : Ud + *R

We may view the measure

by letting

a(cp) = 2[6a(t)cp(t)

: t E

Ud].

( 2 . 4 . 2 ) PROPOSITION:

a

For

Ud

and

t E Ud]

is finite.

that i s ,

a(

is

)

as

Then

aboue. U cp Z

+

S-continuous o n

Z[ 16a(t)

suppose

a(cp)

implies

Z

I

:

a(+).

*BC(IRd).

PROOF : Since

q(x) 1 +(x)

infinitesimal

L

>

for

Icp(x)-+(x)l

all

x

f o r all

in

x

*IRd, in

*IRd.

there is an Then

Section

147

2.4: Weak Standard Parts

The

S-continuity of

means that we can associate

a

associate saw

in

a

with a countably additive measure

the

measure

last

can be

section how

represented

every

by

We can also

BC(Rd).

with a n element of the continuous dual of

a

Td. We

on

countably additive

such

and

a ' s

a

this

Bore1

section

explores one aspect of the opposite direction.

(2.4.3) EXAMPLE: Let

6 a : Td + [O.l]

let

Consider

p

:

the

BC(IRd)

+

standard

9.

IR

be

then

Jqdp

p(lim

the indicator

externally

defined

given by

p(q)

The norm of

p

c o u n t a b l y additiue m e a s u r e o n p(9) =

Hd

be an unlimited or infinite element of

to

Rd

= st(a(

function of

standard

*9 ) ) =

is clearly

{to}.

functional

* st( p(tO)) 1.

that r e p r e s e n t s

and

for

There

p.

is n o

since if

and'we pick the monotone decreasing sequence

9 ) = lim p(p )

by the monotone convergence theorem n n *vn(tO) = 1 for finite This is not possible because

for

p.

n.

while

lim 9,

E 0.

The "problem" with the

a

example Is that i t is carried on the unlimited points.

in this

148

Chapter

2:

Measures 81 Standard Parts

M(Wd)

Let

denote the space of finitely or countably d additive measures on IR . Every continuous linear functional on BC(IRd)

can be represented by a finitely additiue measure and [1972]

Loeb

* finite

shows how

sets.

internal

of

those on certain

We will content ourselves with studying positive

a's

results to

to represent all

on our set

+ a -a7)

Ud

(we can always apply

these

since our modest aim is construction of some

interesting processes, not representation of all possible ones.

(2.4.4) DEFINITION:

Ud

Let

be

6a : U d + * [ O . m ) say

We

a

a[A]

has

near-standard

A

It follows that if

and

let

carrier prouided

that

for

a

i s an

S-tite measure i f

a[Ud]

has near-standard carrier i f and only

a

This is a'more compact way of saying i t , but

we can do even better: Then by (1.2.25).

*IRd

has near-standard carrier.

a

a(Ud\O d ) = 0.

of

containing only unlimited points o f

S 0. We say

i s finite and

subset

be a positiue internal weight function.

each internal set

Ud,

* finite

a

a

Od

is Loeb. s o

Ud\Od

is Loeb as well.

has near-standard carrier i f and only i f

a(U d\O d ) = 0. The generalization of spaces can be

found

this notion

to arbitrary Tychonoff

in Anderson and Rashid

[1978]

and Loeb

[1979a] as well as in the special extension treated in chapter 5 below for the path spaces

C[O,l]

and

DCO.11.

In non-locally

compact spaces a distinction must be made between "limited" and "near-standard'' or "unlimited" and "remote," and i t is worse

than

that

enters.

because

the

"Baire"

vs.

"Borel"

distinction

also

(Loeb [1979a] has a nice universal measurability result

in Borel sets.) from

149

2.4: Weak Standard Parts

Section

the

These technicalities distract the uninitiated

central

infinitesimal

analysis

substantially harder for the initiated).

(and

aren't

Moreover, this form is

useful to us in the description of the law of a process. The next result says that

S-tite measures are weakly near See Stroyan 8r Luxemburg

countably additive standard measures.

lo]

[1976. chapts. 8 &

for the infinitesimal functional analysis

jargon; we will show that the infinitesimal relation holds.

PROPOSITION:

(2.4.5)

If

ud c

*Rd,

pa = a

0

S-tite

measure

the

countabLy

is

the

(weak-star)

part

of

that

a.

*finite

on a

then

st -1

standard cp €

is a n

a

additive

measure

a(M(Wd) ,BC(Wd))for

is,

set

each

standard

BC(Rd) c(t)Wt)

%

J r(x)dlJa(x).

PROOF : First of all, what does near-standard carrier have to do with it?

Clearly,

a

0

st-'

is always a Borel measure-but

notice that i t is zero for example (2.4.3). whereas does not "see it" as zero.

1

uBC(!Rd)

N

We always have of Chapter 1 , since Since .[It1

>

such

that

a(Ud)

n] zz 0 . n

Xp( t)a( t)

and

9

is

a

%

q( t)da( t)

are finite.

limited

and

for standard positive

>

implies

by general results

a[ltl

for B

<

every

infinite

n.

there is a finite n]

>

a(T d ) - B .

m

Thus,

2:

Chapter

150

n udl

a[od

Measures & Standard Parts

=: a[adl.

= a[st-'(~~)l

so

N

= Jod cp(t)da(t)A

standard

bounded

continuous

function

is

S-continuous at each (near-standard or) limited N

finite

and

this means

t;

N

= cp(st(t)).

cp(t)

or

neighborhood of

t.

is constant

cp

on

the

infinitesimal

Combining this with the last remark. we see

N

that =

= cp(st(-)).

q(*)

J cp(st(t))da(t). =:

(2.4.6)

cp(x)dpa(x).

of

Variables

1

=

cp(st(t))da(t)

Now we know

a].

The Change

shows that Zcp(t)a(t)

[a.e.

q(t)da(t)

(2.3.5)

Theorem

cp(x)dpa(x),

(Notice that

r

we

so

st-'(Borel)

see

that

is Loeb.)

PROPOSITION:

Td C *IRd

Let

a sequence

be

* finite

o f positive

{ak : k

and let

internal

€

alN}

be

T.

functions on

The

following are equivalent: ak

a) additive cp E

converges weakly

finite Bore1

to a

measure

p,

standard that

is,

countably f o r each

BC(Rd). S-lim Zcp( t)ak( t) = Jcp(x)dp(x).

b)

For each internal extension

Td.

sequence o f functions on

such that f o r all infintte am (st-'(

1)

= a (st-l(

1)

{ak : k

€

*a}

to a

there exists a n infinite

m

<

n.

an

is

S-ttte

n

and

= p.

PROOF : (b) definition

implies

(a)

principle,

by

the

since

elements are within epsilon.

last

all

result

and

sufficiently

the

internal

small

infinite

2 . 4 : Weak Standard Parts

Section

151

Notice that we do not assume that the finite

k:

the linear functionals

be finitely additive when The

converse

suggestions: compact

approximated S-titeness.

The by

are

= ZLp(t)ak(t)

S-tite for may only

is finite.

is

is regular,

p

set.

part

k

ak(Lp)

ak

left so

compact continuous

as

an

exercise

with

these

i t is approximately carried on a

set's

indicator

functions.

function can be This

leads

to

The continuous functions can distinguish different

Bore1 measures.

152

CHAPTER 3: PRODUCTS OF HYPERFINITE MEASURES

Let and p[U]

U

6u

:

V

and

V

* finite

be

be

+ *[O?)

= Z6p(u)

internal

= 26u(v)

u[V]

and

the procedure of chapter 1 to complete

product

measures

p

and

Alternately,

we

measure u.

may

and

p

construct

-+ * [ 0 , m )

functions

with

We may apply

0.

and then form the

u

with

the

U x V = {(u.v)

on

first

weight

limited in

associated

p x u

6p : U

sets and let

the

:

hyperfinite

u E U. v E V}.

internal

product,

letting = bp(u)6u(v).

6n((u.v))

then form the

* finite

extension,

This is a measure on

= p[A]u[B]

T.

for internal

property of p x u.

measure

A c U.

* summation.

-u = L6n

and the hyperfinite

U x V

B c V.

The measure

and we know

n[AxB]

This is just a simple T

is an extension of

but i t is a proper extension (in general) and yet still

has a Fubini-type theorem. last statement.

This chapter simply explains

the

We only work with bounded hyperfinite measures.

(3.1) Anderson’s Extension

We begin with a brief description of the complete product of two bounded hyperfinite measure spaces (V.Meas(u),u).

The

reader

can

find

the

construction in Royden [l968], for example.

(U,Meas(p),p) details

of

and this

Section

3.1:

153

Anderson's Extension

(3.1.1) DEFINITION: The bounded

complete

p x u

product

hyperftnite

measures

U

on the

is

V

x

unique

two

of

countably

addtttue extenston (Caratheodory extension) of the function

p(A)u(B)

= p x u(AxB)

A x B.

rectangles Meas(u).

We

shall

A

for

defined for measurable

tn

denote

Meas(p)

the

complete

contatntng the measurable rectangles by

%(p

and sigma

B

in

algebra

x u).

To apply Caratheodory extension one must show that i s countably additive

p

X

u

in the case where a countable union of Anderson [1976] observed:

disjoint rectangles is a rectangle.

(3.1.2) PROPOSITION: The

complete

Meas(=).

where

product

measurable

r = p x u

sets,

%(p

X

u)

t s

the internal product.

of

the internal product.

Moreover, the restrtctton.

the hyperftntte r = p x u .

extenston

T

t s an extenston of the complete product of the

separate hyperftntte extensions

p

and

u.

PROOF :

A

Take

in

Meas(p)

and

B

(1.2.13) that there are internal sets that

p[A

v

C]

= u[B v D] = 0.

in

C

#eas(u). C_

U

and

For internal sets,

We know by

D C V.

such

154

3:

Chapter

r[C

Products of Hvperfinite Measures

D] = ~[C]*U[D],

x

so that

T[C

Moreover, we k n o w

is a complete measure and

T

p x

u[(A

(A x B)

because

[(A U C)

x

v

(B v D)]

x

= p[C]u[D].

D] = p[A]u[B]

x

(C

B) V (C

x

while

<

u[F]

1.

v

C)

x

(B

U

D)] U

and both measures of a rectangle with one

E,F

sets

D)] = 0.

[(A

g

D)

(For each

z e r o measure component a r e zero. internal

X

so

such

that

B[E

F]

x

<

B

>

E 2 N. F 2 M B

0, and

there are

p[E]

<

a**).

(3.1.3) DEFINITION: Let

C'

C E U

V.

x

V

= {v E

T h e sections o f

C

are:

C},

f o r each

u E U

(u.v) E C}.

f o r each

v E V

: (u.v) E

and

Cv = {u E U

Let sections o f

fU : V

:

f : U x V -+ [ - - O D , @ ] f

are:

4

[-"."I

be

giuen b y

f'(v)

f o r each

a

functton.

= f(u,v.). u E IJ

and

fV : U +

[-m,@]

giuen b y

fv(u) = f(u.v).

f o r each

v E V.

The

E

3.1:

Section

155

Anderson's Extension

Our next result is a prelude to a Fubini theorem.

(3.1.4) PROPOSITION: Let

Loeb(U).

algebras of

Cu

C

Loeb(V)

E

E

Loeb(U

and

Cv

If

b)

f

V).

x

fv

p, v

Let

and

If

c)

is

V

x

fU

Loeb(U

x

V)-

Loeb(V)-measurable

ts

be a s above.

B

f

is

Loeb(U)-measurable.

then f o r almost all

-

v

a n d for a l m o s t all

If

d)

sets a s above.

[-m.m]

-B

C E Meas(r).

Cu E Meas(u)

d e n o t e the L o e b

then each section satisfies

measurable, then each sectton and each

V)

x

Loeb(U).

E

U

:

Loeb(U

*f i n i t e

the respective

If

a)

Loeb(V).

:

U

x V

f o r a l m o s t all

tn

u

ts

[-m,m]

fU

U,

V.

in

u

in

U.

Cv E Meas(p).

r-measurable. then t s

u-measurable

and

f o r a l m o s t all

v

in

V.

fV

is

p-measurable.

PROOF : Part a) is proved by observing that the collection of all

S

sets

sy

E

U x V

Loeb(U)

such that for all

u.v

Su E Loeb(V)

both

and

hold, is a sigma algebra containing the internal

sets. Part b) follows easily from part a). Part

c)

can

be

internal and satisfy

shown by

using

r[C v D] = 0.

(1.2.13).

Let

D

be

W e k n o w that the sections

of a n internal set a r e internal, so i t would b e sufficient to prove that

u[C'

v Du] = 0

for almost all

prove this for a n arbitrary null set.

u

in

U.

We will

156

ChaDter

N

satisfy

sequence of

internal

Let

finite

m.

3:

Products of HvDerfinite Measures

r[N] = 0

Wm

sets

Wm

and let

>- N

be a decreasing 2 r[Wm] < l/m , for

with

By summation we see that

because

N E Wm,

Since

the outer measure m.

T h i s means

-~ {

{u : u[NU]

that

u:

>

l/m} C {u

-u[NU]

>

l/m}

:

u[Wi]

>

<

l/m

for every

the outer measure

c{u

:

l/m},

-u[NU] >

so that

finite

0) = 0.

Hence almost all sections of a set of measure zero themselves have measure z e r o , so this concludes the proof of part c). Part d) follows easily from part c).

= {v

:

f(u.v)

<

r} = {v

:

fu(v)

<

r}.

{(u.v)

:

f(u.v)

<

r}U

157

(3.2) Hoover's Strict Inclusion One of the most interesting hyperfinite probabilities the uniform infinite shows

type

*finite

that

the

#

= 1/ [U].

6p(u) set.

V

u

U.

in

an

The following example of D. N. Hoover subsets

internal

measurable with respect to and

for all

is

p x

or in

1).

are uniform probabilities.

U

of

x

V

are

not

all

when

%(p x u ) .

U

This description of Hoover's

example comes from notes of Keisler and uses the following basic facts

of

probability.

(Independence

arguments

give

a

more

intuitive. but less direct proof.)

(3.2.2) CHEBYSHEV'S INEQUALITY:

(R,P)

Let

f

:

R

+

*R

be

a

be internal.

in particular. i f

m = E[f]

* finite For any

probability

m

in

*R

and

and a

let

>

0.

and t f we denote the uariance

158

3:

Chavter

f

of

V[f]

by

2

E[(f-E[f])

=

Products of Hvverfinite Measures

1.

then

PROOF : g(o) = (f(w)-m)2

Let {w

1 b} =

: g(o)

2 a},

If(w)-ml

:

{GI

b = a

and let

2

.

Since

Markov's

g

+

= g

and

inequality gives

the result.

U = {t E

Let 6t = l/n

*R

for an infinite

set o f internal functions 6p(t) = 6t

= 1/2".

for

all

so both

<

: 0

1. t = k6t. k E *IN},

*

1.

n

in

w :

U + {-l,l}.

V

and

Let

V = R

while

t,

U

<

t

where

R = {-1.1}'

be the

U = U

take

We

and

6u(w)

and

= 6P(o)

have uniform probabilities.

Now

6 r = 6 p . 6 ~ = 6t.6P.

(3.2.3) HOOVER'S HALF: W e c a l l the set

H = {(t.o)

Hoouer's

half

respect

to

H

E Loeb(P

T x R.

of

P x v.

x

R)

and

It but

H

is

= 1)

: U(t)

is

not is

it

-1's

example,

oh(k6t) = (-1)

greatest

let

integer

and

less

1's

'picture' of

lined up over the

h+k- 1 I--+

than

internal.

with hence

r-measurable.

We suggest that the reader draw a sequences of

measurable

or

,

where

equal

to

[xi x.

U

x

R

Y-axis. denotes

as

For the

l < k < n .

Section

1

<

<

h

3.2. Hoover's Strict Inclusion

2".

The signs alternate

159

in the first column, change

every second time in the second column, every third row in the third, and s o on.

=

r[H]

Show that

$.

(3.2.4) LEMMA:

If

S

T

i s an internal subset o f

internal subset o f

R,

r[H

then

n(S x A)]

A

and

i s an

1 =: 5 at[S]P[A].

PROOF : = p[S] =

6t[S]

Note

#

[Slat

We may assume

*

6t[S]

0. We

define =

g(0)

&- (Z[o(s)

m

: s

s])2

E

CSl so

that in case

in

S)

B2(t)

then from

S x {w} E H g(w)

= 6t[S].

[Sl = 6t

:

We

for all

develop g(o)

s E S]+Z[w(s)w(t)

+

Z S#t

s

separate

is:

: s # t

in S])

Z o(s)o(t)

sum is zero because

e =

= 6t

0

Applying Markov's inequality to

s o we let

= 1

0

[Sl The second

g.

The expected value of

6t Z6P(0)(Z[w2(s)

=

w(s)

(Note the relationship between

(0.2.8). (0.3.12) and

simple estimates here.)

E[gl

(that is.

fi and obtain:

it

g.

runs

through all of

we see that

R.

160

3:

ChaDter

<

P[g

Thus

for

P[A']

=

R ' = {a : g(o)

n']

P[A n

2

PCA].

Therefore, for each

'[(S

X

Products of Hvuerfinite Measures

a]2 1 <

-

a} and

A ' = A fl R '

X

Also, f o r

we

have

A'.

in

A'.

in

x {A})

a

f l H16t 3 "[(S

{X}\H]Bt

x

and

r[(S x A ) fl H]

2

r[(S x A)\H]

while

r[(S x A)]

= r[(S x A ) fl €I]

+ r[(S

x

A)\H].

This proves the assertion.

(3.2.5) PROPOSITION: Hoover's

haLf.

H C T

x R,

is

p x u = 6t x P-

not

H 6 %(p x u).

measurable.

PROOF : Since (6t x P)[H]

is an extension of

T

p x u,

if

H

is measurable,

z.

= 1

The last lemma extends easily to measurable rectangles by (1.2.13): for measurable

S'

and

A'

take internal

S

and

A

Section

3.2. Hoover's Strict Inclusion

with

6t[S v S'] = P[A

=: r[(S x A ) fl H] p[S'

measure also

P[H]

1 ~ ( 6 tx P)[S

A']

x

extends

Z

v A']

= 0.

A]

x

%(6t x P).

to

1

= ~ ( 6 tx P)[H]

T[(S'

so

1 5 (6t x P)[S'

Z

n H]

x A')

= T[(S'

161

x A'].

The

on measurable rectangles if

so

x A ' ) fl H]

H

is

measurable,

a contradiction.

= r[H].

(3.2.6) EXERCISE:

Give another proof that Hoover's half

H

is nonmeasurable

in the complete product along the following lines. A

and 6t x P[S

E R x

are

A]

Z 0

internal

sets

A

because

S x A E H .

and

n

{A E

If

R

S C U then

= 1)

and

section

(4.3).

: h(t)

t ES

P[A]

<

TI P[A(t)

= 13.

by

independence, see

tES Either 6t x P[S the

S x

is finite or

A]

Z

0.

#

1/2

,I'[

so

in either case,

Use the internal computation to show that

6t x P-inner measure of

is one.

<

P[A]

H

is zero and its outer measure

162

(3.3) Keisler’s Fubini Theorem

Even

though

strict extension

is a

T

of

general, we have already seen that sections of functions are respectively

or

u-

in

p x u

*-measurable

p-measurable. (3.1.4).

We

can also do iterated integration.

(3.3.1) KEISLER’S FUBINI THEOREM:

where

6 r = 6 p - 6 ~ be

( l J . 6 ~ ) . (v.6~) a n d

Let

and

p

f : U x V +

are

u

[-”.”]

a)

fU

b)

F(u)

c)

JF(u)dp(u)

is

as a b o u e

If

limited hyperfinite measures.

is

r-integrable, then

u-integrable

= [f(u,v)du(v)

p-a.s., is

p-integrable,

= Jf(u.v)dT(u.v).

PROOF : Recall that the last part of the proof of (3.1.4) showed that if f

N

U x V

f

is bounded, then

g(u.v)

%

f(u.v)

for

sections means that For

these

G(u)

is a

We treating

r[N]

has

g ‘

u-lifting of

f+

the

and

f-

= E[g(u,v)6u(v)

F(u)

general

]= 0 p-a.s. g

If

by (1.3.9).

= 0. The fact about null

r[N]

is a

U

r-lifting

(u.v) Q N ,

p-lifting of

obtain

u[N

then

has a bounded

G(u)

U’S,

= 0.

:

a.s.

fU

v E V]

%

p(u).

[fudu.

so

and

case

from

the

bounded

case

by

separately, using linearity and the fact

Section

that

3.3: Keisler’s Fubini Theorem

there

are

bounded

163

fk T f+.

functions

namely ,

fk = min[f+.k]. We assume that

bounded, for each is

f

k

:f

is

Fk

is

fk = min[k,f].

and

u 4 Nk,

so

N = UNk k

if

Since

Nk

p-null set

u-integrable for all

convergence, for

Each

0

there is a

u-integrable for

and

>

has

fk

is

such that

f;:

p-measure zero

u 4 N.

By monotone

u 4 N,

p-integrable and satisfies the theorem, s o

by monotone convergence again. left side tends to and

s

fdr

Jf(u.v)dr(u.v)

Since

f

is

as well, hence

=

F(u)

r-integrable. the is

p-integrable

ss

[ f(u.v)du(v)ldp(u).

The general case follows by linearity since we know i t now f+

for

and

f-.

(3.3.2) EXERCISE:

For 0

<

r

<

p,u,r

1.

a s above, let

Prove that

A E U

x

V

be

r-measurable and

164

is v

3:

ChaDter

u-measurable.

Products of Hvperfinite Measures

(Clearly one can reverse the roles of

u

and

in Fubini's theorem.)

When we

study

stochastic processes below we

[O,l] x R.

product

algebra or

R

fact, -**.wl)

where

[O.l]

carries either

the Lebesgue algebra and

R

will

sequences

and at

consist

of

internal

times we will

((wo."'.wt-at):(wt."'.O1)).

will

have

the Bore1

is hyperfinite.

want

to break

This

results

w = (w

R in

a

In

O'w6t'

into pairs a

three-way

product,

[O,l] x R 1 x R2'

with one classical factor and two hyperfinite factors.

We can

obtain results about three factors f r o m four factors,

A x B x U x V.

where by

U

and

taking

B

V

a r e hyperfinite and equal

to

a

one-point

A

and

B

a r e arbitrary, Thi s

space.

technical

convenience makes the results easier to state.

(3.3.3) A HIXED TYPE FUBINI THEOREH: Let

(U.Meas(p).p)

and

(V.Meas(u).u)

hyperfinite measure spaces a n d Let be the internal products.

Let

U

x

(A.d.a)

V

be bounded and

and

T

=

v

(B.3.P)

u

x

be

arbitrary bounded measure spaces a n d consider the product

A x B x U x V.

3.3:

Section

Keisler's Fubini Theorem

H E A

If

a)

x B x

U x V

measurable, then, except f o r a

concLuston about measurable

set.

p

secttons

If

B x U x V

F(b.v)d(P

f

is

the

same

( d x Meas(v))-

is

H(b,v) a. s .

p

a.s.

x u) =

x

is

x u)-nuLL

f (b,v)(a*u))

are

u).

( a x p ~ r ) -

is

f (b,v)(a*u)

x u(b,v).

v)

(p

for a

(resp.

p x

f)

(or a function

f : A x B x U x V + I R

(a.u)d(a

j.

draw

(d x d x Meas(r))-

a

then, except

integrabLe. then the sections LntegrabLe,

may

f (b,v)(asu)

A x

(a x p)-measurable. C)

We

secttons o f

x r)-measurable,

the

E H}

u.

H E

If

b)

the

function;

measurable 0.s.

(a x

u-nulL s e t , the sections,

( d x Meas(p))-measurabLe.

are

( d x 3 x Meas(r))-

is

E A x U : (a.b.u.v)

= {(a.u)

H(b.v)

165

the

(p

is

are

(a

integral.

F(b.v)

x u)-tntegrable.

Sf

and

a.b.u.v)d(a x x I). Eloreouer, ( (d x ?& x Meas(r))-measurable, then F

tf is

( 5 x Meas(u))-measurabLe.

First w e r e f r e s h the reader's memory about the distinction between

the measurable

sets

of

a product

of

measures a s

43.1.2). which is a l w a y s complete, and the sigma algebra where

d

and

9

are

sigma algebras and

respect to measures isn't mentioned.

in

d x 9,

completeness with

T h e collection of pairwise

disjoint finite u n i o n s of measurable rectangles

A x

B.

with

A

Chapter

166

in

d

B

and

9

in

Products of HvDerfinite Measures

3:

forms an algebra of sets.

d

s i g m a a l g e b r a c o n t a i n i n g t h e s e is d e n o t e d

x

T h e smaLLest

3.

A useful set-

theoretical fact about this situation is the next result, which the reader can find in Hewitt and Stromberg [1965. 21.61.

(3.3.4) THE MONOTONE CLASS LEMMA:

V

Let

V.

d

be a s e t a n d

be a n a l g e b r a o f subsets o f

T h e s i g m a a l g e b r a g e n e r a t e d by %

family

o f subsets o f

V

d

is

the smallest

that c o n t a i n s

d

and

is

m o n o t o n e c o m p l e t e , t h a t is. s a t i s f i e s a)

if

Fm

E %,

Fm E Fm+l. f o r

then

if

Fm

E %,

Fm 2 Fm+l.

then

U F m € % and

b)

n F~

for

E 5.

The collection of disjoint unions of measurable rectangles is an algebra, so

this result

says roughly d x 3.

property of rectangles is true in

that a monotone

The Monotone Class

Lemma is the most convenient tool used in proving the incomplete classical Fubini theorem.

PROOF OF (3.3.3): The class of sets

{H

E d x

9

x

Meas(=) 8

:

(avo

E

Meas(u))[u[Vo]

(v e Vo 3 H(b.v)

E d x

= 0

Meas(p))]}

3.3:

Section

167

Keisler's Fubini Theorem

is a sigma algebra because countable unions of null sets are null and complements, unions and intersections commute with taking

W

E

sections.

Meas(r),

by (3.1.4) factor.

H = C x W

If

then

= Cb x Wv

H(b.v)

C

for E

E d

d x Meas(p)

x 48

and

a.s.

u(v)

and a simple sigma algebra argument on the first

This proves (a).

Next we prove a special case of (c) for indicator functions as a lemma to establish both (b) and the general case of (c).

H E d

We wish to show that i f = (a x p)[H(b,v)],

q(b,v)

(a x p)-measurabLe,

then

d x 3 x Meas(r)

when

H(b,v)

is not and

The proof of this is based on the

Monotone Class Lemma (3.3.4). in

and we define

(9 x Meas(u))-measurable

is

cp

x Meas(r)

9 = 0

letting

Jqd(/3 x u) = (a x p x r)[H].

sets

48

x

We claim that the collection of

that

satisfy

this property

is a

monotone class containing disjoint unions of rectangles. function

(48

V(b,v)

= a[Cb]p[Wv]

x Meas(u))-measurable

theorems combined.

H =

when

C x

W.

The

and

is

by the classical and hyperfinite Fubini

Moreover.

those theorems imply the second

part of the property for each factor as well.

Since disjoint

finite unions of measurable rectangles produce disjoint sums, we have the property holding on an algebra.

A monotone l i m i t of

sets with the property also has the property by the Dominated Convergence Theorem. d x 9 x Meas(r)

Thus,

(3.3.4) shows

that all

X

r)-measurable.

such that

H'

then there is an

(a x /3 x r)[K v H] = 0 .

containing

in

have the property.

Part (b) follows from our special case, because i f (a x /3

sets

the difference.

H

E (d x

K

is

9 x Meas(r))

or, in other words, a null By

our property

above, the

168

CharJter

integral of the sections of section of

K

3:

Products of Hyperfinite Measures

K v H

is z e r o and

differs from the section of

H

so

almost every

by a null set.

Part (c) follows from the special case because any positive f

integrable

is a monotone

fn 1 f.

indicator functions, measurable

a.s., s o

= [fn(b.v.*)d.

Convergence

SSfn

l i m i t of

The sections of all

Theorem,

then so

F

fn

are

are.

If

Fn(b.v)

Fn 1 F

by

the

Monotone

is

measurable.

Last,

f

the sections of

(a x p),

linear combinations of

so the second property is satisfied. JF = JJf, Finally, we may decompose any integrable f into positive and =

JFn

negative parts to obtain the general case of (c).

169

CHAPTER 4:

DISTRIBUTIONS

(f2.P)

In this chapter we let

* finite

be a

probability

space and adhere to the expected value notation (1.5.4) rather than

the

general is:

probability Another

measure 'the

notation.

study

of

One

invariants

"definition" of probability is:

hyperfinite

spaces

equivalence by

equal

distributions

"definition" of

of

distribution'.

'measure theory'.

On

imply measure-theoretic

reshuffling the points of

the space: moreover,

This chapter only gives the

this can b e extended to processes. basics of distributions.

(4.1) One Dimensional Distributions In accordance with customs in probability we shall call an

X

internal function a

P-measurable

uartable.

: R

3

function

*R

Y :

a n tnternal random uartable and

R +

a measurable

[-m,m]

random

The general results of Chapter 1 apply to lifting and

projecting random variables.

(4.1.1) DEFINITION:

X

Let

*

R + R

be

function of

dtstrtbutton

F

:

*R * * [O,l]

:

internal.

X

-

the charactertsttc

function

f

:

*R

the

cumulattue

tnternal

functton

gtuen b y

F(x) = FX(x) = P[{o

and

ts

The

*E

:

X(W)

function o f given by

< X

x}]

t s

the tnternal

Chapter

170

= E[e

= fX(u)

f(u)

R

is

integral with respect to JeiuxdF(x) = E[eiux],

F

* finite the internal Stieltjes * finite sum, for example, is a

in fact

they are

F

standard part of the distribution absolutely

or

continuous

some

the

same sum.

The

may be discrete, singular,

of

each,

depending

on

the

We already saw in Chapter 2 that every

infinitesimal jumps.

* finite

Bore1 measure has a

iuX,

F(x).

the Fourter transform o f

Notice that since

Distributions

4:

representation.

To standardize distribution functions, we need to view a limit

function defined for all

y

>>

b

for Ig(y)

R and for every

€

all

-

Y

<

bl

B.

E.

>>

satisfying

If

g

y

*

0, there is a

8

>>

(y

>

x

x).

x

<<

y

<<

0

we

w 2 z

and

w

Z

z

z

Z

x

then

N

F(x)

F(x)

as aboue. deftne

u

= F(x)

have

= S-lim F(y),

YlX

for

x

in

such that g(w)

(4.1.2) PROPOSITION: and

such that

x + 9,

(Also see Appendix 1.)

X

we say

is internal, the reader can show that

and whenever

For

is a

= b

this is equivalent to the existence of a g(z) =: b

g(y)

and

x

S-lim g(y) Y JX

if

If

from above with standard tolerances.

R.

Z

g(z).

Section

171

One Dimensional Distributions

4.1:

N

F

The function

is increasing. right continuous.

takes

ualues between zero and one and satisfies

N

N

F(x) = P[X

X

( f o r the projection o f

5

X]

already defined).

PROOF : N

N

X

is

P-measurable

and

5 x] = l i m st P[X

P[X

<

x+l/m].

m e x

Increasing

makes the sets larger, hence the probabilities N

larger

in both

cases.

Right

continuity

F

of

is monotone

convergence of measures. N

We have not shown that

F

is a distribution function in

the strict sense because we have not shown that or

P[X

>

for every infinite n.

n] Z 1

P[X

<

-n]

Z

0

I t is easy to see that

N

1x1

this is equivalent to

P[

surely."

easy

is also

It

N

=

a]

= 0

or

to see that

"X it

is finite almost is equivalent

to:

N

lim F(-m) = 0 and lim F(m) = 1. The next result lists these mmand more equivalent conditions-they mean that F is near the N

standard distribution

F

(compare Greenwood and Hersh [1975]).

(4.1.3) PROPOSITION: The following are equtualent (notation as above):

(a) (b) and

x

n.

t s finite almost surely.

For positiue infinite

n.

lim X+"

F(x) = 0

and

031

in

-

F(b) Z 1. (b')

b

~ ~ 1 x< 1 *IR.

l i m F ( x ) = 1. X 4 "

= 1. F(-b)

Z

0

4:

Chapter

172

(c)

* finite

The internaL

Distributions

dF(x)

measure

on

*IR

has near-standard carrier and

N

dF

st-l = dF

0

as a BoreL measure.

and

(d)

For each standard bounded continuous

(e)

The function

f(u)

(f)

The function

f(u)

is

= E[exp(iuX)]

st(f(u))

BC(IR).

S-continuous at zero.

is

if

'p E

S-continuous for aLL

u

u

i s standard.

PROOF : (a) i f f (b) i f f

(b') is the point of the remarks preceding

the statement of the result. (b)

implies

(c)

is

clear

since

the

dF

measure

of

positive or negative infinite sets is less than the measure of an

infinite

M =

* sup[x

: x

F(M) =: 0.

interval

E N] For

containing

them,

say

(-m,M)

with

in the negative case, and that measure is the

second

part

of

(c).

notice

that

N

dF{st-'(-m,b]}

= S-lim F(b+l/m) = F(b)

and

use

the

classical

a

Lebesgue-

m-yo

representation

of

a

positive

Bore1

measure

by

Stieltjes measure. (c) E[q(i)]

implies

= r-p(x)d:(x)

(d)

by

(2.4.5) and (as

a

the

classical

general Stieltjes

which follows from a simple partitioning argument.

fact

that

integral)

Section

4.1:

173

One Dimensional Distributions

(d) implies (b):

Let

and apply the Monotone Convergence Theorem to that for finitely positive

<

E[vm(X)]

~ 1 x 1>

n]

B

“N

for all

n

>

-

E[vn(X)].

there exists finite

a

m.

We see such that

in

Hence, for every infinite integer

0.

(b) implies (e):

If

u 2 0

when

1x1 <

1/m. exp(iuX)

when

1x1 >

l / m E[exp(iuX)] .

PIXI > l/mz 0. then

and

Z 1.

Since

1

Z

so

lexp(iuX) f(u)

“N

I

= 1

even

f(0).

(e) implies (b) follows from the classical result (4.1.4) which follows next.

(4.1.4) INEQUALITY: F([-u.ulC)

<

6’“

au

where a = l/inf[[1

-

(1-Re f(v))dv

1-

sin t

:

It1 1 11.

PROOF OF (4.1.4):

uJyU(l-Re

f(v))dv

1 u

-

Jy’I(

1-cos (vx) )dF( x)dv

Jur/” ( 1-c o s( vx ) ) dvdF ( x )

Chapter

174

4:

Distributions

= a 1 F([-u.u]~).

(b) implies (f): The set

1x1 >

{a :

Let

6

be positive and infinitesimal.

l/a} has infinitesimal probability. s o

If(u+b)-f(u)

I <

<

E[ lexp(iuX)

(exp(iax)-l)

I]

E[ (exp(i6X)-l

I]

0.

This concludes the proof of (4.1.3).

The next

result

is a

simple

"universality

theorem"

"representation theorem" for a single random variable.

or

Kelsler

[1980] has given an interesting generalization of this idea for stochastic processes.

(4.1.5) PROPOSITION:

R

Let

BP(o) = 1/ functton.

#

be

[n]

*ftntte

a untform

and l e t

G(x)

probabtLtty

space,

be a standard dtstrtbution

X

T h e r e t s an tnternal

:

R * *IR

such that

u

G = FX. PROOF :

For a finite natural number

m

define

= G(-m):

p -m

ph = G k ]

-

<

<

= 1-G(m). m +1 Use these probabilities to form a partition of the unit interval

Gp!?].

for

-m2

h

m2:

and

p

Section

4.1:

One Dimensional Distributions

2=o'

u

Let

{u,}

be a samp e of

these

i=-m

-m

points without repetitions ( n case some R

Since

infinite

* finite

is

u : R + *[O.l]

onto

natural

~

<

<- u(w) ~

there

number,

pi = 0). interne

bijection

l < k < n .

for some

is an

kn'

the points

R = {w 1 , w 2 , ~ * * , ~ n } ,~(w,)

u

175

For

n.

= n'

example,

1.

=

X,(w)

Define

if

whenever

uh. ,Since repetitions have been removed from the

these intervals are finite and

Uh'S.

< Xm

"1

<

;Z

ph,

for

<

k]

G[i]

+ Gk]

-m2

<

<

h

m

2

.

Define the sequence

<

um = max{P[i

so

is

om

Xm

-

internal and

:

infinitesimal

<

-m2

for

j

<

is still infinitesimal. m and let F = F X , so

for

<

-m2

j

<

h

<

Let

with

m2

m

X = X

m

m

finite

Robinson's Sequential Lemma, there is an infinite u

<

h

m

m.

By

so

that

for this infinite

infinite.

Since

G

m

is a N

standard distribution function

G(-m)

Z

0 and

G(m)

Z

1.

so

F

N

equals

G

for all

x.

Recall that

F(x)

= S-lim F(y). Y lx

see

(4.1.2). This concludes the proof, but we want to add a few words of

caution about

S-limit

the

convention, hence i f then

G(x')

know

other

x

is standard and

- F(-m)

F([x],)

x' =

F([x],)

Z

x'

[XI,

G(x)

Z

G(x) = st F([x],).

words,

but

-

>

x

=

*min[k

two

difficuit

as

internal the

reader

infinitesimal amount. S-lim F(y) Y 1X whenever

exists, X" Z

x

distributions can

see

only

for

:

%

.

x,

m in

standard ,

x.

.

G = F

with

by

x'

with

G(-m) = G(x),

N

Comparing

The l i m i t

is standard and right continuous by

I f we take

G(x).

%

Distributions

that we have just used.

G

above is pointwise and

we

4:

Chapter

176

is

translating

more

jumps

an

I t is easy to show that i f

then

and

there

x" 2

exists then

X I .

x'

F(x")

Z

x

such

F(x')

Z

These difficulties come up in a more serious way

Z

that

S-limit.

for paths in

Chapter 5 and for Stieltjes integrals in Chapter 7 where we show

[XI,

(roughly) that the be a small enough

trick always works provided we let

infinite number

(so you miss

m

the whole gap

between translated jumps).

(4.1.6) PROPOSITION: Let Suppose

(n,P)

be a u n i f o r m

that internal random

*f i n t t e

probability space.

X

uartables

f t n t t e a.s. a n d haue dtstrtbutton functions N

respecttueLy.

Y

F

and

=

G.

Then

there N

internal bijection

(I

:

R

are

G,

N

F

sattsfytng

and

R

such that

X(o)

is

an

N

= Y(o(o))

0.s.

PROOF : An internal bijection

(I

preserves

P

because all points

have the same weight. . . . N

Since

F = G.

for each finite natural number

m

there is

xh

a sequence -m2

<

h

177

4.1: One Dimensional Distributions

Section

<

m

2.

xh =: h/m

such that Let

xh

for

denote the sequence with any repetitions

- F(X~-~)0

F(xh)

removed so that

F(xh) =: G(xh).

and

x

(always keeping

and

-m x 2.

say).

Let

m

and

Ah = #

We know nearly

[A,]/#[n]

the

urn #

min[#[nh].

#

Z

: x

h

we

Rh

of

points.

<

Y(w) ~

xh}.

because the distributions are

each

from part

[A,]]

<-

~

[R,]/#[R]

For

same.

injection

{w

may

to part

choose of

an

Ah

internal

pairing up

Doing this for all the finite number

of pairs only leaves an infinitesimal proportion of points left over and we may take points. finite

We

urn

to be an arbitrary pairing of those

P[IX(o)

have

-

Y(um(w))l

Extend the sequence

m.

(I

m

2 2/m] =: 0

for

each

to an internal sequence of

internal transformations and apply Robinsonn's Sequential Lemma to

IX(w)

find

-

an

<

Y(u,(w))l

an infinite Keisler general

infinite

2/m

m

satisfying

nearly surely.

Let

the

inequality

u = um

for such

theorem

for

m. and

Hoover

have

stochastic processes.

variable onto another.

a

reshuffling

This result

just

shuffles

very one

Chapter

178

4: Distributions

(4.1.7) INVERSION FORMULA:

F

Let

* [O.l]

*IR

:

be

internal, increasing, and

Let

F

F o r example.

e iux dF(x).

=

f(u)

could be

the distribution o f internal

f

its characteristic function.

f i n i t e a.s. uariabLe a n d

F

If

S-continuous a t

is

- F(a)

F(b)

Z

>

v

then f o r any infinite

the two f i n i t e numbers

1

infinite and

if

f(u)du.

iu

1

p

(I(a.j3)

Z

Z 0

if

I(a.P)

I(a.P)

is negative infinite and

a

if

a

b,

-iua-e-iub

1

We need facts about the integral %

<

0.

PROOF :

I(a.p)

a

a

Z 0

and

j3

p

and

=

$ p

Jl

sin w dw . 7

is positive

is positive infinite)

are infinite of the same sign.

First, we rewrite the integral term above using the integral formula for

2=

f(u).

[ -iua-e-iub-,

iv -v

iu

-m

eiux dF(x)du

iu(x-a) --v

-

-e iu

J ~ ( ~ - sin ~ W) w -m

-v(x-b)

iu(x-b)

dwdF(x),

dudF (x)

Section

4.1:

by transfer of classical formulas. infinitely near continuous at x =: b:

for

1

a

the net

and

<<

a

b

a

The inside

<<

x

b

dF(x)-contribution

to

b.

dw-integral is

and since

we may ignore the cases

dw-integral is infinitesimal when interval from

179

One Dimensional Distributions

is negligible. x

F

is

x =: a

S-

and

The inside

is finitely outside the

Hence our original integral is within

an infinitesimal of

[ (4.1.8)

dF(x)

= F(b)

f(u)

and

are characteristic functions of

g(u)

the internal random variables

X

hyperfinite probability space

R.

g

and

Y?

F(a).

EXERCISE:

Suppose

and

-

are

and

If

Y

defined on the uniform

f(u) =: g(u)

and both

f

S-continuous, what is the relationship between

X

180

(4.2) Joint Distributions. Laws and Independence (n,P)

In this section we continue to let probability space.

We

R.

random variables on

{X(t)

let

T}

: t E

T

The index set

be a

* finite

be a family of

and the family itself

may be either internal or external.

(4.2.1) DEFINITIONS: Let

{X(t)

variables.

the

: t E

T C T}

(b)

(X(t)

be a f a m i l y o f tnternal. r a n d o m

the

: t E

T}

joint

(c)

dtstribution

internal

= T G T.

function

of

dtstribution

measure

of

function

of

is

the

: t E

{tl.*-*.tm}

is

f o r a n tnternal subset

{X(t)

T}

For e a c h f t n i t e s u b s e t

(a)

{X(t)

: t E

T}

A E + IRm ,

joint

charactertsttc

is

fT(u)

= E[exp(iu*X(T))]

m

where

u = (u

- 0 .

.urn) a n d

u*X(T) =

B uiX(ti). i=1

Let

(Y(t)

: t E:

P}

be a f a m i l y o f m e a s u r a b l e r a n d o m

v a r i a b l e s o n an a r b t t r a r y p r o b a b i l t t y s p a c e . L e t

T

T

Section

4.2:

181

.Joint Distributions

be f i n i t e .

T h e definition o f the distribution functions

and characteristic functions o f finite subfamilies o f are formally the same a s above. measure

(Y,T)

of

is

the

Y

T h e joint d i s t r i b u t i o n BoreL

measure

giuen

on

BoreL-measurabte rectangLes b y :

The measure

.

FT

is related to

by repeated Stieltjes

We could proceed in a manner similar to the last

integration. section

pT

We would define

-

FT = S- lim FT(x) y .lx

J

and show that i t

d

is the standard distribution of the projected finite family if and only if the random variables are finite almot surely or i f and

only

(pT

st-')

if

pT

is

could

S-tite.

be

related

Then to

the

the

Bore1

measure

Lebesgue-Stieltjes

.w

integrals of

fT

could

dFT. be

Finally,

related

to

S-continuity and standard parts of finiteness

and

function of the projected random variables. this out for two reasons:

the

characteristic

We shall not carry

first, i t is similar to the ideas in

section 4.1, second, the effect of the distributions is easier to work with. so (4.2.6) is the main fact we need. Recall the definition of S-tite measure from ( 2 . 4 . 4 ) .

(4.2.2)

PROPOSITION:

I f each

X(t)

is f i n i t e a . s . ,

then each

pT

is

S-tite.

PROOF : Suppose that

pT

is not

S-tite for some finite

T. Then

182

4:

ChaDter

A

there is a n internal set of infinite points,

E

Distributions

*IRm\Om,

such

that P[(X(tl),-**,X(trn)) E A]

Let

* inf[max()aj(

b =

finite point

a'

is

0.

= ( a l , * * - , a mE) A],

: a

. I This number

A.

bound for

>>

b

in

a max-norm

lower

exists and is infinite because no

A

A

and

is

internal.

The

probability above is smaller than the following sum

X(t,)

Hence, either an contradiction

that

a

is not finite

sum

finite of

8.8.

or we have the

infinitesirnals

is

not

infinitesimal.

(4.2.3.1)

DEFIIVITION:

Let

{Y(t)

variables.

.

: t E

T h e Law

P} of

Law( Y T) ,

is

dimensional

real-valued

the

be a family o f measurable random the family

collection joint)

of

of

random variables.

all

the

distributions

(finite from

the

family.

If we select the Bore1 measure representation (cf. (4.2.1)) we can take

Law(Y.T)

I

{uT

:

T is a finite subset of

a}.

4.2:

Section

Joint Distributions

* transform

This notion automatically has a f a m i l i e s of

for the internal

random variables, but we usually do not wish t o

*Law

consider the whole

(4.2.3.1)

183

even in this case.

DEFINITION:

Let

{X(t)

U}

t E

:

be a f a m i l y o f i n t e r n a l r a n d o m t +

T h e f a m i l y (or indexing

uariables. external.

The

o f the f a m i l y of

S-law

S-Law(X.U).

is

the

collection

X(t))

may be

random variables,

of

all

the

finite

distributions from the family.

X(t)

Even if

is an internal family,

*Law(X.U)

external subset of

S-Law(X,U) = {pT

{X(t)

*Law(X.11)

T is a finite subset of T}.

:

(4.2.4)

T

:

U}

to be an unlimited

*finite

set.

is internal,

T is a

*finite

subset of U}.

DEFINITION: Let

uariables

X

measures

U.

Y

Let the

2

and

indexed

X pT

with

: t E

= {+

is the

given by (cf. ( 4 . 2 . 1 ) )

Notice that we do NOT allow I f the family

S-Law(X.11)

and

be

families o f

by

T

2 +,

respecttuely, for

with

internal

be a f a m i l y o f m e a s u r a b l e

same

internal

index

set

U

and

random

distribution

T

finite in

random variables

Borel

distribution

Chapter

184

measures laws

of

vT

T

for

X

f i n i t e in

and

Z

4:

Distributions

W e shall. say that the

'IT.

are

infiniteLy

close

and

near-standard.

S-Law(X,U) 1 S-Law(Z,Y),

i f aLl their f i n i t e dimensionaL distributions a r e

S-tite

and pairwise haue the same weak-star-standard-part,

W e shall say that the Laws o f

X

Y

and

a r e infinitely

cLose

S-Law(X.U)

i f f o r each f i n i t e

T E T.

Law(Y,U).

Z

X

pT

is

S-tite

and

We studied the relationship between Bore1 measures and the

inverse standard part in section (2.3).

Now we may a p p l y that

to distributions.

(4.2.5) PROPOSITION:

I f each

X(t)

is f i n i t e a.s., then

N

S-Law(X.T)

=: Law(X.T).

Section

4.2:

Joint Distributions

185

PROOF :

By(4.2.2)

each

has finite carrier or is S-tite. s o we -1 pT 0 st is the distribution measure

pT

only need to show that N

uT

of

X. Checking on simple intervals, we see

= S-lim PIX(tl)

<

a1 + 1 &

*** 8

X(tm)

k= PT

Once

0

st-'[(--

the Borel measures

all x

UT

- 0 .

and

pT

<

am

1 + r;]

x (--.am 11.

0

st-1

agree on the

intervals, they must agree on a 1 Borel sets.

We could rephrase equal and infinitesimal laws by looking at characterizations in terms of distribution functions or characteristic functions, but the following is the most useful characterization for us:

( 4 . 2 . 6 ) PROPOSITION:

{X(t)

Let

uariables on a

: t E

Y}

* finite

denote the projection o f

(R.P).

be a family o f internal random probability space and let

X(t)

on the hyperfinite space

The following are equivalent:

(a)

each

X(t)

t s

z(t)

finite a.s. and

u

S-Law(X.T) =: Law(X.T).

4:

ChaDter

186

(b)

for each finite

Distributions

m} = T E U

{tl.***.t

and each

standard bounded continuous real-ualued function

1p

of

m

real uariabLes.

PROOF :

(a) implies

I f the

laws.

(b)

X(t)

is

essentially

our

a r e finite a.s., then each

S-tite and (2.4.5) shows (b) since ^. .-1 x only i f st xt E B, so p : o st = pT. law is

of

close

*measure

in the

definition

X t E st

(b) implies (a) because we can approximate function

Ia

of each rectangle

A = (-=,al]

a sequence of continuous functions 1 and pn(x) = 1 if x < a + -. j j n m

0

<

cpn

x

<

Condition

(B)

the

- 0 .

1

-1

i f and

indicator

x (-m.a m 1 lpn

4 Ix

means

that

with (b)

by

(4.2.7) DEFINITION: Let

X 1 . - - * . X m be a finite set o f random variables.

They a r e said to be

independent i f

one of the following

equiualent conditions holds:

the

joint

distribution

function is

the product

of

the

separate distrtbutions. (b)

The joint distribution measure i s the product o f

the indtuidual dtstributton measures.

4.2:

Section

(c)

187

Joint Distributions

The charactertsttc functtons multiply:

(d)

For

bounded

continuous

real-valued

functtons

m'*

(P1."'

The

m-fold

extension

of

the

inversion

formula from

the

last section can be used to prove that (c) implies (a) and (b). The equivalence integration. Notice (d).

of

(a) and

Condition

(b) is

through repeated

(a) implies

(d)

by

Fubini's

that (c) is essentially a complexified ei8 = cos 8

+ i

Stieltjes theorem.

special case of

sin 8.

(4.2.8) DEFINITION:

A famtly of

random uartables

{X(t)

:

t

E T}

ts

said to be an independent family t f each fintte subfamtLy

ts independent.

Our next result is a basic fact that we shall put to work in the next section.

Chapter

188

4:

Distributions

(4.2.9) PROPOSITION: Let and

Y

(n,P) are

independent

distribution

functions

f

functions function of

g,

and Z =

probability space.

internal

F

random

G

and

and

If

variables

X

with

Characteristic

respectively. then the distribution

X + Y

5 z] = J

P[X+Y

*f i n i t e

be a

is:

G(z-x)dF(x),

the c o n u o l u t i o n ,

-m

and the characteristic f u n c t i o n o f

]= f(u)g(u).

ECeiu('+')

2

is:

the product.

PROOF : The characteristic function follows easily from (4.2.7)(d),

Moreover this formulation easily extends to any finite number of independent variables:

the characteristic function of the sum

is the product of the characteristic functions.

The convolution formula is easy enough in the setting because values, hence

X

P[X+Y

and

5

23

Y = ) [P[Y X

only take on

<

z-x]P[X

= x]

* finite :

* finite sets of

X = x].

189

*Finite

(4.3) Some

Independent Sums

Random variables (or

indicator functions of events) that

are functions of separate factors in a product of probability spaces are independent.

P = P

X P2,

Then

P[X

= Pl[f

<

<

while

X(ol.w2)

<

y] = P[{wl

x & Y

I

x]P2[g

independent.

For example, suppose

<

y] = P[X

= f(ol) :

and

<

f(ol)

<

x]P[Y

1 x R2

so

{02

X

and

= g(02).

Y(w1.02)

x} x

y].

R = R

<

: g(02)

and

In this section we give some examples of

Y

Y}] are

* finite

extensions of this idea.

1

6t = n -

Thruout the section we let in

*IN. U = {t

W={wE

E

*R

<

E *I 0,

<

t

1).

*I N : 1 < w < n ” )

R = {o : T

6P(o) =

= k6t, k

: t

n = h!

for infinite

* W)

-, ~

= internal functions from

the uniformly

U

* finite

weighted

into

W.

probability

CRl on

R.

(4.3.1) PROPOSITION:

f(t,w)

Suppose

*R .

tnto

The

* tndependent, from

P{o

U

:

X(t,o)

internal

* [O,m],

<

xt

x

= f(t.ot)

W is

ts a n internal. functton

then

xt for all. t) = n[P{X(t)

In particular. we may many factors. so

X(t.o)

famiLy

that is, t f

tnto

T

i s an internal function from

take

x

t

=

m

<

xt} : t

E

TI.

for alL but ftnitely

ChaDter

190

<

P[X(t,)

x1 &

*** &

X(tm)

<

4:

Distributions

m Xm]

TI P[X(tj)

=

I

Xjl.

j=1

PROOF : Each separate constraint on the t-th factor,

<

{o : X(t.o)

so

=

TI

w

x {w E

w

:

<

* transform

xt

f(t.w)

is only a condition

I

Xt} x

xt for all t} = n[{w

:

* finite

f(t.w)

<

rectangle,

xt} : t E U].

of the cardinality of a product of factors.

is the internal product of the uniform probability on '[TI-times.

n w. s>t

s< t

that the net internal constraint is a

{o : X(t.o)

By

Xt}

<

X(t.w)

W

P

taken

The product formula follows from this since the

measure of a rectangle is the product of the measures of the factors

.

(4.3.2) EXAMPLE: Let -1

p

:

W

-

[Compare to (0.2.11).]

on one half of

p(ot) = f(t.ot)

{-l,l}

W

be an internal function that equals

and equals

+1

on the other half.

forms an independent family and

Then

Section

is

a

4.3:

sum

191

IndeDendent Sums

of

independent

factors.

Hence,

by

(4.2.6) i t s

t/6t

.

characteristic function.

E[exp(iuB(t))]

= {E[exp(iu

a P(w))]}

We simply compute:

E[exp(iu

a P(w))]

1 i u a = z[e

+

e-iual

= cos(ua)

We thus obtain the characteristic function

-= 2

E[exp(iuB(t))]

By (4.1.3).

=: [l

-

2

3 6t]t’6t

2

z e

the inversion formula and the classical computation

-=

that the charcteristic function of the normal law is

-x-

2

iux e 2 t d x = e

we

arrive

at

the

conclusion

of

De

e

2 2

-= 2 2 ,

Moivre’s

(0.3.12) by a simpler i f less direct route.

limit

theorem

This idea can be

generalized to prove a more general central l i m i t theorem.

Chapter

192

(4.3.3) EXAMPLE: Let

4:

Distributions

[Compare to (0.3.6).]

a

be any standard positive real number. Define the n for 1 I w I and ~ ( w )= 0 internal function r(w) = 1 1 otherwise. We know that a6t - n P[r(w) = 11 5 a6t and n n is infinitesimal, letting p = P[r(w) = 1]/6t, we since n n

+

-<

have

p 1 a.

By

(4.3.1).

f(t.ot) = r(ot)

the family

is

independent and

is a sum of

independent random variables.

By

(4.2.6).

its

characteristic function.

and the inside term,

E[exp(iur(w))]

= e iu p6t + e0(1-p6t)

= [I + p(eiu-1)6tl. We see that

E[exp(iuJ(t))]

1 [l

+ a(eiu-1)6t]

ea(eiu-l)

t/6t N

Again we arrive at a result we have seen before, (0.3.7). simple

indirect method

once we know

function of a Poisson distribution is

that

by a

the characteristic

Section

4.3:

193

Independent Sums

W

-

Ie

iuk e-at (at)k k!

,at(eiu-l)

k=O

The ideas in the last two examples can be extended to give a general representation formula for "infinitely divisible laws" A

along the lines of the classical LCvy-Ito formula, but combining the

continuous

and

discontinuous

parts

and

using

sums

of

infinitesimal Bernoulli trials instead of integrals with Poisson measures.

We shall not give the details of that representation.

We give one general result that we shall use in an example in the next chapter and sketch the representation for the Cauchy process in the closing exercise.

(4.3.4) PROPOSITION:

X(t)

Let

6X(s)

F(x).

Let

characteristic

= C6f(u)l

then

:

t/6t

.

6f(u) = 1

0

<

6f(u)

function

X(t)

If

+ 6t+(u)

continuous for f t n t t e

<

s

t , s E U],

* independent

are a n tnternal

distribution, the

= 1[6X(s)

where the

faintly w i t h the same

= ECexp(iu6X)I.

X(t)

of

s o that

f(t.u)

ts

is f i n t t e a.s. for f t n t t e

+(u)

wtth

f t n t t e ,and

t.

S-

u.

PROOF : We know

f(1.u)

formulation o f positive finite

8 m.

is

By the

6-5

S-continuity. we know that there is a standard such that

m

S-continuous by (4.1.3).

qf(l.u)l

>

lf(l,u)[ = [ l6f(u)

m]I

z1

for

1.1

<

"-

equals either zero or one for all

exists and finite

u

8.

For each -1 S-limlf(1.u)

Im

according

to

4:

Chapter

194

0

not.

Hence

the

Distributions

limit

is

one

for

whether

f(1.u)

1.1

Since the square absolute value of a charcteristic

<

9.

Z

or

function is also a characteristic function, we may apply (4.1.3) S-limit is a standard characteristic function

to see that the is

(it

S-conntinuous at zero).

The

is continuous,

S-limit

takes the value one and only can take at most one other value, zero: therefore i t is identically one, s o finite

f(1.u)

0

for any

u.

We let

so that we have

= [6f(~)]'/~~

f(1.u)

We know that when

Moreover.

by

*C .

is finite in

$

Robinson's

1/6t .

= [l+at$(u)]

Sequential

Lemma.

this

identity continues to hold on an infinite disk

$(O)

know that 6f(u)

= 0

and that

$(u)

is

approximate

l+l

is an internal charcteristic function.

Since

S-continuous and noninfinitesimal. i t follows that finite and

S-continuous for finite

First. suppose finite

l$l 0

<

= R x

<

The

ul.

uo

at

uo)]

-

1

$(u,)

*arc <

u1

(a

-m)

where for

We

because

f(1.u)

is

$(u)

is

u.

I$[

lies outside $[O.ul]

S R.

*continuous

5 R

for some

first crosses the boundary

>

Re[$(u)] 0

<

u

<

uo.

log[min(lf(l.x)l

:

This is because

Section

) u ( ' e x

4.3:

f(1.u)

Z

<

uo).

for

When

= (q+i2kr)

*arc

u

for

has

u.1

I+(u)I v

Z

in

<

R

(and because

[O,uo],

*

f(1.x)

we must have

infinitesimal and an integer

q

$(u) k.

+

is

S-continuous on

cannot be finite.

S-continuous for finite

u

[O.uo]

0

- +(v) k = 0.

and i t follows

+

Similar reasoning shows that and since

for

Since the

bounded below, i t is easy to show that

Re[+]

This means that that

195

IndeDendent Sums

is

$(O) = 0. we are done.

(4.3.5) EXERCISE:

6X(t)

Show that the terms infinitesimal a.s.

(Hint:

in the last proposition are

(4.1.4). )

(4.3.6) EXAMPLE: We may view

W = nn

* finite

as a

product.

The set of

functions from an m-element set into an n Cn/ml-element set has n --.m m elements whenever m h , where n = h! as above. Any n n are both finite m h is such that and n Cn/ml

<

-

<

infinitesimal. that

x = kax,

such

X'S.

so

Let

ax = l/&.

k # 0, k

*Z

we may view all

with values in a set with Since

in

Consider the values of

n/n Cn/ml

may deine an internal

r(x,w(x))

and w

n [ n/m 1

in

po i n t s

1x1

G 3. There

W

as functions

=

{ 01 .,

x

are

m

w(x)

as above we

family of functions

probability otherwise

such

.

is infinitesimal for each

* independent

x

p(x)

196

4:

ChaDter

where

= p(x) =

p(-x)

-ll - 6t(l+r).

Z

0.

t, s E

a].

with

L

Distributions

X

We let

and

The

characteristic

where

6f(u)

<

: 0

C(t,o) = P[7(os)

function

of

s

<

C(t,w)

equals

is the characteristic function

of

[Sf(u)]

t/6t

~(w),

II {1+2p(x)[cos(ux)-1][l-P(x)l}. x>o

=

(4.3.7) EXERCISE: Show

1

6f(u) = l+bt[

that

rcos(ux2 )

cos(ux -1 2)

Then use the fact that that

[6f(u)]

t/6t

characteristic dFt(x) =

1

T

Define

e-tlul.

function

The

of

9

Z

0.

1 dx

last

the

=

1.-

to conclude

expression

Cauchy

is

the

distribution,

2 2 dxt

t +x

(4.3.8) EXERCISE: The

u

-' 1 ax++

X

x>o

[Compare to (0.2.11).]

construction

of

(4.3.2)

is

certainly

not

unique.

Section

4.3:

197

IndeDendent Sums

and also define

W

We may consider

Vu,

as the product

n

( 2) V = {1,2.--*,n } .

and of

W

onto

VU ;

*bijections

Of course there are many

fix one.

u :

W

+

VU .

U = {-l,l}

where

Define

and

Each

p

. i

defines an infinitesimal random walk:

B.(t.o) J

Show that

* independent and

p2

= p,.

= Z[pj(os)fi

{pl(os)

families.

: s E

<

: 0

U}

s

and

Show how to select

PIBl(l) >

t. s E 'U].

{p2(ws)

Show how to select u

so

are all independent. What is

<

0

I

B2(1)

>

O]?

v

that

: s

E

s o that

p1.p2,p,

a)

p, and

are =

p3 p4

ChaDter

198

For u :

W

+

each

W

pair

so that

(4.3.9) EXERCISE: Use

the

(i. j)

show

Bi(t.u(o))

that

4:

there

Distributions

a

is

bijection

= B.(t,o). J

[Compare to (0.3.5).]

methods

of

(4.3.3.

6,

8)

to

independent (approximate-Poisson)

jump processes

four (possibly) different finite

ak;

construct J,(t.w)

k = 1.2,3,4.

four with

199

CHAPTER 5: PATHS OF PROCESSES

This

chapter

studies

the

hyperfinite evolution scheme. properties

for

paths

of

processes

over

a

The idea is to find corresponding

stochastic processes

defined

on

internal processes on the infinitesimal time line

[O,l]

and

T.

Hyperfinite evolution will always be relative to the space

R = W

T =

{w

:

H

W. internal}

+

where

H = {t E

*IR

: t

<

= k6t. k E *IN, 0

t

<

1)

and 6t =

-n1

n = h!

for some infinite

in

*IN

and

W

= {k E *IN

:

1

<

k

<

n”}.

We take the uniform internal probability with weight function

A function

X(t,w)

:

T

x R

4

*IR

a function-valued random variable.

XJt)

= f(t)

with

f(t)

= X(t.w).

can also be thought of as w +

Xu,

where the section

This approach requires us to

consider some simple spaces of functions.

(5.1) Hetric Lifting and Projecting Let

(U4.p)

be a metric

distance function or metric.

p

space entity, :

M x W

[O.m)

M , p E 6.

satisfies:

The

If

5:

Chapter

200

1.

p(x.y)

= p(y,x)

;

2.

p(x.z)

<

+ p(y.z)

3.

P(X,Y) = 0

p

p(x.y)

for

x.y

in for

;

1

and

2

M. x.y,z

M.

i t is called a s e m i m e t r

[In that case the set of equivalence classes = 0)

in

x = y.

i f and only i f

only satisfies

Paths of Processes

forms a metric space under

p . ]

x

P

= {y : p(x

C .

Y)

Our main examples of

metric spaces are as follows:

(5.1.1) EXAMPLES: M = IR d = {(x

:

1'X2'"''Xd)

x

j

E IR},

1

d j=1

M = C[O,l] on

= continuous real-valued functions defined

[O,l].

the uniform norm. M = D[O,l]

,

= the right-continuous real-valued

functions with left limits on

[O,l],

Example (c) is explained in greater detail below, especially in section 5.3. Recall that a metric space Cauchy sequence in

M

M

converges in

is called c o m p l e t e if every

M

and is called s e p a r a b l e

5.1

Section

20 1

Metric Liftinp: and Projecting

if i t has a countable dense subset.

All three of the above

examples are complete and separable.

A

topological space is

said to be a Polish space if the topology is induced by some complete separable metric. Some useful extensions of these examples are to consider C(IR)

or

IR

domain

the continuous real-valued functions with

C([O,m)),

or

In these cases the metric is the metric

[O..m).

of uniform convergence on compact subsets (no longer a norm). One can also extend the domain for for

or

C

D

or allow the values

spaces to lie in a complete separable metric

IR.

space instead of One

D[O,l]

These spaces are still Polish spaces.

slightly useful

semimetric example is

Lo[n].

the

mesureable functions with the semimetric

p(x*y) =

This

semimetric

y, 1+ x(0)-y(0)

measures

dP(w).

convergence

in

probability-two

functions are close if they are close except on a set of small measure.

(5.1.2) DEFINITION: Let

y

in

*M

x

in

aM

(M.p)

stp(y) = x.

€

9L

be a standard metric space.

is near-standard f o r such that

p

i f there is a standard

p(x,y) =: 0.

In this case we d e f i n e

the standard part

standard points is denoted

For example. i f

A point

M = Rd,

of

y.

T h e set o f

near

* nsp( M ) . then the near-standard points of

5:

Chapter

202

*Eld

are

just

limited.

Paths of Processes

y = ( y l , y 2 , - - - . y d ) with

those

Yj E 0

each

x = st(y) = (st(yl).st(y2).***.st(yd))

In that case Ix-yl z 0.

satisfies

Another simple characterization of near-standardness is:

(5.1.3) PROPOSITION:

A function

y(r)

*C[O,l]

in

i f and only

the u n i f o r m conuergence metric finite and then

is near-standard

r

S-continuous. that i s , i f

y(r)

Z

y(s)

y(r)

if

s

Z

for

in

is

*[O.l],

a n d both are finite.

PROOF : Let

0

<

11 = 6

r

x(st(r)) y(s)

x(r)

Z

be

0.

Z

x(r)

Z

y(r)

Then

y(r)

let

x

x(r)

Z

y(r)

and

x(s)

%

x(r)

by

X ( . )

=: y(r).

Ix(s)-y(s)(

Therefore

* sup[

the

<

Ix(s)-y(s)

I

If

for all

<

:

and

Thus,

s

<

is standard,

in

s

standard

11

function

we

have

1).

Now,

Y(r) z Y(S)

and

every

r

Appendix

(see

y(s)

: 0

y(s).

r = st(s)

letting

for

e

Z

finite

continuous

r.

for standard

Z

x(s)

standard

construction, x(s)

Z

is

I

(See Appendix 1.)

S-continuity

Thus

sup[ Ix(r)-y(r)

x(st(r))

x(r)

otherwise

by

S-continuity.

be

suppose

know

and

x(r) = st y(r).

given by

we

and

and both are finite.

Conversely,

that

standard

Z

by

* [O.l].

positive

so B.

0.

For a generalization of this result and its relation to the Ascoli-Arzela

theorem,

see

Stroyan

and

Luxemburg

[1976.

(8.4.43)]. Other

characterizations

of

near

various spaces of paths are given below.

standard

points

in

the

W e will also need an

203

5.1 Metric Lifting and Projecting

Section

internal

generalization

of

this

notion

for

hyperfinite

infinitesimal

random walk

processes, but give the standard case first. Here B(t.w)

is the basic we have

idea.

seen above in

almost surely finite and the

next

section;

w + Bo

view the map

is the function

already

(0.3.12).

etc., is

[We will prove this in

S-continuous.

from

know

R

that

is

it

*C[O,l]

into

= B(t.o)].

Bw(t)

s tp(Bo)

0 .

finite

by

[where the section

In this case then, for almost

exists

and

by

the

last

result,

= st(B(r.o)).

stp(Bo)(r)

X

Let P-almost

:

R

*M

surely near

= 0

P[A]

we

(0.2.10).

We could f i l l i t in piecewise linearly in order to

(0.3.12).]

a1 1

The

be a function.

w

Q

X

say that

i f there is a set

standard

such that i f

We

A E R

X(o) E ns ("M).

R. then

is

with

When

P

X

is almost surely near standard we define the m e t r i c p r o j e c t i o n N

X

of

by choosing any

a E IN

and letting

X(o) = st,[X(o)],

u

when

X(o)

is near standard and

X(w) = a

otherwise

(5.1.4) THE HETRIC PROJECTION THEOREM:

(i2.P)

Let (M.p)

X

:

R

be a

*f i n i t e

p r o b a b i l i t y s p a c e a n d let

be a standard c o m p l e t e s e p a r a b l e metric space. +

*M

is

internal

and

is

almost

surely

If near

N

X(o) = st,(X(w))

s t a n d a r d t h e n the metric p r o j e c t i o n P-measurable.

Moreover.

if

X(o)

near

is

is

standard

for

u

o Q A

with

A

internal and

P[A]

= 0, then

X

has

compact range.

PROOF : Measurability is simple.

Take

b

in

M

and

F;

positive.

ChaDter

204

N

<

{ w € Q : p(X(w),b)

The difference between

Paths of Processes

5:

U {w E R :

and

e)

m

<

p(X(w).b) set

is

in

e-l/m)

A.

is contained in the null set

the Loeb algebra since i t

The latter

is a countable

union of

internal sets.

The points

standard is

part

always

internal

an

of

compact,

see

set

o f near

&

Stroyan

standard [1976,

Luxemburg

(8.3.11)].

(5.1.5) DEFINITION:

Y

Let

M

0

:

be an internal function.

Y = st (X)

if

null sets,

to

for the

so

X

We say

[PI.

a.s.

P

X

be a function and let

:

R

*M

---)

Y

is a lifting o f

This depends o n

if needed we may say

X

and

p

is a

P

up

P-Lifting

metric.

p

The result above shows that an a.s. near-standard internal function is a lifting of its projection.

(5.1.6) THE HETRIC LIFTING THEOREM:

(R,P)

Let be

a standard

K C

*M

be

complete

: R

*M

lifting with values in

measure space, let

separable

internal and

Y

function

* finite

be a

P-measurable

X

: R +

space and

stp(K) = M.

S-dense.

is

K.

metric

(M.p)

then it

let

If

a

has

a

K.

PROOF : This is very similar to the scalar case (1.3.9). slightly different approach. subset of

M.

Since

K

is

Let

{z,)

Here i s a

be a countable dense

S-dense. for each

zk

there is a

Section

yk

in

5.1

K

Metric Lifting and Projecting

with

For each finite

yk.

m

fixed the :L

E

A:

let

For

each

1

k

< k.Extend

P[A>n:]

with

< ,I) 1

p(x.yh)

:

M.

are a Bore1 partition of

= Y-'(L;).

A:

Let

IN.

in

m

s o choose an associated sequence

= {X E M : k = min[h

:L

For

= zk,

st(yk)

205

yk

choose an

n:

and

internal

to internal

m2 sequences

using

countable

comprehension

extension of yk in K. m X ( 0 ) = a off the union of internal internal

P[p(Xm,Xn)

Xm

with

sequence

>

p]

<

P[p(Xm.Y) and for

choose m

<

n.

keeping

Xm(u) = yk

Let

n.:

(0.4).

Then for each

> --I1 an

< .;1

Such an

m

Extend

infinite

Xn

R:

on

n lifts

the and

there is an

Xm such

Y.

to an

that

206

(5.2) Continuous Path Processes Let

X(t,w)

x R

U

:

*IR

4

be an internal function. R

view this as a function from

+Xu

w

T

Xu,

these sections,

the path

almost surely finite and with

P[A]

when

w

= 0

e A.

The

by considering the

= X(t,w).

X

of

We call one of

=: (Xw)(s)

standard

parts

of

finite

6 U : t

max[t a

is

xu

function

{stX(t,w) X(r,w)

finite

and

S-continuous

is

A C R

t =: s

for

E 0

functions are standard continuous functions on that

Xu

We say

o.

at

S-continuous, if there is a set

(Xo)(t)

and

*IR

into

(Xw)(t)

where

F(T)

into the internal space

of all internal functions from section map

We may

U

in

S-continuous

[O,l].

and

Suppose

[r]

let

=

*

<

on

X(r,o) = stX([r],w)

The equation

r].

For

[O,l].

I t =: r)

each

r,

the

is only a singleton. s o

defines

set

of

numbers =

stx(st-'(r),w)

is well defined.

(5.2.1) DEFINITION: Let have

X : U x R

finite

* *R

S-continuous

projection o f

X

be internal and almost surely paths.

The

continuous

path

i s the function

N

x

:

[O,l] x R

+

IR

N

giuen b y

X(r.o)

= ~t[X(st-~(r).o)],

when

Xu

is finite

N

and

S-continuous. and

X(r.w)

= 0. otherwise.

w

We

know

from

Appendix

1

that

the

paths

of

X

are

continuous. but moreover. the assignment of paths is measurable.

Section

207

5.2 Continuous Path Processes

X

We may also describe the relation between

and

I

by

N

= st X(t,o)

X(st(t),o)

(5.2.2)

a.s

THE S-CONTINUOUS PROJECTION THEOREM: Let (a)

X

and

The

." X be o

map

-

a s in (5.2.1).

Then:

N

Xu

R

of

C[O,l]

into

is

P-measurable.

(b)

For each

[O,l].

in

r

the random uariable

N

Xr

is

P-measurable. N

(c)

X

The distributions o f

q(x1.*--.xm)

t s

m

function o f

uary continuously, i f

a standard bounded

continuous

r l = st(tl).***.

uariables a n d

t

r m = st(t,). N"

,r 1 ,***.X m)].

E[q(X

(c') T h e map the

r +

continuous.

X

:

2.

[O,l] x R

[O.l]

of

conuergence

N

(d)

E[c(Xtl.-.*.X m)]

then

-

Lo(P)

into

in

probability

IR

is

with

semimetric, i s

(Borel[O.l]

x

Meas(P))-

measurabLe.

PROOF :

Part (a) follows from the metric projection theorem (5.1.4) by extending

X(t,o)

to be piecewise linear between the points N

of

T.

Then we may identify

uniform norm (up to a

P

We can argue on evaluation map

x

-P

and

stp(X)

where

p

is the

null set). X(t.w)

x(r)

the uniform norm. so

X

of {x

:

for part (b) as follows. C[O,l]

x(r)

<

into a}

IR

The

is continuous in

is open.

By part (a),

208

ChaDter

N

{w

:

<

a} = (XW)-’({x

<

x(r)

:

{o : X(s,,w)

A

<

1 5 j

finitely many indices

9

Paths of Processes

N

Xu(‘)

a})

Part (c) is proved as follows.

where

5:

B

m.

is measurable.

First, let

s

j

Z

t

j

for

Then

1 5 j 5 m}

X(t.,o): J

X

is the null set off which

is

A

S-continuous.

Let

be a bounded continuous standard function. and consider the

internal probability:

This

probability

is

infinitesimal

contained in the null set

A,

since

is

it

internal

and

hence

The internal set of tolerances.

contains the external set of all positive infinitesimals, thus 8.

i t contains a standard positive

and

r j = st(tj)

and

(5.-t J

j

Finally,

I <

812.

if

then

qj = st(sj) E[v(i(q’s))]

N

z ECv(X(s’s))l

IE[v(G(s’s))]

and

-

z E[q(X(r’~))l*

so

Compare this with

(4.2.3).

E[v(X(t’s))l

<

E[v(X(r’s))](

B.

u

but

note

the

change

in

notation,

X(t)

= st(X(t))

with

Section

5.2

209

Continuous Path Processes

N

in (4.2.3). while now we index

nonstandard time

t

standard instants

r Z t.

Part

(c’)

is

a

special

case

of

part

(c)

with

since

the

E[cp(X.Y)]

semimetric of convergence in probability is given by where

X

is the standard bounded continuous functon:

q~

N

X

We prove part (d) by showing that step functions.

min[s

This is based on part (c).

= X([kS].w)

X,(t,o) E

H

is a uniform limit of

for Also let

k9]

: s )

k9

<

t

m,(o)

<

Let

(k+1)9

9

>

and let

0

[k9] =

where

= max IX(t.o)-X9(t,o)l. t

If

w C A

and

9

Z

then

0.

m,(w)

2

0. Since

P[A] = 0 ,

is infinitesimal almost surely and thus also nearly surely

m9

by (1.3.10). P[m9

2

E]

<

If so

q.

and

B

are standard positive tolerances

q

there must be a standard

property (consider the internal set of

9

with

this

that work).

This

to zero in probability, s o

for a

9’s

N

means

and

mg

tend

mS N

sequence of

8’s.

0 a.s.

mg

N

When

is

9

= st(X,(t.w)).

standard

kS

for

and

r,t

positive

we

let

X,(r.w)

N

<

(k+l)S,

so

N

lX9(r,w)-X(r,w)l

N

N

.( mS(w)

0

a.s.

P.

Each

X9

is

(Borelx P)-measurable.

This proves (d).

The next (0.2.10).

result

(0.3.12).

shows (4.3.2)]

that Anderson’s random has

walk [see

S-continuous patha.

That,

together with all the machinery w e have built up, shows that i t is infinitely close to a conventional Brownian motion.

Later we

210

5:

Chapter

will give a general

Paths of Processes

S-continuity theorem for martingales.

(5.2.3) THEOREM: Anderson’s infinitesimal random walk

B(t.w)

almost

= 8 [ a p(os)

surely has

:

0

<

S-continuous

s

<

t. step 6t]

paths.

Its projection,

.Y

B

: [O.l]

x

stationary

R -+ IR.

has

independent

continuous

normally

paths

a.s.

distributed

and

has

increments,

that i s , N

(stationary) the distribution o f

(a)

depends only on

(b)

(independent

<

[G(r+e)-B(r)]

a.

increments) i f

0

<

N

r m 5 1.

ro

<

<

rl

--*

N

{B(rj)-B(rj-l)}

then the family

ts

independent. N

(c)

P[B(r)

(normal)

,(

a] =

ZGF

r,

--X

2

e 2r dx.

PROOF :

To prove an

e-8

S-continuity we begin by estimating violation of e =

condition with

2 m

are finite positive integers.

:R

= {a

: (ilk)[

Since

the

differences

between

-n1

where

and

m

Define the internal set

max B(t.o) k+ 1 n t n

E< <-

8 =

and

k

min B(t.o)] k+l

>

}:

;;
max

and

min

1 -time n

on

intervals at worst occur for separate sample sequences

o.

n

Section

5.2

21 1

Continuous Path Processes

<

P[fl:]

nP[[ max

B(t.w)

1

-

o
min

B(t.w)]

o
If the difference between a max and min for then

Bw

either goes up by at

w

>

t].

is m o r e than

2

m’

least half that or down by at

least half, starting from zero: the last probability is

<

nP[ max IB(t).l 1

o
B(t)

$1 > $1 >

+ nP[ min

B(t)

<

1

-$I.

o
Any sample

w

up to

sample choices making

t

B(;)

1 < n 1

is followed by an equal number of

- B(t)

positive and an equal number

negative, so the above probabilities are

<

2nP[B(;;)1

= 4nP[B(--) 1

> --I1

+ 2nP[B(--) 1

< ---I1

> --I 1

Finally a simple calculation shows that

S-llm P[n:] = 0 for each fixed m. ncontinuous paths are contained in the samples from so

that

The

S-dis-

Chapter

212

5:

Paths of Processes

~ = u n n : m n and = sup(inf

P[A]

This

proves

that an

B(6t) 2 0.

B(t)

= 0.

P[n:])

is

a.s.

S-continuous.

Since

S-continuous path is finite as well. N

B

Continuity of

A

now follows off

as in (5.2.2).

The stationarity comes simply from the defining sum for

-

B(r+e)

B(r)

has the same form as the sum defining

Independence of

the increments follows from

B;

B(e). (4.2.3) and

(4.3.2).

(5.2.4) REMARK ON WIENER MEASURE:

One definition of Wiener

W

measure

on

C[O,l]

measure

is:

the unique Borel

satisfying: 2

(a)

W{x

: x(r)

<

--X

-

a} =

e 2r dx

4ziG

-m

<

<

and

(b)

if

0

<

variables

ro

x

-

<

rl

0 . -

(x(r

J

rm

<

1,

)-x(rj-l))

then

the

random

form an independent

family. One can show that

W

is given by

W(E) = P[{o

where

E

is a Borel subset of

B(t.o)

associate measure

P

into

:

B,

E

C[O.l].

E}]

Roughly the idea is to

with piecewise linear paths and transfer the

*C[O,l],

so the formula above is

Section

5.2

213

Continuous Path Processes

W(E) = P [ s t -1 (E)] P

analogous to the construction of (2.3.4).

(5.2.5) DEFINITION: Turo functions +

IR

are

Y

:

[ O , l ] x R + IR

P-indistinguishable

P[A] = 0 , such that when for all

in

r

o

if

e A,

2 : [O,l] x R

and

there then

A.

is a set

Y(r,o) = Z(r,o)

that is. if their full paths agree

[O.l],

almost surely.

-

Y

Let

x

:

u

x

R

[O.l] x R

:

*IR

be

4

be

IR

internal

S-continuous paths.

Then

X

function and

a

and

a.s.

haue

let

finite

S-continuous path

is an

u

Y

lifting o f and

Y

o f (5.2.1)

are indistinguishable.

Suppose almost all paths of The projection several

X

prouided that the projection

properties

if

[O.l].

are continuous on

theorem (5.2.2) tells us

measurability

question is:

Y

it

Y

that has

How little can we assume about

a

Y

must have lifting,

the

and still have

a lifting?

(5.2.6) DEFINITION:

A

Y

function

stochastic process o n the section

Yr

is

:

[O.l] x R

[O.l]

-

IR

if for each

is r

called in

a

[O.l]

P-measurable.

The answer to the above question is that we will assume

Y

5:

ChaDter

214

Paths of Processes

is a stochastic process.

(5.2.7) THE

S-CONTINUOUS PATH LIFTING THEOREM:

I f a stochastic process continuous

paths,

then

Y

: [0,1]

has

it

x

an

R + IR

a.s. has

S-continuous

path

1 t f t ing.

PROOF :

Yo

Assume that

P[A] = 0. We want to show that map into

C[O.l].

{o € R\A

o

U

That is, i f

uniform norm, then

-

is continuous except for

:

Yo

Yo

is

U}

is

with

P-measurable as a

is open in €

A

in

o

C[O.l]

with the

P-measurable.

This

will allow us to apply the metric lifting theorem. We {o : a

know,

<

since

<

Y(r.o)

b}

' Y

each is

C[O.l]

%(a.b;r) = {x : a

set

is

continuous. measurable

closed Also,

with

in

C[O.l]

(Yr)-l[a.b]

respect

measurable,

P-measurable.

dimensional cylinder set in

This

is

to

Define

that

the

one-

by

<

x(r)

b}.

since

the

evaluation

= Y;l[%(a.b:r)].

cylinders.

Yo

so

The

closed

is

is

ball 'in

C[O.l].

by continuity. are measurable.

Hence inverse images of closed balls in Finally, separability of

C[O.l]

C[O.l]

(rational

coefficient polynomials are dense by Weierstrass' approximation

Section

5.2

Continuous Path Processes

theorem) means balls, s o Let

Yo K E

that open

sets

215

are countable unions of closed

is measurable.

*C[O.l]

be the set of piecewise linear continuous

T,

functions with corners at points of S-dense for the uniform norm and in internal functions defined on lifting with values in

K

S-continuous path lifting by

T.

so

is internal,

correspondence with

1-1

By

K

Yw

(5.1.6).

and the restriction to

has a

T

is an

(5.1.3).

(5.2.8) EXERCISE: Wiener measure is unique, yet we had considerable choice in N

constructing are

B(t).

distinguishable.

Show that You

N

B1(r)

should

and

B2(r)

construct

from (4.3.8)

similar

examples

based on (4.3.9) when you read about decent path projection in the next section.

216

(5.3) Decent Path Processes

We shall say that a function

x

:

[O.l]

+

IR

is a decent

if i t is continuous from the right and has left limits.

path

We denote the space of decent paths by

D[O,l]

= {x

:

[O,l] -+ IR:lim x(q) exists and lim x(s) = x(r)). qfr s lr

("Decent" is a free translation of

the Alsatian word cadlag,

which

American

sometimes

slips

into

North

Cf.

articles.

Dellacherie and Meyer [1978, p. 901.) Skorohod Later,

introduced a number

Kolmogorov

gave

a

metric

of

topologies that

induces

interesting of those topologies and makes separable

metric

space.

Kolmogorov's

on

the

most

a complete

D[O.l] metric

D[O.l].

is

given

as

follows.

A strictly increasing LtpschCtz function onto

all

[O,l]

is caLLed a time d e f o r m a t i o n o f

time deformations

of

[O,l]

so

p

in

[O.l]

The set of

A[O.l]. A[O.l]

The is:

that small deformations have secant lines of slope near one.

Notice that pn

[O,l].

is denoted

measure of the amount o f deformation for

from

p

6(p-l)

= 6(p)

and if

6(pm)

tends uniformly to the identity map.

tends to zero, then

Section

217

5.3 Decent Path Processes

(5.3.1) DEFINITION: The Kolmogorou metric D[O.l]

is giuen by

( f o r Skorohod's

topology) on

=

k(x.y)

Of course the uniform metric is also defined on since

sup)x(r)-y(r)l r

is finite when

x

and

y

D[O,l]

belong to

D[O,l]. The instant theory of

r = 1

D[O.l];

plays a somewhat artificial role in the

1

for example, jumps at

cannot be shifted.

This technical annoyance disappears in the study of

D[O,m).

but at the expense of other difficulties with the metric. infinitesimal analysis of like that of

D[O,l].

D[O.m).

when

r

when

r

is l e s s than

The

is finite, is just 1.

We begin with some examples to illustrate how the metric

works.

The

first

one

infinitesimal amount-the

shows

that

we

can

shift

jumps

an

uniform metric would not allow that.

This will make our infinitesimal Bernoulli processes like the approximate Poisson process

J(t)

from

(0.3.7) near standard.

(5.3.2) EXAHPLE: Define a sequence of functions by:

This sequence does not converge in the uniform norm, in fact,

218

Chapter

suplx,(r)-x,(r)

I

= 1

m # n.

for

Paths of Processes

5:

Consider

the

time

deformations

With

these deformations we have

xm(pil(r))

1 = xm(r) = I[-2 .1].

The figures show the functions and deformations:

.. ..

I

-.

1

We see that slope,

-&Z

6(pn)

0

f

when

n

-

is infinite, since the extreme

k

1.

Therefore,

k(xn.xm)

%

0

and

xm

xco.

The next example shows how the measure of deformation can prevent undesirable convergence.

(5.3.3) EXAWPLE: The sequence order to make

x

in

1 + --I 1 = I[1z. 5

lxn(r)-xm(p(r))l

<

1

like the one in the following figure:

does not

converge.

In

we need a time deformation

5.3

Section

219

Decent Path Processes

P *

-m1

/

..1......./.

1 2

infinite

m

n = m

If we take

and

slope.

sequence.

This

We don't

infinite, then the short segment has

m

prevents

want

xm

from

forming

a Cauchy

to converge either. because i t is 1 tending toward the indecent path I ( 3 ) . Our next

*D[O.l] x

it

task is to say which

internal

functions

y

in

are a Kolmogorov infinitesimal from some standard path

in

D[O,l].

After we have done that, we want to see how to

P

view functons on

*D[O.l].

as near functions in

Lifting and

projecting will be done between internal processes on

[O,l]

standard ones on

H

and

similar to the continuous case of the

last section. By

transform

of

k.

of

the definition

p

there is an internal increasing function infinitely

x(r)

",

1

near

y(p(r)).

deformation When standard

x x

such

That and is

postive

y

is,

up

for

all

to

an

in

in

D[O.l].

infinitesimal

such jumps occur at standard times. this, we see that finite jumps of when

k(x.y) Z 0.

k(x.y)

%

0.

with secant slopes

r

in

* [0,1].

infinitesimal

time

are u n t f o r i n l y close.

standard

jumps

that

if

it

cannot

time-being

have

two

standard,

By the last paragraph and y

are finitely separated

The next result gives three ways to say that

noninfinitesimal jumps are finitely separated.

5:

Chapter

220

Paths of Processes

(5.3.4) PROPOSITION 8 DEFINITION:

Let y

has

*

D[O,l].

be an internal function in

y

We say

S-separated jumps if it satisfies the following

equiualent conditions: (a)

For euery positiue infinitesimal internal

an * a *

1

<

rj

(b)

* finite

<

rn = 1

j

<

<

<

r

with

For euery standard s Z r

that

s.

then

rl

<

r2

<

s1

Z

r

and

if

6

s

rl

for

r2

and

r

Z

then

s

y(s)

Z

and

for

1

Also.

y(sl).

j

Z

y(6)

positiue

any

<

m,

< r < rj+l. j PROOF (of equivalence):

and

= 1

with

and for

there exists a

E,

8

r 1 < * * * < rm

<

if

L.

r

(a) 3 (b):

Z

there exists

y(rl) =: y(r2)

positive

<

O = r

with

if

For euery standard positiue standard

s3

(O,l),

in

wheneuer

n.

~ ( 1 - 6 )Z y(1-L)

infinitesimal

<

<

for

is a positiue infinitesimal, y(0)

and

(c)

<

6.

y(r) =: y(rj)

r

such

>

(rj-rj-l)

0 i j

for

rj+l.

there is

0 = rO < r l <

sequence

satisfying

n.

6,

(rj-rjvl)

ly(r)-y(rj)l

0

<

j

<

sequence

a

<

a

>

when

m.

If (b) fails, there are three times

s 1 Z s2 Z 53

and

Y(S1)

*

8.

Yb,)

and

Y(S2)

s1

*

<

s2

Yb3).

6 = s3-sl Z 0. Condition (a) cannot hold f o r this 6. Failure of (b) at an endpoint implies not (a) by similar Let

reasoning. (b) 3 ( c ) :

If (b) holds and

a

is standard and positive,

Section

ro = 0 and

let

condition

‘1

N

‘2

(b) rj+l

N

whenever

s j + l = st

to

find

and

q1

<

rj+l

t

’j+l q2

<

r

<

-

such

sj+l

then

t =: ‘j+1*

8 = min(r

2

ly(r)-y(rj)l

rj+l.

and

such that no sequence of y

:

A 1.

that

.i j-1

)

Use

whenever

y(ql) S y(q2)

then -r

E }

Y(rj+l)

while

=: Y(t).

0.

If (a) fails, there is an infinitesimal

(c) 3 (a):

right

* inf{r N

Condition (b) implies that

rj

221

5.3 Decent Path Processes

variation.

6

6-separated points has infinitesimal 1 * Let B = 5 st( inf{max[ ly(r)-y(rj) I :

rj+l. etc.] : r -r j

j-1

>

This internal

6)).

bounded below by all infinitesimals. hence

B

>

0.

inf

is

Condition

(c) fails for this epsilon.

Notice that an our condition of ‘has

S-continuous internal function satisfies

S-separated jumps.

Perhaps we should say

S-

S-separated jumps, i f i t has any, and otherwise is

continuous‘. function

y

Besides having separated jumps, a near-standard

y E stil(x)

must

take

finite

particular only jump finite amounts.

values,

and

in

This is clear from the

untform approximation

(5.3.5)

where

p

x(r)

y(p(r)).

f o r all

r.

is a n t n f t n i t e s t m a l t i m e d e f o r m a t t o n .

6(p)

Z

0.

This

approximation also makes i t clear that

(5.3.6)

x(r)

= S-lim y(s) s lr

meaning. for every standard positive

e,

there is a standard

Chapter

222

8.

positive

<

ly(s)-x(r)l

such

r

if

that

Paths of Processes

<<

<

s

r+8.

then

Condition (b) is an infinitesimal way

a.

left and right

5:

to say

S-limits exist.

Our discussion of finiteness and separation of jumps proves one implication of the next result.

(5.3.7) PROPOSITION: Let

be an i n t e r n a l f u n c t i o n i n

y

*D[O,l].

Then

y

is n e a r - s t a n d a r d f o r K o l m o g o r o u ' s m e t r i c i f a n d o n l y i f

y

takes finite ualues a n d has these are

the only

with

y

internal functions

euery standard positiue D[O.l]

S-separated jumps.

k(x,y)

<

Moreouer.

s u c h that f o r

t h e r e is a s t a n d a r d

B,

x

in

a.

PROOF : First we show that an internal finite can be approximated within be

standard

<

*--

<

rj

and

rm = 1 r

step

<

with

<

m

8

*

0.

by standard functions.

and

r -r

. i j-1

>

take a

8

r j + l , from (5.3.4)(c).

function

j

positive

a

and

x,(t)

satisfying

= st(y(r,))

xa(l) = st(y(1)).

the piecewise p(sj)

0

sequence

and

for

has

Let

0 = ro

ly(rj)-y(r)l

sj = st(rj)

Let

<

<

5.

&

r1

<

for

and define a

when .i+l are separated by

S j < t < S

Since the

linear function

= rj

S-separated function

sj

p

with corners at

sj

6 ( p ) =: 0.

p (r)-p( s l z 1 . r-s

Also,

lx,(r)-y(p(r))l

<

8.

so

k(xB.y)

We finish the proof by showing that

<

B.

DC0.11

is complete in

5.3

Section

Kolmogorov's

xm's

D[O.l]

If

metric.

functions with the

<

k(y,x,)

form a Cauchy

then

k(x,.xn)

xm

infinite

approximated

n)

by

a

is

2-m-1,

a

0

Z

If we knew

<

k(xn.y)

standard

2

=: 0.

sequence

of

standard

-

k ( x , . ~ ~ + ~ )< 2-m,

for infinite

k(y,x,)

so

sequence

then

sequence.

standard sequence) and also small

223

Decent Path Processes

n

xm

so

x,

in

(in the extended

-n-1

(for sufficiently

In other words, any is

near-standard.

y

(See

Stroyan & Luxemburg [1976, (8.4.28 & 291.) Here is the proof of completeness. there is a time deformation

k ( x , . ~ ~ + ~ ) < 2-m

Since

such that

p,

and

<

6(pm)

For each fixed

2-.

the sequence of repeated compositions

m

0

P m * P m + l O Pm.Pm+2 (n

compositions)

O

forms

Pm+l a

P,."'.

O

Pm+n

uniformly

...

convergent

O

Pm

sequence

of

S-continuous when

n

functions; in fact,

m+n

k =m

so that the iterated compositions remain is infinite. above

shows

Let

am = l i m pm+, n* -m+ 1 6(am) i 2 .

suplxm(am-1 ('))-Xm+l(am+l('))-1 r and

xm

0

om1

I

0

***

o

Also.

p,

and notice that the

u

m

= a m+lPm'

= suPlxm(r)-xm+l(Pm(r))

r is uniformly convergent, say to

x,.

I <

2-

Since

SO

Chapter

224

* 0.

6(am)

k(xm.x,)

+0

once we know

Paths of Processes

5:

x,

E

D[O,l],

that is.

has left and right limits.

This follows easily from condition

(b)

that

of

(Notice

(5.3.4).

condition

directly from an approximating sequence,

so

jumps even if i t didn't start out near such a

(c)

is

x,

inherited

has separated

y.)

(5.3.8) EXAMPLE:

Consider the internal process

given in example (4.3.3) and (0.3.7).

* [0.1]

a step function on t

r

<

t+6t.

Then

i f and only i f

by letting

J(r.w)

is finite and has

J(r)

does, while

J(t)

We can extend

J(r)

J(t.w)

= J(t,o).

is in

*D[O.l].

This stk(Ju).

Later we want t o describe the standard part directly in

terms

t. To show that jumps of

The quantity of

for

S-separated jumps

will mean we can form the Kolmogorov standard part

of

to

J.

(5.3.9)

rl

For each

J(t)

are separated, define

is the random time you wait for the first jump in

t

P[T~ > t]

f,

= "independently draw

= [1-p6t] (t/6t) N

.-pt

,-at.

t

zeros"

5.3

Section

An

225

Decent P a t h Processes

especially

important

feature of

this waiting

time

is

that

there is n o "aging":

PITl

(5.3.10)

>

t+SIT1

>

S]

= "independently draw

t

more zeros"

= [1-pst]

= PITl

Note:

A'.

>

t 6t

t].

For nonempty internal sets the probability of

P[AIA']

= P[A

A

given

ll A']/P[A'].

T h e next jump is just like the last:

P[TZ

>

t+SIT1

=

S]

= "independently draw

= PITl

>

t 6t

zeros"

t].

Finally, by (5.3.9). for sufficiently large infinitesimal

1

(5.3.11)

P[T~ i At]

%

At,

a

or equivalently,

S-lim

P[T~

<

At]/At

= a.

At10

for small finite time the rate of jumps is approximately Let

Am

--.

1 At =

f o r some finite natural number

of sample sequences

w

m.

a.

The set

w i t h two or more jumps during a

At

5:

ChaDter

226

interval and

J(l)

finite consists of

Paths of Processes

the following samples

summarized in the table below. First. consider probability

{o

: T~(w)

1 13

P[J(At)

by

1,

probability of a jump after

Thus

=

[

T

>

is

The

p(At),

independent of

= 0

is approximately

J(l-At)

<

Denote the

At].

1-At & J(l) 2 1 1 .

the probability of our

sample with indecent jumps is

5

~

Second, we may eliminate

[ T ~

1-At

and the probability that

e

At}

p(At).

paths with a jump right before

J(l-At)

<

second piece of

p(At).

the

Next we suppose the

N

jump is o . k . ,

first first:

<

[At

T~

but

<

the second one

is too close

1-At & r 2 5 rl+At & J(l)

to the This

1 23.

probability is

1

P[s2 5 Tl+At

I

T1

= S]PITl =

S]

At<s
1

= p(At)

P[rl = s]

2 11

i p(At)P[J(l-At)

At<s
Our final example before the table is the case where are

T~

[At

<

T~

At-separated,

<

T2-At

probability is

<

but

1-2At & -r3

T~

<

is

r2+At & J(l)

too

close

2 31.

T~

to

and r2: This

Section

5.3

1

227

Decent Path Processes

P[T 3
2> T1+ A t l T 1 = S]PITl =

T

S]

At<s
1

-

P[T 3St 2+Atlr2-t]P[T2=tlT [ A t < ~ < l - 2 A t ] [s+At
= p(At)

11 P[T2=t

I

T

1--S]P[T

1 =S]

-S]p[T,=S].

1-

s t

The inside term.

1

P[T 2-tlT1

= S] =

s+At
At
Again, the outside term,

PITl = 1 At
BAD SAMPLE (0):

C T ~< A t 1

(1):

[ T ~

1-At & J(l)

(2):

[At

T~

(3):

[At

> < <

T~

S]

$ (l-f?-a).

p(At)[l-e

-a,2

PROBABILITY

<

p(At)

6 1-At & T~ < T ~ + A ~ ] < < T ~ - A< ~ 1-2At 8 T~ < T ~ + A ~ ] 6

P(At)

N

L

13

N

~(At)[l-e-~] ~(At)[l-e-~]~

k (k+l):

[ j=1 &

T

>

-At

. i

The

~ &- rk~

T

& T

<+ T

~

(k+l)st

(3)rd was above.

The

innermost

1 p[Tj+l

~ ~

<

Paths of Processes

1-At

+ A8 ~J(l)

2 k+l] 5

~(At)[l-e-~]~.

estimate is obtained in the same manner as

the

t

5:

ChaDter

228

term

= tj+1 I T ,

has

probability

p(At)

is less than

= tjl

while

each

(l-e-a).

j

Thus the total probability of two or more jumps within

l"A,l

I

p(At)[ea+l1.

asymptotic t o has finite

J

We

know

as

(At)a

At

(5.3.11)

from

p(At)

finitely tends to zero. flAm = A .

S-separated paths except on

satisfies the next definition with

At,

is

Since

we see that

6t = At.

(5.3.12) DEFINITION:

X

Let The

* [O.l]

U

:

At-sample x

R

W e say that

x

R

+

of

*IR

X

be is

internal and the

internal

let

A t E U.

function

on

g t u e n by

X(At.o)

,

for

X(kAt.o) X(1.o)

,

for

,

for

X

0 5. r

<

kAt < r r = 1.

2At

<

min[l.(k+l)At]

a.s. has a decent path sample or

J

D-sample

Section

5.3

A

i f t h e r e is a set At 1 bt

229

Decent Path Processes

in

T

P[A] = 0

with

s u c h that the

and a n infinitesimal

Xtt

At-sample paths

are

o Q A.

Kolmogorov-metric near-standard f o r

To be specific we will refer to the sample mesh by saying

X

has a

At-decent

path

The

sample.

function

XAt w

is

of

the

completely determined by its values on

T A = {t E T

Notice

that

k E

I t = k*At. f o r some

indecent

paths

sequence in (5.3.3)] c a n have

[like

the

I"+}

U (1)

extension

D-samples provided oscillations

or close jumps are confined. For example, if X(t,o) = 1 when 1 t = - and zero for all other t, independent of w . then a 2 At-sample which avoids -1 has X A t t 0. Sampling will be needed

later

to

assure

*martingales

that

have

decent

projections, see (5.3.25).

(5.3.13) DEFINITION:

X

Let

be i n t e r n a l a n d a.s. N

The function

IR

X(r,o)

At

= stk(Xo

)(r)

have a from

D-sample

XAt.

[O,l] x R

into

is the d e c e n t p a t h p r o j e c t i o n o f the s a m p l e o f

As remarked above in (5.3.6). we know

N

Also note

X(l)

= st

X(1).

X.

5:

Chapter

230

(5.3.14) DEFINITION: x :

Let

* CO.11

-

postttve Cnftntteatmal.

Xh(r) =

{

*IR

Paths of Processes

be internal a n d Let of

s € UA], 0

<

The formard

<

hZ[x(s)At : r s 1 1 1- F xh(lx(1). r = l

<

r +

f.

<

r

<

1.

At

1 yauerage

be a

x

r

<

is

1-

abbreuiated

r+l/h

1

xh(r) = h

x(s)At

s=r

step A t

Our next result says in detail almost a continuous map from

D[O,l]

that forward averaging into

C[O.l].

is

We make use

of this in the projection theorem for decent paths by reducing i t to limits of continuous paths.

(5.3.15) LEMMA: Let

x.y

€

*D[O.l]

be Kolmogorou near-standard and

sattsfy

k(x.y) 1 0 .

and

a positiue infinitesimal a n d let

At

Let

the forward auerages. (a)

xh

(a)

xh(r) yh

yh(r)

be a f i n i t e natural n u m b e r

xh

and

yh

be

Then

is f i n i t e a n d Z

h

* C0.l).

S-continuous on

f o r alL

r.

0 5 r

<

1.

xh

and

d i f f e r b y a u n i f o r m metric tnftnitestmal.

= l i m st xh(r). h e

(c)

stk(x)(r)

(dl

stk(xh(r)) =

r < l - - .h1

st,(x)(s)ds

for

standard

Section

23 1

5.3 Decent Path Processes

PROOF : (a) then

xh(q)

and

xh(r)

finite bound for (b)

If

Averages of finite functions are finite.

Let

x.

xh(q)

so

1

Then

for

inequality.

We

-

xh(r)

1 yh(r).

Let

the

>>

e

infinitesimal 0

Z

proof

0

by by

be

a

the

and

<

0= r

<

ly(r)-y(rj)l

5

if

IY(P(~)-Y(P(P~)I

so

<

r1

< rm

**-

showing

standard

= 1

<

when

r E [rj V Pj.rj+l A P ~ + ~ ) .then

0

with

2

is an infinitesimal time deformation.

pj

positive

<

8.

V Pj)-(rj

[(r,

9

= P

E [Pj.pj+l).

ly(r)-y(p(r))l m

that

>> r -r > . i j-1

Define

r E [rj.rj+l).

time

triangle

tolerance and apply condition (c) of (5.3.4) to obtain and a sequence

r

times the

2lr-ql

an

[y(p(r))Ih

complete

Z

xh(r).

x(r) 1 y(p(r))

deformation.

[y(p(r))lh

differ at most by

q

-1

0

9

(rj). When

Since

P

A pj)]

j=1 Z

0 and thus

lyh(r)-y

0

ph(r)l

standard tolerance we see that (c)

Since

Since

e.

yh(r)

y

0

ph(r).

for

= S-lim x(s).

stk(x)(r)

is an arbitrary

B

each

standard

s lr e

>>

0.

there exist

Istk(x)(r)-x(s)l

For any finite

(d)

h

r

s1,s2,

<

such that

The standard path

e

r+

Z

s1

<<

for

1

<

stk(x)

such that

s2

s1

<

s

<

s2.

s2,

is Riemann integrable, s o

Chapter

232

this

is

just

Riemann

the

standard

5:

Paths of Processes

infinitesimal characterization

(see Stroyan & Luxemburg

integrals

[1976])

of

plus

the

triangle inequality and a simple estimate with time deformation as in the proof of (b) above.

(5.3.16) THE DECENT PATH PROJECTION THEOREM:

X

Let

haue

infinitesimal

At

path projection

in

6t

X.

of

path

sample

some

for

N

>

T h e map

(a)

At-decent

a

71

X

and let

be the decent

Then:

R

of

o +

into

D[O,l]

P-

is

measurable. (b)

For each

Zr 2

(c)

is

r

in

-

the random uariable

P-measurable.

[O.l] x R

:

[O,l],

IR

is

(Borel[O,l]

x Meas(P))-

measurable. (d)

T h e distributions o f

%

if

is

V("'"''Xm)

are right continuous,

m

continuous function o f 1

<

j

<

m,

that

such

are i n rj z tj

standard

a

variables and

CO.1).

>

bounded

there is an

r +q j

1

for

r

j'

q Z 0

<

j

<

m

t

implies

E[q(Xt'.***.X

m)]

E[q(%rl,*--,?rm)].

Z

PROOF : Let

XAt

be as in (5.3.12).

number and let (Notice that (5.3.15) each except at

Xh(r.o)

Xh

let

h

be the forward

be a finite natural r1 a v e r a g e for

only depends on the sample of

X,(r.o)

r = 1.

(5.2.2). Therefore

is finite and

gh

Zh

X

XAt.

itself.)

S-continuous off

A.

By so,

is a continuous path projection as in

is

233

5.3 Decent Path Processes

Section

N

(Borel x P)-measurable for each for

each

A

outside

0

h.

and

N

X(r,o) = lim Xh(r.w) h+

Since

each

2

r.

is

This proves (c) and (a) and (b) follow

(Borel x P)-measurable. from (c).

t

(d)

S-lim X j = t lr

We know

j

A,

such that

8(w)

1

<

<

j

A€ 2 A

there exists a positive =: X(tj.o)

%(rj.o)

For each standard

m. with

<

P[A,]

a(€)

off a certain

P-null set

j

o 4 A

hence for

Zrj

if

B

>

r +9(o)

i.

0.

infinitesimal

<

t

rj

Z

for

there is an internal

call

t;

= max{$(o)

: w 4

A€} =:

0.

and take an infinitesimal 8 greater than a countable family of 1 has zero a(€). say a(;). Then the set A' = n A 1 /P probability, and for o E A '

r +9 j

<

tj =: rj,

1

<

N

j

5 m

imply

X(r.w)

=: X(t,w);

in other words, the measurable projection of

q(X

tl

.-**.Xtm

t

equals

q(grl,-**,~rm) a.s.

Since

q(xt'.---.x

m) E

sLl(P)

the integral of its projection is infinitely close to its sum.

(5.3.17) EXERCISE: Let

J(t.w)

be the Bernouilli process infinitesimally

from a Poisson as in (0.3."). distribution (from both sides).

Prove that

J

far

is continuous in

Give an example of an internal

separated jump process with a left discontinuity in distribution.

234

5:

Chapter

Paths of Processes

Our next example needs the following standard result.

We

include the (*-transformed) proof for the reader's convenience.

(5.3.18) SKOROHOD'S LEMMA: Let

X(t) = B[6X(s)

{6X(s)

where

: s E

Y}

:

0

<

s

*

is a

<

X(0) = 0.

step at].

t,

independent f a m i l y .

Then

PROOF :

p[(X(t)(

> €1 2

>

>_ p[(X(t)l

B

>

& maxlX(s)(

2 ~ 1

slt

1 p[(X(t)(

>

€

&

=

T

S]

slt

2

1 P[lx(t)-x(s)l

<

E

<

ElPCT

81

T

=

S]

slt

2

1 PCIX(t)-X(s)l

= s1

slt

2 {I - max ~[lx(t)-x(s)l

> €1)

S
'j

P[T

=

51

sSt

>_ {I - max ~ [ I ~ ( t ) - ~ ( s ) l > e l } ~ [ m a x l ~ ( s )>l 2 e l . s
sSt

(5.3.19) EXAMPLES: Let sum of Let

X(t) = B[6X(s)

* independent

for

6(X)'s

6f(u) = E[exp(iu6X)]

infinitesimal a

: 0

s

<

t.

step st],

X(0) = 0. be a

all with the same distribution.

be the characteristic function of the

increment, cf.

finite

<

(4.3.5).

S-continuous

If

function

6f(u) = 1 + 6t $(u) $(u).

then

the

Section

5.3

characteristic

= [l+6t'(u)] a.s.

function

(4.1.3).

last remark.

X(t)

of

=: et'(u)

6 ' t

by

235

Decent Path Processes

is

for

>>

t

(4.3.4)

f(t,u)

X(t)

S-continuous and

Proposition

0,

is finite

is the converse of

the

These processes are infinitely close to stationary

independent

increment

standard

processes.

We

have

seen

the

examples of Brownian motion. the Poisson process and the Cauchy process

(4.3.6-7)

X(t)

moments, e.g., arises

this

above.

way

by

Observe

SL1(R).

C

that

the

for

has

no

X(t) = ct

Deterministic drift 6X(s) = c6t

taking

latter

all

s

with

probability one.

Also, a deterministic process is independent iuct of anything else and f(t,u) = e in this case.

The decent

following

paths.

Skohorod's

We

results show only

prove

lemma works.

More

that the

internal

easy

part

'versions' have to

indicate how

'general' sampling results with

weaker hypotheses and conclusions are proved later.

(5.3.20) LEMMA:

be

a

: 0 < s < t , step at]. * independent identically

X(t) = B[6X(s)

Let

sum

of

infinttesimal

increments

finite a.s. f o r

0

<

aboue is ftnite and

6X(s)

1.

t

such

X(0) = 0 . distributed

that

or equivalently.

S-continuous.

X(t)

ts

'(u)

as

Then the paths o f

X

are ftnite a . s .

PROOF : I t won't do to have

infinite: up.

We

(4.1.4):

T

X(t.w)

finite i f

X(t+bt,w)

is

is uncountable so sets of measure zero could build

estimate

the

easy

term

in

Skorohod's

lemma

using

236

Chapter

>

P[IX(s)l

Since

$(v)

Z

for

0.

probability

e]

above

<

ae

<

aeJO

>

P[max(X(t)(

Z

>

0.

s/6 t

]dv

l/€

v

Z

(el'(v)

I-l)dv.

when

0,

is

2n]

Paths of Processes

[l-Re[l+Gt$(v)]

is

E

infinite

infinitesimal.

max {P[lX(t)l > b]} Z 0 whenever b O
So

5:

0 whenever

the

Therefore

is infinite. By (5.3.18) 6X(s).

n

is infinite.

For finite

t
m

natural numbers

A m = {a

:

let

maxlX(t)

I >

2m)

and

A.

A.

and

= tl A m .

t
The paths of

x

are finite off

PIAo]

= 0.

This

proves the lemma.

Next. we show that the paths of at the endpoints. epsilon.

Let

e

>>

X

a.s.

are

S-continuous

This time we use Skorohod's lemma with small 0 be standard.

since the distribution of

X(t)

-

X(s)

is the same as

X(t-s).

Section

237

5.3 Decent Path Processes

Using (4.1.4) again, when

>

P[lX(s)l

el

<

<

s

we see

s/6t

{l-Re[l+6t$(v)]

ae

}dv.

s o that

In

whenever n is max P[ IX(s)l > E ] Z 0 s
particular,

infinite.

max IX(s)l > 281 tends to zero as O<s
of

P[

S-limits.

S-continuous

at

zero

A

a.s.

similar

argument

m

tends

X

are

yields

a.s.

continuity at one. In

order

to

make

a

sum

of

stationary

independent

infinitesimal increments compatible with the measurable theory of time evolution (and notational conventions) in the remaining chapters we shall suppose that

X(t) = ][6X(s)

= 0 and

X(0)

: 0

<

s

<

t. step at]

where there is a fixed internal function

We must select course. classical

7

carefully so that

In chapter 4 we made laws

can

be

7

2

:

W

4

*IR

and

is finite 8.s.. of

the vague claim that all the

represented

this

way,

but

we

have

explicitly shown that Brownian motions. Poisson processes and

238

5:

ChaDter

Paths of Processes

Cauchy process (4.3.6) do arise this way.

The latter kind are

integrable and so they offer certain

not

which

will

we

sidestep

"semimartingale."

by

showing

(See (7.3.3).)

technical

that

such

jumps.

'natural classical

is

a

The paths of semimartingales in

general must be sampled for infinitesimals separated

X

an

There we also prove a general

semimartingale lifting theorem.

have

challenges

Our next

undefended

object' has a

>

6t

in order to

claim

says that a

At

'nice internal version'.

shall not defend the claim because we must work with

We

At-samples

for other reasons below.

(5.3.21) PROPOSITION:

X(0) = 0

Let step

be

at]

X(t) = B[-r(us+6t)

and

a

sum

of

7

-

W

:

*W.

finite a.s. on

T.

and

path projection

its decent

tndependent

o

<

5 ro

is

an

increment

<

r1

Gn denko Gihman

***

<

and

wishing

Kolmogorov Skorohod

technical help with motivation

process

for

%(r)

is a

[O.l],

on

and s.

to explore

D[O.l].

introducing

X(t)

t,

is

the

is.

<

of

r+s 5 1.

this claim

is referred

classical

and

if

5 j 5 m)

distribution

0 5 r

Ethier

stationary

that : 1

[1954, 19681 for

[1974] or

<

dt-decent path sample

{%(rj)-%(rJ-l)

family

s

identicaLLy

Assume that

has a

only depends on

reader and

X

r l 5 1.

tndependent

[%(r+s)-%(r)]

The

Then

<

6X(t,o) = -r(ut+6t).

distributed infinitesimal increments for a singLe internaL

0

:

* independent

Kurtz

laws

to nd

[1980] for

This claim is not used except as semimartingales,

where

different

Section

5.3

239

Decent Path Processes

estimates play the role of Skorohod‘s lemma.

(5.3.22) DEFINITION:

Y

Let

X

:

U

[O,l] x R

:

R + *IR

x

path sample. D-Ltfttng

be

Y

be

a

function

internal and a.s.

X

We say

of

IR

4

ts a

if

the

Y

has a

haue a

and

Let

At-decent

path ltfting o r

At-decent

2

D-projection

Y

and

are

tndtstingutshable.

I f we show that of

that

lifting

k*At. k E IN,

At-lifting, then the restriction

of

to multiples

is a

infinitesimal

vt =

the same projection.

This

another

vt-lifting with

seems a little silly, but we’ll see later that we sometimes need to make

At

>

the lifting.

6t

in order to maintain some side condition on

The Projection Theorem says that

works for every infinitesimal

2

At

6t.

Y

Recall the Doob’s definition (5.2.6): process i f each section,

Yr.

the projection

is a stochastic

P-measurable.

is

(5.3.23) THE DECENT PATH LIFTING THEOREM:

If a stochastic process decent

paths

A t 2 6t

tn

in

U,

D[O,l],

Y

has a

Y

:

then

CO.13

x

a

for each

4

IR

a.s. has

tnfinttestmal

At-decent path ltfttng.

PROOF : Assume

Yw E D[O.l]

We want to show that the map into

D[O.l].

(1) each

Yr

What is

-

except for

we

w

have

P-measurable

YW

to and

w

in is

work

A

where

P[A] = 0.

P-measurable as a map with

are

(2) evaluation

the

facts:

y + y(r)

240

ChaDter

from

D[O.l]

IR

into

5:

Paths of Processes

is Bore1 measurable.

The second fact can

be proved by forward integral averaging in a manner analogous to

(5.3.16).

Evaluation of

each average

is continuous and

the

evaluated averages tend to the evaluation. Let

{q,}

For a fixed

enumerate the rationals in x

in

J(m.e:x) = {y

V

Let

D[O,l]

for some

such that

q

€

and

Iqj-fjl

€

<

0

<

A[O.l].6(p)

6(p)

<

E.

1

<

j

for

8 ,

q 1 = 1.

with

define the set

f = ( f l , - - - , f m ) in

with

p

>

a

3 p

:

be the set of points

fj = x(p(qj))

of

D[O,l]

[O,l]

Let

I

m.

a

IRm U

such that be the set

U

The set

is

open and since evaluation is measurable,

{W

In other words, and

x.

:

y(qj.W)

E U}

is

Y;l(J(m.e;x))

P-measurable.

is

P-measurable for each

m.

E

The lemma following the proof shows that open balls in

the Kolmogorov metric are in the sigma algebra generated by the J-sets.

Since

D[O.l]

is

rational values are dense), Let

K E *D[O.l]

separable

Yo

(step

is measurable.

be the set of

*right

step functions with jumps at the points of The set is internal, 1-1

functions with

continuous internal

Y

(if anywhere).

S-dense for the Kolmogorov metric and in

correspondence with internal functions defined on

(5.1.6).

Yo

has a metric

restriction of the lifting to

lifting with values in

T

is a

II. K.

By The

6t-decent path lifting.

Section

5.3

24 1

Decent Path Processes

The effect of the next technical lemma is that the s algebra generated by

D[O

9-sets is the Bore1 algebra of

The lemma finishes the proof of the lifting theorem.

(5.3.24) LEMMA:

{y E D[O.l]

:

k(y.x)

<

<

then

= U[n J(m.8-8;x) 9 m

E)

:

n

8 E Q

(O,E)].

PROOF :

If

k(y.x)

rational

<

6(p)

8.

a-8

a.

there

so

and

is

a

<

ly(r)-x(p(r))l

hence in particular for rational Now we

that

1

I

<

j

6(pm)

m.

<

a-9

and

Since any

deformation

B-9

for all

A

6(p)

<

with

p

r

[O.l].

in

r.

Fix

8

and suppose

choose a time deformation Ix(pm(q,))-y(qj)I

6-9,

*extension

<

p,

for

6-8.

of

the

p,

is Lipschitz, and the

pn

A

p = pn.

infinitesimal hull

6-8

<

6(pn)

for some positive

time

in the

pn

A

deformation with

m

For each

sequence satisfies

<

a-8

show the other inclusion.

y E ll B(m.6-8;x). m so

<

k(y,x)

E--8.

(0.3.22). is a Also. we know

standard

time

Ix(pn(qj))-y(qj)l

for all standard rationals, in fact, for

1

I

j

<

n,

A

infinite.

For each of these points either

<

Ix(;(qj)-)-y(qj)I

a-8

and

or k(x.y)

<

6-8.

I a-8.

so

Ix(p(qj))-y(qj)

suplx(;(r))-y(r)I

I

n

I 6-8

This proves the inclusion.

Internal processes that almost surely have left and right S-limits always have decent path samples.

Chapter

242

Paths of Processes

5:

(5.3.25) THE S-LIMIT LEMMA: Let

X

be an internal process and suppose there is a

set

A

all

r E [O.l].

R

= 0

P[A]

with

S -1im X(t,w)

Then there ts a posttiue

X

that

has

a

S- lim X(t.w)

and

ttr

A'

uA

path

exist.

tlr vt = k-At

infinitesimal

vt-decent

e A , then for

w

such that i f

sample with

such

projection

j i ( r , w ) = S- lim X(t.o). uA tlr PROOF : Y(r.o) = S-lim X(t.w)

Let set

A 1 2 A . and

S-limits. for each 1

-m

and

zero otherwise.

Y.

path lifting of

Since

Z

w E

Let

Z(t.w)

X

and

A 1 , for a Loeb null

Am E U A

m E IN. there is a

such that Am 1 1

max IX(kAm) - Z(kA,) I V IX(1) - Z ( 1 ) 1 ) > 1
comprehension.

,]< ;;. UA

-

N

using

The probability inequality above remains true up

to some infinite

n1

smaller

n.

infinite

while by

Ak

..,I

Robinson's

r;

remains true up to a sequential

.

Since Z has a fit-decent path n vt-decent path sample. We know X(t) Z Z(t)

X

be a fit-decent

have the same right

P[(

Extend

vt = A

for

lemma.

Let

sample, i t has a for

t = k-vt, s o

has decent paths for a sample along multiples of

vt.

This

proves the lemma. Our

next

topic

is

lifting

(continuous

or)

decent

processes with a side condition of uniform integrability.

path

Section

5.3

243

Decent Path Processes

(5.3.26) DEFINITION:

{Xa

A family o f random variables. is E

called

>

EC

IXa(W)

(w) = 1

tf

a.

Ixa I > m l

IX,C.)l

Q

M

in

II{

Ixal>m}

provided

IN

that

L1(P)

for every

m 1 M

such that f o r a l l

<

(w)]

>

IX,(u)l

( T h e indicator

8.

=

and

m

0.

if

i m.1

integrability

Uniform

X Z = X I

integrable

there exists

0.

a n d all

I

uniformly

a E A}

:

{

means

converge in

Ix,lm

a.

infinity, uniformly in integrability

of

a

that

L 1 -norm to

the

Xa

truncations

m

as

tends to

I t is not hard to show that uniform

{Xo}

family

is

equivalent

to

the

two

conditions: (a) there is a (b) every

L 1 -norms

the b

the

of

family are

such that for all

u n t f o r m L y bounded,

<

E[lX,l]

a.

the family is u n t f o r m L y absolutely B

>

measurable

0

there

E

A

set

8

exists

R.

if

P[A]

>

0

<

b,

conttnuous.

such

8,

and

that

then,

for

for

for every

all

a.

lxallAl < e * If and each family

We

{Ya Ya

{ya

A}

: a E

want

S-integrable. Lemma (1.6.2) implies that the

is

: a E

to

is a n internal family of random variables

A}

is uniformly integrable.

show a

real-instant-sections

correspondence

of

a

sections of a corresponding stochasttc process

X

:

stochastic

between

the

process

and

internal process.

[O,l] x R

-B

R

We

family the say

of

time

that a

is untforrnly tntegrable

Chapter

244

if

the

family

of

{Xr

sections

Paths of Processes

5:

: r E

[O.l])

is

uniformly

integrable.

(5.3.27) THE

A

S-INTEGRABLE DECENT PATH LIFTING THEOREM:

stochastic

Y

process

x R +

[O,l]

:

R

is

uniformly integrable and a.s. has decent paths if and only i f there is an infinitesimal

path lifting

T.

X

the section

Y

for

Xt

is

At

in

U

and a

At-decent t = kAt

such that whenever

in

S-integrable.

PROOF : X.

For each

%(r)

= stX(t)

Suppose there is such an is a

t = kAt =: r

such that

(1.6.2) implies that the whole

Y

uniformly integrable, hence Conversely, suppose has

decent

paths.

The

Y

family

r

in

= Y(r).

{stX(t)

[O.l] a.s. : t E

is

is uniformly integrable and a.s.

family

of

sections

and

left-limits-

continues to be uniformly integrable by Fatou's lemma.

M(e)

U}

Lemma

is.

sections,

Let

there

be a function such that i f

m ) M(t).

then

Section

5.3

Let

Z

:

T

x

R + *R

be a

and define truncations for each

while f o r each finite

These

truncations

any

= Yn(st(t))-)]

t,

= 1.

lit-decent path lifting of n

in

Y

*IN,

n,

continue

appropriate senses and

For

245

Decent Path Processes

Zn

to

lifts

P[st(Zn(t))

have

Yn

decent

paths

for finite

n.

= Yn(st(t))

so that

Hence we see that the internal sets

or

in

the

st(Zn(t))

ChaDter

246

n’s

contain all finite n

n[I

: p € IN]. P saturation (0.4). €

past

p

Paths of Processes

5:

when

p

is finite.

Let

The countable intersection is nonempty by The process

decent path lifting of

X = Zn

is an

S-integrable

Y.

(5.3.28) EXERCISE:

If D-lifting

Y of

is an

8.5.

continuous

Y.

show

that

X

process is a.s.

and

X

is a

S-continuous.

Logically, we could have done without section (5.2) and derived those results from this exercise.

Re-state (5.3.26) for an a.s.

continuous uniformly integrable process.

247

Lebesgue and Bore1 Path Processes

(5.4)

We continue to work on the hyperfinite evolution scheme defined at the beginning of the chapter, P,

..*.1}.

etc.

The main

results

R = W",

T = {6t,26t,

of

section are

this

extensions of Theorems (2.1.4) and (2.2.6) to the time variable of a stochastic process. [O.l].

Let

multiplied

denote

6t

by

complete product of

6t@6P

denote

6t@6P(t.o)

6t x P

Let

counting

and

6t

let

internal

= 6t*6P(w)

and let

on

T

denote

its

measure 6t

denote the (conventional)

the (ordinary) measures

the

extension.

denote Lebesgue measure on

internal

the constant

hyperfinite extension.

X

Let

measure 6t@6P

with

P.

and

6t

weight

Let

function

denote its hyperfinite

(See Chapter 3.)

Our first result is the analogue of the theorem for one variable that says standard parts of internal sets are closed. Thruout the section we need the half-standard-part function

:

s

T

x

-

R

[O.l] x R ,

s(t.o)

= (st(t),o).

(5.4.1) LEMMA: G

Let

= (st(t).w).

Y

x

R

T h e set

be

an

s(b)

Internal I s

set

and

(Borel[O.l]

s(t.w)

x Loeb(R))

-measurable.

PROOF :

Let endpoint (t.w)

E b}

{I,}

be an enumeration of the standard open rational

intervals and let

in

IR.

Let

Em = {o : 3t E

*Im

with

em AC

where

em

so

= ([O,l]

E

(t,o) € &

since

it

which makes

r e Im

If

we still have r Q Im

A.

(Bore1 x Loeb)-measurable.

then

o E Gm.

(1; x E m ) .

denotes the complement of

is

Im.

C

em u

x

for the following reasons. r

5: Paths of Processes

Chapter

248

t E

is

s(t.w)

E

* Im

(t.0)

Im

(tm.o) E 1 .

such that that

r E

E E ,

the

t E

* Im

and

(t.w)

then

(r.0) E

nEm

E

8.

E

= {r}.

decreasing.

choose

Extend

:I

x &:.

and for the

*

tm

Next, select a sub-subsequence

is monotone

Ir-tkl

interval,

satisfies

(r.0)

nIm

and

If

In the final case where

For the opposite inclusion, suppose subsequence with

has

* Im

[O.l] x dm.

has

open

= (r.0)

m.

and fix

E s(&)

standard

t E

is internal,

This construction gives

(r.0)

a

but some

(r.0)

and no

Let

Em

Each

Im

tk

so

to an

tk

internal sequence and consider the internal set of indices

{n

An infinite

*I:

E

n

(Vk

€

*#)[O <

k

<

E &

n 3 [(t,,~)

from this set satisfies

(r.0) = (st(tn).o).

This proves the lemma.

Let

(M,p)

be a metric space entity,

our results for functions

X

:

C0.11 x R

+

M . p E 91.

M,

We state

since i t is no

Section

5.4

Lebeseue and

249

Bore1 Path Processes

harder than for real values.

(5.4.2) DEFINITION: Ciuen a

X

function

= st Y(t,o) P o f x.

6t x P

a.e.

[O.l]

M

Y : T X R +

function

:

*

R

x

+

such

M.

an

internal

X(st(t).w)

that

is called a Lebesgue

lifting

Implicit in the definition of Lebesgue lifting is a weak notion of almost Y(t.o)

Y(s.w)

“N

If

S-continuity.

“N

= st(s)

st(t)

= r

then

at least a.s. in the product space.

X(r,w),

(5.4.3) THEOREM: (M,p)

Let entity.

K

Let

st (K) = M.

if

Y(t.w)

= stp(Y(t.o))

complete

:

U.S.

and

T x

separable

metric

space

S-dense internal subset o f

*M ,

[O.l] x R + M

is

X(r.w)

function

(hxP)-measurable function

a

be a n

A

P

be

only R +

6t x P .

has a Lebesgue Lifting in

K

if

:

there such

that

is

an

X(st(t).o)

that

is.

if

internal

and

only

if

K.

The proof of this result, given below, is based on the two variable extension of Theorem (2.1.2) as follows. if

Y E [0,11

-1 (Y). = s

where

x

n. s

then

{(t.w)

E

Y

x

Notice that

n: (st(t),o)

E

Y)

is the half-standard-part map from above.

250

5:

Chapter

Paths of Processes

( 5 . 4 . 4 ) PROPOSITION:

For

each

9

set

c

[O.l] x R ,

the

following

conditions are equiualent:

(a)

In

Y

(b)

s

(c)

s

this

( A x P)-measurable.

is

-1 -1

(9) is

(6t x P ) - m e a s u r a b l e .

( 9 ) is

6tC96P-measurable.

case,

preseruing,

the

s(t.o)

map

(at x P)~-'(Y)

= (st(t).w)

is

measure

= A x P(Y).

PROOF :

Y = [a.b] x A.

If = n[a-

m

-.m 1

1 b+ -I

A,

so

(A x P)-complete

S i n c e both

s

= h x P(Y).

6 t x P(s-'(9))

is a

x

A

where

-1

is internal,

is

(9)

then

s

-1

measurable

(9) and

The set

sigma algebra

Bore1 x Loeb.

containing

measures a r e complete this shows that

(a) implies

(b). Condition (b) implies (c) by (3.1.2). (c 3 a):

Suppose

s

-1

(9)

a r e chains of internal sets such

that

%? = u'&,

6 t @ 6 P ( ' & )+ 6t@6P(3)

E s-l(Y).

= 1.

is

E

bt@bP-measurable. T2

E

- - a

W e k n o w that

J1

and

J = UJm E s('&~)

s

so there

-1

5D2

E

c (9 ) ,

and

~(9,)

I

measurable.

Also,

1 = 6t@6P(%) + 6t@6P(11)) -1

<

6t@6P[s

(s('&)]

+ bt@bP[s -1 (S(11)))l.

0 . 0

and are

Section

From

5.4

Lebesaue and

the beginning

of our proof h x P[s(%)]

(3.1.2) we know that

J.

same for

1

and s

<

Y

25 1

Bore1 Path Processes

[before

(a) implies

= 6t@6P[s-'(s(%))]

(b)]

and

and the

Hence,

h x P[s(%)]

+ h

is measurable.

<

x P[s(J)]

+ h x

h x P[Y]

P[YC]

Again, the first part of our proof shows

is measure preserving.

This proves (5.4.4).

PROOF of (5.4.3):

X

Let

( h x P)-measurable

be

and define

Z(t.w) = X(st(t).o).

By

(5.4.4).

2

is

6t@6P-measurable.

st (Y) = Z a.s. P Conversely, suppose stp[Y(t.w)]

2

has a

Y.

metric lifting

(t,w)

in

d

stp[Y]

is

(6t@aP)-measurable.

{(t.w)

(5.1.6).

By

a

(at x P)-null

: St [Y(t,w)]

P

E

U}\#

= X(st(t),w)

set.

Since

U

Let

= {(t.w)

A.

: X(st(t),o)

E

: X(st(t),w)

E

: st [Y(t.o)]

is

U}\N

X(r.o) E U}

:

= {(t.w)

{(t,w)

for

is internal,

be open in

E s-1 ((r.0)

By completeness and (5.4.4). X

Y

except

P

E

U}

€

U)

u

N.

( A x P)-measurable.

Notice that re have neither assumed nor concluded that

X

5: Paths of Processes

Chapter

252

is a stochastic process in the sense of (5.2.6).

For reference, here ( A x P)-setting.

is

the analog of

(2.1.4)(b)

in the

We state a slight extension for vector values.

Extensions of this result are useful in the study of stochastic processes.

(5.4.5) S-INTEGRABLE LEBESGUE LIFTING THEOREM: Let

Eld,

be an integer. let

d

p 2 1

and let

x R

Eld

an

internal

llXll

has

Y

(6t@GP)-S-integrable

Lp(A

x

x R +

and

that is, if and only i f

P)

denote the norm on

A function

be real.

E

U

:

II II

:

[O.l]

i f and only i f there is

*IR

such

has an

llYllp

that

= X(st(t),w)

st[Y(t,o)] X

X

is a.s..

SLP-Lebesgue-lifting.

We leave the proof as an exercise.

If the vector values

bother you, first show the scalar case and then use the max 1 (*~ d * * , x)II = maxlxjl. norm on md, I I .

(5.4.6)

PROPOSITION: I f

X

has a

At-decent path lifting

Y.

a Lebesgue lifting f o r the internal measure

counting on with

T

(t E

: t = kAt,

k E *IN}

times

Y

is

A t x 6P

of

then

At

crossed

6P.

PROOF : By extending kAt

and

(k+l)At.

Y

to be

Y(kAt)

we may assume that

for points of

Y

is a

t

between

&it-lifting. We

5.4 Lebesaue and

Section

wish to show that

X

Since measurable,

Z.

Borel Path Processes

= st Y(t,o)

X(st(t),o)

253

a.s.

At@BP.

has a decent path lifting, i t is = st Z(t,o)

X(st(t),w)

so

Also, for a.a.

a.s., for some internal

1 lim st Y(t+--)

0,

(Borel x P)-

= X(st(t)).

It follows

m-rO

that, except for a null set. for every

I

in in P 1 (Y(t+--.w)

{n E

*IN

such that i f

-

<

Z(t,w)l

: Vm E

*#.m

1 -. P

<

in

m

is finite and

m

I

>

m

then

P’ Therefore, the internal sets

. Choose n E nI[m P‘nP 1. P is a Lebesgue lifting of X.

Y(t)

there is an

2 p] 1

m .( n 3 (at x P)[IY(t+;;;)1Z(t)l

P -

contains an infinite and hence

p

n

then

I p} 1 Y(t+;) 1

(5.4.7) PROPOSITION: each

For

B

set

E [O,l]

the

x R.

following

conditions are equiualent: (a)

9B

(b)

s

(c)

s-l($)

(BorelCO.11 x Loeb(R))-measurable.

is -1

(9)

is

(Loeb(H)

is

Loeb(H x R)-measurable.

x

Loeb(R))-measurable.

PROOF : (a)

A

E

Loeb,

implies

(b):

If

8 = B x A,

for

then Henson’s theorem (2.2.6) says

x A € Loeb(U)

x Loeb(R).

Finite

rectangles also have this property.

disjoint

Loeb(T)

x

Loeb(R).

is

trivial

-1 -1 s (9) = st (B) unions

of

such

A simple application of the

Monotone Class Lemma (3.3.4) shows that every has s-1 (9) E Loeb x Loeb. (b) implies (c)

B E Borel and

because

9B E Borel x Loeb

Loeb(T x

n)

3

254

Chapter

(c) implies (a):

We know that

5:

s-l($)

T x R

derived from internal subsets of

Paths of Processes

(See (2.2.1) and the following remark.)

CS-'(~)]~

and

are

by Souslin operations. Let

and

9's

for internal

and

Borel x Loeb

are

By (5.4.7).

$'s.

s(sUln)

and

S(*uln)

sets and since

and

it

suffices

to

intersections of analog

Borel x Loeb.

9 Let

commutes

S

Separation

Theorem

x R.

holds

the for

if

?k

and

aC

(BorelCO.11 x Loeb(R))-measurable.

F = C0.11

x

We claim that

If

that

the family o f compact x internal rectangles.

is a compact subset of

R.

(2.2.3)

The necessary separation theorem says:

is

decreasing

sets (compare (2.2.4)) and

R

and let

5

be the family

K

all disjoint finite unions of rectangles

A

with

The proof of the first fact is similar to that of

are analytic ouer then

that

internal

Lusin's

of

(2.2.4).

show

{Kn x An}

[O.l]

1

&

and

A E

&

x

A.

1

where

&

K

of € I

is an internal event

is a semicompact paving of

C0.11

has the finite intersection property, then

Section

{K,}

both

x

{An}

and

nKn

[O.l],

n(Kn

Lebesvue and

5.4

An)

# 0

255

Borel Path Processes

have the property.

nAn

R.

and by saturation on

By compactness on # 0.

Therefore,

Theorem (2.2.3) now gives the separation result

# 0.

stated above. The reason for proving Proposition (5.4.7) is to give the two-variable

analog

(2.2.8) (which

for

says

that

st-l(B) E Loeb(P)). measurable. Loeb

following

g : T

by

function

B E Borel[O,l]

R

(2.2.8).

measurable

Also,

t % s

and

f(r) = g(st-'(r))

is

well

g

is

is

Borel is

infinitesimally

Conversely. i f

and

Borel

is

g

g(t) = g(s).

implies defined

if

only

then

measurable

by

We did not give this version of (2.2.6) in Chapter 2 ,

(2.2.6).

but instead showed how to replace was

and

of

g(t) = f(st(t))

given by

S-continuous (but usually not internal). Loeb

if

version

f : [O.l] + R

Suppose

Then

measurable

the

S-continuous

everywhere

with

and

respect

satisfied

g

with an internal

= st(h(t))

f(st(t))

to various

h

hyperfinite

that almost

measures

(see

section 2.1 for Lebesgue-like measures and 2.3 for Borel-like ones).

The corresponding

internal path

lifting

is postponed

until Chapter 7.

(5.4.8) PROPOSITION: Let

G : [0.1] x R

B

be

any

function.

The

following are equiualent conditions: (a)

G(r.o)

is

(Borel[O.l]

x Loeb(R))-measurable.

(b)

G(st(t),w)

is

(Loeb(T) x Loeb(R))-measurable.

(c)

G(st(t).o)

is

Loeb(P x n)-measurable.

Chapter

256

Paths of Processes

5:

PROOF :

The

above

result

<

% = {(r,o) : G(r,w)

Apply (5.4.8) to the sets

can

be

extended

a}.

(Borel x P)-

to

measurable functions by the following

(5.4.9) PROPOSITION:

A is

(BorelCO.11 x Meas(P))-measurable

indistinguishable

measurable

function.

(Borel[O.l] A

set

from

x

(Borel[O.l]

a

That

is,

Loeb(R))-measurable w C A.

such that if

G

H,

function, Loeb(R))-

x

there

exists

and a

a

Loeb(R)-null

then

H(r,w) = G(r.o). PROOF : By the Monotone Class Lemma (3.3.4) and that

if

9 E (Borel[O.l]

Loeb(R)-null that if Let

w

set

Q A,

Hm

A

x

and a

Meas(P)).

-

be a sequence of simple

functions such that

Hm

H

measurable functions r.

Let

G = lim sup G

Gm m'

Am

such that Off

Ipw

UAm,

x

=

Loeb(R))

is

a

such

Yw.

(See the proof of

Hm's.)

and simple Gm = Hm

G = H

there

(Borel x P)-measurable

everywhere.

(1.3.9) for an example of constructing paragraph there are null sets

then

9 E (Borel[O,l]

then the sections agree,

(1.2.13) we see

By the first (Borel x Loeb)-

off

for all

Am r.

for all

257

(5.5) Beyond

and Scalar Values

[O.l]

This section briefly describes extensions of the results o f this chapter.

It can be

skipped without

There are two kinds of extensions:

loss of continuity.

the range and the domain.

Extending the path spaces to have values in a complete separable metric space presents no serious difficulties. paths

*C([O,l],M)

in

standard in

*M.

*D([O.l].M)

or

Near-standard

take values

rather than just finite values.

near-

Things are a

little more abstract, but the ideas are pretty much the same. Extending the domain for continuous functions from IR

[O.m). on

x

IRd

or even

in

*C([O,m),M)

finite rather D([O,m).W)

presents no problem either. look the same on

than in

[O,l]

* [O,l].

x(t)

Conditions

except

The extension

t

D[O,m)

is more technical.

(5.5.1) DEFINITION: Let

(M,p)

Let

C([O.m).M)

C0.m)

i n t o IN}.

be

is

space.

a continuous function from

A metric for measuring u n i f o r m conuergence

on c o m p a c t s u b s e t s o f

W e shall catl

a complete separable metric

= {x : x

c

[O.m)

is for

x.y E C([O,m),M).

the compact convergence metric.

to

is

or

5:

Chapter

258

Paths of Processes

(5.5.2) EXERCISE:

Let

x,y

E

with values in

*C([O,l],M) be *M. Show that

c(x,y)

%

functions

0, which we write as

P

C

x

*continuous

internal

y.

%

i f and only i f

* C0.m).

in

M = lR.

%

y(t)

for all finite positive t

P

(x(t)

%

y(t)

means

p(x(t),y(t))

give an example of an internal

all finite

t.

your

*C[O.m)?

x

in

but

The notion of x

x(t)

is

= 1

x(T)

x

0.)

For

0

for

x(t)

with

Z

T.

for some infinite

Why is

S-continuity is nearly the same as before,

S-conttnuous

at

b

in

*[O,m)

if

t

b

%

implies

P

x(t)

%

x(b).

(5.5.3) PROPOSITION:

A

x

function

conuergence near-standard for each

-

t.

ftntte t

t f and only

and

In thts c a s e ,

x

is

stc(x)

tf

compact

ts

x(t)

E

S-continuous

ns (*M) P

at

i s giuen externally by

P

result

plays

the

same

role

as

(5.1.3) does

The next result is the extension of (5.2.2).

need to extend the hyperfinite evolution scheme.

(5.5.4) FRAMEWORK FOR HYPERFINITE EVOLUTION OVER

C0.m):

Let

T = {t where

each

st (x(t)).

This C[O.l].

t

fintte

*C([O,m).M)

tn

E

*R

: t

= k.8t. k

E

*M.

t

1 < z}

for

First we

5.5

Section

259

Bevond rO.11 and Scalar Values

1 6t = n'

-

for some infinite

n = h!

in

*IN.

Let

R = W

T

= { w : w is an internal map from T into W}

where

W = { k €

*I N :

l
Let

P = uniform internal counting measure on R ,

If

*M

x : T -

*M

standard values in

is

S-continuous

for finite

is well-defined as a map from

takes

p-near-

then

t.

c0.m)

and

into

M.

Denote this map

u

by

x(r),

the continuous path projection o f

x.

(5.5.5) EXERCISE: Show that the map just described is a continuous (standard) function.

(5.5.6) PROPOSITION: Let

X

:

H x R

-+

Suppose there is a set

tf

o

e A.

then

Xw(t)

*R

A t s

be

E R

an

with

internal

function.

P(A) = 0

such that

p-near-standard

and

5:

Chapter

260

S-continuous for finite path

T.

in

t

T h e n the continuous

g(r,w) = st ~x(st-l(r).w)l

projection

is

P

(Borel[O.m)xMeas(P))-measurable C0.m) x R

Paths o f Processes

as

a

function

from

M.

into

To prove the extension of the

S-continuous lifting theroem

(5.2.7). we need to know the fact that separability o f the range

M

makes

C([O.m).M)

C([O,m).nU)

separable.

Completeness of

M

also makes

complete.

(5.5.7) PROPOSITION:

X

Let

: C0.m) x

R

suppose that for a.a.

an a.s.

+

w.

M

Xu

be a stochastic process and €

S-continuous internal

C([O.m).m).

Y

E

T

x R

T h e n there is

-*M

such that

N

Y

X

and

are indistinguishable.

T h e internal process

Y

N

is called an

S-continuous path lifting

of

X.

Y(r,w)

We now sketch the ideas in extending the decent paths to c0.m).

More details can be found in Ethier and Kurtz [1980].

(5.5.8) DEFINITION: Let

D([O.m).M)

Let

lim x(s) slr ltmtt

(M.p)

= x(r)}.

be a complete separable metric

=

{X

: [O.m)

the space

functions taking

of

values

M

space.

I

lim x(q) exists 8 str right continuous w i t h left

+

in

M.

T h e Kolmogorov

metric ( f o r Skorohod's topology) is defined as follows.

5.5

Section

A time deformation o n increasing

LO.m )

Lipschitz

AC0.m)

is

C0.m)

continuous

[O,m).

onto

deformation o f in

26 1

Beyond r O . 1 1 and Scalar Values

T

=

mapping

T

:

{T

is

T

E AC0.m)

and

: r 2 01.

T'(r)I

u

>

0

time

Finally, Kolmogorou's metric for

For

[O.u].

is

C0.m)

-U

inf

T

define

corresponding roughly to Kolmogorou's metric for

k(x,y) =

a

T h e measure o f deformation for

6 ( ~ )= ess sup[llog

x,y E D([O,m),LN),

a strictly

function

AC0.m)

Let

C0.m)).

i s

k(x,y;~.u)e

du.

TEA[O, m)

D([O,m),UI)

T h e space

is a complete separable metric space

b e c a u s e w e h a v e assumed that

IN

is.

H e r e is the Ascoli-type compactness theorem for

(5.5.9)

PROPOSITION: Let

Then

DC0.m).

y

y

be

an internal function in

*D ( [ O . m ) . U I ) .

is near-standard for Kolmogorou's metric if and

only if (a)

for each finite standard i n

(b)

*M ,

r

in

*[O.m),

8 .

is near-

and

for each standard positiue positive

y(r)

u

and each standard

there exists a standard positiue

Paths of Processes

9

and an increasing sequence

-*-

<

rk = u

p[y(r),y(rj)]

PROOF

5:

Chapter

262

with

rj+l-r

<

when

a,

.I r

>

j

8

<

r

0 = r0 < ’ 1 < such that

<

rj+l.

(C) :

For each

m

in

1. 9

choose a n associated

let

E

1 = 2m

and m

m

>

rj+l - r j

and

u = log(2m)

0 I .I

8.

I

and

k(m).

Define a standard step funcion

where

sj

finitely function

<

<

= st(rj)

for

0

separated,

the

following

internal

6 ( ~ )3 0.

T h e map

satisfies

T

linear between the corners

j

k(m).

S i n c e the

r7.s

piecewise T

are linear

is piecewise

sJ’

and T(r)

= s-S k(m)

+

‘k(m)’

for

r

>

sk(m).

W e also k n o w

and ~(y.xm:T.u)e-udu

<

1 2m’

This proves

that

y

is “pre-near-standard“

in the terminology

of S t r o y a n 8 Luxemburg [1976. 8 . 4 1 . so by completeness of is near-standard.

D, y

Section

5.5

263

Bevond rO.11 and Scalar Values

We shall not give a proof of the converse implication

(5.5.10) DEFINITION:

X

Let

:

U

R

x

-+

*M

be internal.

has a

At-decent path sample or

there

is

a

A 5 R

set

infinitesimal

2

At

6t

D-sample ouer

T

a.s. if

[O,m)

P(A) = 0

with

in

X

We say

and

an

such that the extended step

paths for

0

X(kAt.o)

,

for

kAt 5 r

for

r

are Kolmogorou-near-standard

%(r,u) = stk[X

projection o f

<

,

X(l/bt,o),

Then

<

X(At,o)

At

the sample of

2At

<

(k+l)At

1/6t

*D([O.m).M)

in

(r,o)]

2

r

is called

when

LI

E A.

the decent path

X.

(5.5.11) PROPOSITION: If

an

internal

function

X : U x R

-+

*M

has

a

N

D-sample.

then

its decent

path

projection

X(r.o)

is

(Borel[O.m)xHeas(P))-measurable.

The lifting half for decent paths is as follows.

(5.5.12) PROPOSITION: I f a stochastic process

Y

: C0.0) x

decent paths then f o r euery infinitesimal there is an internal

X

:

H x R

+

*M

R

+

M

a.s. has

A t 2 6t

such that

in

X

T

has a

u

At-decent

Y.

path sample and

Such an

X

is called a

X

is indistinguishalbe f r o m

At-decent path lifting o f

See the proof of (6.7.1) below.

Y.

ChaDter

264

5:

Paths of Processes

(5.5.13) PROPOSITIONS:

A

stochastic

Y

process

: [o,m)

x R

uniformly integrable (on compact interuals has

a.s.

decent

infinitesimal

X

(finite)

s = kAt,

U x R

* + M Xs

and

if

U.

in

At

lifting

:

paths

there is a of

is

only

Y

such

-,

ui

is

[O.m])

and

for

if

some

At-decent path that

for each

S-integrable.

F i n a l l y , the analogs of (5.4.8 & 9) say the following two things.

First, a

[Borel[O.m)xMeas(P)]-measurable

G.

Second, i f

M

is a complete separable metric

space, we have the following

(5.5.14) PROPOSITION:

Let

G : [O,m)

following

are

= G(st(t),w) K(t.w)

x R +M

equiualent

be

any

conditions

is defined for finite times

equals any fixed

H.

[Borel[O.a)xLoeb(R))-measurable

is indistinguishable from some function

function,

a E M

function. where

K(t,w)

t E 0 t l

f o r infinite

The

t.

(a)

G(r,w)

is

[Borel[O.m)xLoeb(R)]-measurabLe.

(b)

K(t.w)

is

[Loeb(P)xLoeb(R)]-measurable.

(c)

K(t,w)

is

Loeb(TxR)-measurable.

U

and

265

CHAPTER 6: HYPERFINITE EVOLUTION

In this chapter we begin the study of time evolution of stochastic processes.

The basic internal framework is the same

as in Chapter 5 except that now we add zero to the internal time

T.

line,

(This

is

done

for

Let

h

previsible processes.) 6t = 1 and n

technical E

*IN

be

reasons

involving

infinite,

n = h!,

-.

T

= {t E

W={kE

*IR

I (3k

= k6t. 0

€ *lN)[t

<

<

k

n]}

*I N : l < k < n n )

R = Wn = { w : U

+

W I

w tnternal)

R.

P = uniform internal counting measure on

#. 1

6P(u) =

with weight function

CRl One

contemporary

random

processes

algebras

called

general is a

way

to have

sigma

algebra

a

describe

indexed

time

time-indexed

"filtration."

"everything random up to time the

to

t"

by

The

idea

evolution

family of of

this

is

of

sigma that

is measurable with respect to There

t.

is

an

obvious

combinatorial way to restrict an internal random process on our internal many

s

R E T

to the internal times

the

T,

have the same standard part,

may not be obvious what begin

t E

chapter

with

this means a

look

at

but since infinitely st(s)

"in standard the

= st(t),

terms.'.

difference

it

We

between

266

ChaDter

determining an event at a time event during st-l(r)

the instant

Y

in

t

in

r

6: Hyperfinite Evolution

and determining an All

[O,l].

t's

in

may be involved during the instant.

The

remainder

of

the

chapter

gives

respecting lifting and projection theorems'. which

the

time

properties

of

internal

the

basic

'time

That is. i t shows

processes

correspond

to

filtration properties of their standard parts and vice versa.

(6.1) Events Determined at Times and Instants A partial

schematic 'picture' of

R

is the tree of choices in Figure (6.6.1).

in case

W

= {-l,l}

A particular

o

is

one branch.

......

T 0 6r

26t

t

Fig. (6.1.1)

1

Section

6.1

267

Events Determined at Times and Instants

(6.1.2) NOTATION: For

R

u E

and

t E

Y

let

= (U0,U6t,'".U

Ut

t

) = u

IUCO.tl

the restricted sequence of choices in

U

to

0

from

t.

Let

[ ut ] = { w E R : w t = u t} .

the internal set o f all samples that agree with ttme

=

(0

t. E

For

n : 3

A E

A schematic If

A

A E R.

any

let

up to

u

C A I t = U{[ht]

h E A}

:

A with a t = ht).

and

ut

Cut]

are shown in Figure (6.1.1).

is an internal subset of

R.

then

[At]

is also

internal by the internal definition principle.

(6.1.3) DEFINITION: An internal event determined at time

A C R

internal subset

such that

A(t)

subset

denote

determined at

the

A

_C

sigma

R

such

algebra

is an

C A I t = A.

A measurable euent determined at ttme P-measurable

T

in

t

that of

Y

is a

C A I t = A.

Let

t

in

measurable

events

t.

The reader should recall (1.5.6) thru (1.5.11) which show that

A(t)

is a

sigma algebra and describe

internal sets determined at time

t.

it

in terms of

Chapter

268

6: Hvperfinite Evolution

(6.1.4) DEFINITION: Let

r E [O.l]

(a)

w

1 A

For

and define wt = u t

if and only if

u

with

t

E R,

Z

3X E A with X

= {w E R

A measurable event determined during the instant A C_ R

P-measurable set

in

t

r.

(A)'

let

for all

(A)r

such that

= A.

0).

r

is a

Let

S(r)

denote the sigma algebra o f measurable events determined during

r E [O,l]}

{S(r)

The family

r.

is called the

progressive filtration. (b)

T

with

8

w

;u

t

<<

leans

3h E A with h

wt = u t

if and only if

r

(t

<

*o = uo.

&

t

2

r).

For

A

_C

R.

r

for all

when let

r

>

(A),.

in

t

while

0.

= {w € R

;w } .

A measurable event determined before the instant is a B(r)

A

P-measurable set denote

:

the

determined before

sigma r.

E R

(A)p

such that

algebra

The family

of

= A.

measurable

r

Let

events

{ g ( r ) 1 r E [O.l]}

is

called the preuisible filtration.

Notice that the times

A(t).

but that real instants

t E

U

index the sigma algebras

r E C0.l)

index

% ( r ) and

'3(r).

This is a lot of jargon, so here are some simple

(6.1.5) FILTRATION FACTS: (a)

w

t

(b)

w

z u if and only >> r . ;u if and only if

tf

wt = u

a t = ut

t

for

for some

some

t z r.

6.1

Section

Events Determined at Times and Instants

E

I f

(c)

determined at some

E

I f

(d)

E (e)

%(r).

is internal and in

is internal and Cn

t'

<<

r

t

<<

>

r

$(r).

t'.t E

t.

Z

E

then

is

E E A(t).

t Z r.

w a s determined at some

For

269

r

then

E E A(t).

r.

H,

0.

E

[O,l],

the

sigma algebras below are nested a s s h o w n :

c $(r)

A(t')

c A(t)

c

U A ( s ) c %(r) s%r

n

=

A(s) =

s>>r

n %(u). u>>r

PROOF : (a)

I wt = ut)

{t

is internal and contains all

an external set [cf. (0.3.8)]. (b)

I wt = ut}

{t

Thus i t contains a

for all

Thus i t contains a

>>

r.

E

Since

r.

t t

<<

E

t

is internal,

I

{t

First inclusion:

P-measurable. u E M.

implies {t

implies

[Elt = E)

Thus

If

w E

[MIt'

Let

M

and

u

I

[Elt

t

r,

r.

[Elt = E E)

=

is

[Elt = E

when

also contains a

r,

then

;w

M

then

M

for

ut' =

t' GJ

,

SO

Let

satisfy

t"

t'

<<

t"

[wt"] E O(r)\A(t').

Second inclusion: u

;w

=

M E O(r).

Strictness of first inclusion:

then

Z

>>

r.

r.

(e)

<<

t

r.

r.

Z

(E)r = E

We observe that Since

= E

is internal,

also internal and thus contains a

(d)

(E)'

We can easily verify that t

>>

is internal and contains all

an external set [cf. (0.3.8)]. (c)

t

Z

t

by (b) and

If u E

(D), = D.

D

or

w E

[D] t = D.

D

and

t ut = w ,

ChaDter

270

Strictness of second inclusion: u u

for

= o

-

0

<

s

<

t-6t

# [at]

([at]),

u t # ot

and

because

still

satisfies

a.

Third inclusion: Trivial, but

E

s,

such that

r

Z

for

J

1

there is an

the union is also a sigma

M. E .4(sj),

R -saturation. If

algebra by

s,

Hvuerfinite Evolution

6:

s

n U[S~.~+~]. 1

.i

<

s

j

Z

r

then

for example, take

sm,

.i Strictness of third inclusion: Since

[w

s+6t

]# [us].

the

union is increasing.

and

u L o,

[MIS = M

If

Fourth inclusion:

us = us,

then

so

(M)'

when =

Strictness of fourth inclusion:

s Z r

[MIS = M

instance.

for any

s

>>

r.

o €

( w ) ~ t? Ul(s).

hence

$(r)

(M)r = H,

If C I(s)

in this

This fact means that the last two intersections are

equal by virtue of the third inclusion applied to different and

M

M.

Fifth and sixth inclusion and equalities: then

and

If

9's.

M E

fl

.4(s).

o E M

u E o,

and

r's

then, by (a),

s>>r

u s = us,

for some

s

>>

r

and

u E M.

so

(M)r = M.

The next result is a set lifting lemma like Lemma (1.2.13). but adding time.

(6.1.6) PROPOSITION:

(a)

If

F E %(r).

E E A(t) P[F

v E]

then there is an internal set

determined at a time = 0.

t 5 r

such that

Section

6.1

27 1

Events Determined at Times and Instants

D

If

(b)

E

J(r).

>

r

>>

posittue

a

E E A(t)

determined

E

internal

E

there

0 .

<

P[D v El

that

then f o r euery standard

0,

a

at

internaL

time

t

D E J(O),

If

e.

an

is

<<

set such

r

there i s an

P[D v El = 0

8(0) such that

PROOF :

1

Cm

(a)

Let

and

Gi

1 1 P[Gm\Cm]

<

F

For each finite natural number, let 1 1 internal sets such that Cm E F E Gm and 1 t m = [r + -1 cf. (1.1.3). Let m

%(r).

E

be

.;1 <

+ ;1I.

E U

:

t

Cm = [C,]1 t m

E

(Cm) l r E (F)’

= max[t

r

1 tm Cm = [C,] .

Define

Since

>>

tm

r. t

Again, since

Cm

Gm

amd

tm

>>

= F.

By

0 . 0

r,

Gm

(R\Gi)r 5 (R\F)r

determined at

countable comprehension, we may

and

P[Gn\Cn]

C

and

tm

tm

extend

n.

We may take

E = Gn

u cm c cn c

G~

or

c

Cn

n G~

and

u cm E

F E n G ~ .

(R\F),

=

< ;1.

C1 E C2 C and

r

<

E F.

--*

< .;1

P[Gm\Cm]

n tn

because

so

Making

(Cm.Gm.tm)

Choose an infinite

sequences are monotone up to

< ;.1

P[Gm\Cm]

select sequences

internal sequence, (0.4.3). G

E

[n\Gt]

are determined at

2 F,

m.

t

dependent choices, we may

G 1 2 G2 2

Gm = n\[n\Gi]

Define

to an

so that the

<

r + n

and

Chapter

272

(b)

<

P[D\C]

that

cm

D

Let

= [C]

tm

be internal and such 1 = max[t E T : t < r- --I and

1

-1 m

tm = [r-

Let

6 .

C C D

and let

9(r)

E

6: Hvperfinite Evolution

. n Cm = (C)r. Let

First,

;w

u

C.

E

u t = wt

so

for all

m t

<<

and

r

tm

u

= w

tm

in particular.

utm = w tm

inclusion: for the other suppose m.

Then clearly

u

-

and

w

This shows one for every finite

u E (C)r.

Next,

c E n cm

=

(c), c (D),

= D.

m

We know

<

IPICm]-PID]I

which

If internal

<

S-lim P[C,] m*

D

E

a(0).

Cm E D

is internal,

w

P[D],

then for every

u

if and only

Cm E (Cm)o C D.

internal and

<

P[D\C,]

IN

m

in

$.

Since the relation wo

if

there is an

(Cm)o

= uo,

is

and contained in

0

D.

Cn = E.

The next lemma is useful below.

(6.1.7) LEMMA:

(a) A

random

X

uartable

:

R + IR

w i t h respect to the c o m p l e t t o n o f onLy i f there t s a ttme measurable

internal

N

that

Y = X

a.s.

?j

Now we may form an increasing

chain of internal sets determined at sufficiently small infinite

for

m

E = Cm .

We may let

B.

such that

6

so there exists finite

P.

t

random

Z

r

is

measurable

O(r)

if and

and a n

J(t)-

uariable

Y

such

A

Section

273

Events Determined at Times and Instants

6.1

(b)

If

X

1x1 <

c.

is

%(r)-measurable

then there is a

Y

measurable internal N

Y =

x X

If

(c)

a.s.

is

lYl

with

%(r)-measurable

Y

E[I?-Xl]

and an

r

Z

bounded,

c

.U(t)-

such that

P.

then there i s a internal

t

and

and

and an

t Z r

determined at

P-integrable.

S-integrable.

time

such that

t

= 0.

PROOF : N

(a)

?

is

Y(0)

X = Y

Suppose

by

and

then

Y(t)-measurability.

This

0

N

%(r)-measurable.

X = Y

Since

respect to the completion of Next,

X

suppose

X

a.s.,

ut = u means

t

Then

,

so

?

is

is measurable with

%(r).

is

completion and for each

<

as stated.

t

E u.

if

P-measurable and

= Y(u).

Y

a.s. for

measurable

with

k

IN

and

rn

in

respect with

to

the

-22m-l < k

22m define -22m- 1 = @m

Ok = m

:

(0

2m

=

For each sets with obtain

m.

the

9:

refining the

R!

sets

(0

< -

5 X(0)

: 2m

<

X(0)).

form a partition of

k = 0 P[Amk v Om]

internal

1

: X(0) < -2-m

(0

Rk

and such

A:

E

%(r).

that

R

and there are

Apply (6.1.6) to P[hmk v Rm] k = 0.

By

to an internal family of nonoverlapping sets

Chapter

274

and

then

6: Hyperfinite Evolution

discarding

the original overlaps. whose external h k k P[Am n A,] = 0, we may assume that the m'

measure was zero,

are disjoint and all determined at the maximal time, determining the originals.

where

I;

k Rm.

to an internal m' internal random variables determined at time tm the

that

whenever

> -1

<

e.

of

and observe

there must be an infinite nm 1 and j I nm. I y m l< ;

m

I

m

1

PIIYm(o)-Yj(w)I

sequence

sequence

that for large enough finite such

r,

: lkl I 22m]

Ii(w)

is the indicator function of

Extend

-

Let

I[L2m

=

Ym(w)

tm

By saturation, there is an

n

in

2m

n[m,n,].

Y

Let

=

m

Yn

for that

n.

The idea is similar to

(1.3.9). This proves part (a).

k is bounded, some of the Am-sets will be k empty and we can replace the corresponding ?,,-sets by the empty Notice that if

set to make

IYI

2

X

c.

This proves part (b).

The proof of part (c) follows the lines of ( 1 . 4 10). each

m,

truncate

X

Xm

know by dominated convergence that

Xm

using part (b) to a bounded

assure us that

Ym.

E[ Ipm-Xml] = 0.

sequence, for every finite such that for all finite sequence of

pairs

IXm

to make an approximation

p

j.k

{(Ym,tm)}

observe that the internal set

so

in

P'

X

L 1 -norm.

in

We Lift

Boundedness and part (b) that

IN.

in

>

+

m.

For

Ym

S-Cauchy

there exists an

E[lYJ-Ykl]

to an

is an

<

internal

$.

mp 2 p

Extend the

sequence and

Section

275

Events Determined at Times and Instants

6.1

contains an infinite

n

n[m , n ] P P P

intersection The necessary

whenever

P

n *IN

lifting

p

IN.

is in

The countable

is nonempty by saturation ( 0 . 4 . 2 ) .

Yn

is

for a n

infinite

n

in

the

intersection.

(6.1.8) LEMMA:

A function X

: R

R

B -

i f and only i f f o r every

O(r)

the completion o f

is measurable with respect to

there t s an tnternal random ttme

where

t

<<

t

r

Y

uartable

such that

a

>>

0

determtned at

>

P[lX-P1

a]

<

a.

PROOF : Define sets

as in the proof of (6.1.7)(a) except now k v Am] k = 0. Let E 9(r) such that PIOm be

Ri

A:

there exist

a,,,k

disjoint internal sets determined at times k 1 P[Am Ik I <22m

k

Rm]

9

a < 3.

R i E .4(tm) C 9(r).

works P[U{Ai

:

For variables Then each

lkl

<

the

22m}3

large

>

1

-

enough

so

r.

all

so

that

<

a

-

and

2m

converse, choose

is

<<

g.

determined before

Pm

k

such that

r

The internal variable

m

whenever

t m = max[t,]

Let

<<

:t

r

a

sequence

Ym

such that

Zl)(r)-measurable and

u

Ym

4

of

P[lX-?l X

internal

>

i] < i.

in probability.

Chapter

276

A subsequence

1

+

mk to the completion of

X

a.s.

X

so

6: Hvperfinite Evolution

is measurable with respect

J(r).

(6.1.9) EXERCISE: Prove (6.1.7)(a)

X

and (6.1.8) in the case where

:

R

+

M

M.

takes values in a complete separable metric space,

(6.1.10) EXERCISE: A function

internal

:

J(r)

completion of an

X

Y

R + IR

is measurable with respect to the

i f and only i f for each

t Z r, N

determined at time

t

such that

Y

=

there is

X

a.s.

(6.1.11) EXERCISE: The filtration facts (6.1.5) can be completed as follows: (a)

Let

tm

<<

r

be a sequence with

smallest complete sigma algebra containing

(b)

S-lim tm = r.

The

U.U(tm).

The following holds.

9(r)

=

fl

.U(t).

tzr

(c)

The

smallest

complete

sigma

algebra

containing

Section

6.1

277

Events Determined at Times and Instants

(6.1.12) EXERCISE:

If

D

for

E S)(r).

exists an internal approximate any internal

D

E

0

<

€ .U(t)

by a sequence

F.

becomes infinite.)

r.

and i f

such that Em

t N" r.

then there

P[D v El = 0.

from (6.1.6)(b).

P[D v F] = 0, and compare

F

with

(First then with Em

as

m

278

(6.2)

Progressive and Previsible Weasurability

This

section

hyperfinite instants

examines

filtrations

r E [O.l].

special

some

9(r)

basic

and

properties

of

the

indexed by standard

%(r)

In particular, these filtrations enjoy a

"completeness"

property

that

adapted

does

imply

progressively measurable, see (6.2.2).

M

Let

denote a Polish (complete separable metric) space,

M = IR d .

for example,

-

and

9(r)

Recall

the definitions of

:,

%(r).

Xr,

from (6.1.4) and the definition of sections,

from (3.1.3).

(6.2.1) DEFINITION:

A function

(a) %-adapted

if

other words.

say

X

: [O.l]

Xr

each section

u.

w

x Loeb(R)]-measurabLe

A function

t s

[Borel[O.l]

say

X

X

: [O,l]

%(r)-measurabLe:

Xr

is

R

If

%-adapted,

-

M

and

X

is

then we

is said to be

2l(r)-measurable.

x Loeb(R)]-measurabLe

in

P-measurabLe

Cs

= X(r.w).

and

x

is said t o be

X

If

%adapted.

we

t s previsibLy measurabLe.

next

probability

result

spaces.

definition of next

M

is

X(r.u)

then

9-adapted t f each section

our

R

is progressively measurable.

(b)

The

x

r E [O,l]. Xr

t f f o r each

and whenever [Borel[O,l]

X

is In

not

true

that

for

case

general

one

"progressively measurable"

result.

The

section (3.3) above.

reader

may

review

filtrations

usually to

changes

on the

the conclusion of

product

measures

in

279

6.2 Progressive & Previsible Measurabilite

Section

(6.2.2) THE PROGRESSIVE AND PREVISIBLE MEASURABILITY THEOREM:

X

If

(a)

each

r E [O.l],

[O,r] x R

X

If

(b)

is

the restriction

[Borel[O,r]

X

of

to

S(r)]-measurable.

x

is preuisibly measurable. then f o r each

r E [O,l].

is

measurable, then f o r

is progressiueLy

the restriction o f

[Borel[O.r]

X

to

[O.r]

x R

9(r)]-measurable.

x

DISCUSSION: Notice that the hypotheses in both (a) and (b) imply that

X

is a stochastic process

measurable for each measurable.

Y =

is CBorelCO.11 x Loeb(R)]-

: s

<

r & X(s.0)

is in the product of the Borel algebra and {(s.w)

(a),

because case (b), If

wt = ut ,

{w

: s :

{w

<

r & X(s.0)

X(r,w)

X(s.0) E B we know

w

E

B}

E B} E $(r)

X(r.w) :

is

the set

[O,l] x R

E

((s.0)

Xr

is a stochastic process we will show that

B.

for each Borel set

X

since

r.

X

Since

in the weak sense that

9l(r).

B}

Then in case

the claimed property

S(r) 3 9 l ( r ) .

and

while in

B} E 9 l ( r ) .

E

<<

and

s

2 u

and

are determined at time

has

E

t.

t w

for

5

u.

U,

t

in

then whenever

so

that the sections

[YSlt = 9’’.

The last two observations lead us to formulate the

Ys

280

Chapter

6: Hyperfinite Evolution

(6.2.3) LEMMA:

If

9

for each

[Borel[O,l]

is

r

instant

d e t e r m i n e d at t i m e

x Loeb(R)]-measurable

in

[O.l].

U

in

t

the section

then

is

'9

[Borel x .U(t)]-

is

9

and if

measurable.

PROOF THAT (6.2.3) IMPLIES (6.2.2): By the preceding discussion i t suffices to show that

[o,~]x n

Y = { ( s . ~ )E

is in

Borel x J(r).

rational

and

q

Y fl (C0.q) x

n}

n

in

<

r &

is

U

in

t

that

so

r

and

<<

t

q

(Borel x Y(t))-measurable by

E B)

X(S,W)

Keep the same fixed

(Borel x 'iJ(r))-measurable Y = u{Y

: s

<<

B.

Choose a

r.

The set

by the lemma and

(6.1.5).

[O,q) x R : q E [O.r) f l

a}.

Finally.

a countable union of sets

This reduces (6.2.2) to proving the lemma

Borel x J(r).

(6.2.3).

PROOF OF (6.2.3):

R t = w'[O*~)

Let

R

identify

-

-

RCt

and

with

R,

x

:R

under

( (Wo.

W6

Py

be uniform counting measures on

So

* . W1)

P = Pt@Pc

t

Meas(Pt)

the

and

R,

as an internal measures.

association between

so

that we

and

Y(t).

n;,

:

T

x

n:

Pt and

respectively.

There is a natural Consider the class of

sets (9 E Borel x Meas(Pt)

may

correspondence

( w ~ , - - - . w ~ ) ) .Let

(ao.

I

WY[t*ll

=

E Borel x Y(t)}.

6.2 Progressive 8r Previsible Measurabilitv

Section

This

is a

sigma algebra because

the second

interfere with set algebra operations.

S x At.

since

Qz

A E

E Y}

Y

t-determined. whenever

{Y

for some set

Borel x Meas(P)

E

Slz.

be the section over

Y = 9 x Rz

or

:

Borel x A(P,).

Y ( A ) = {(r.u)

let

(r.u.A) is

Meas(Pt) x Meas(Pz)

is an extension of

and

c

[O.l]

x

:

r-section of

then R,.

nt

[O,l] x

E

Since each

(r,a) E Y .

9

factor does not

It contains rectangles

Therefore this class equals

see (3.1.2). Let

Meas(P)

28 1

C Y.

(r,[ot])

The class of sets

Y ( A ) E Borel x Meas(Pt)

for Pc-a.e.

A}

t

is a sigma algebra because countable unions of null sets are null and taking sections commutes with set algebra operations. Y = S x A

If

for

S E Borel

Y ( A ) = S x Ah E Borel x Meas(Pt)

is all of

that

T

€

Borel x Meas(P).

Borel

by (3.1.4).

We

let

h

denote Lebesgue measure

algebra described

if

there

(A x

P)(Y

9 x

nz

E Borel x A(t)

so (6.2.3) and (6.2.2) are proved.

( A x P)-measurable

Alternately,

Thus, this class

Meas(Pt).

x

Y = 9 x il;

then

The important conclusion for us is

By the first paragraph of the proof, and

A E Meas(P).

and

Y

c

exists

if

it

is

in

the

and

say a

complete

product A

in (3.1.2) for the measures

[O,l] x R

T

is

in

( A x P)-measurable

measurable i f and only if from Theorem (5.4.8) that -1 only if s ( 8 ) E Loeb(UxR).

s

-1

(9) is

Y

such is

6t@P-measurable.

we could say that

P.

if and only

3 E CBorelCO.11 x Loeb(n)] so

is

sigma

and

(Lebesgue x Meas(P))

v 9) = 0. Theorem (5.4.3) says that

set

(A x

(A x

that

P)-

We know i f and

P)-

Chauter

282

measurable sets are those in the of

[Borel[O,l]

on

U x R

the

equation

x Loeb(R)].

induces a

p-measurable

p = T

6: Hvuerfinite Evolution

p = [bt@bP

0

s

-1

]-completion

Every internal bounded measure.

T .

[Borel[O.l] x Loeb(R)]-measure, 1.1. -1 . 0 s Moreover, Y C [O,l] x R

by is

if and only if there exist sequences of internal

d1 C C s -1 (9) and s1 r s2 E r (TxR)\s-'(Y) - d2 C 1 such that r[dm U bm] 2 1- .; Such internal measures will be sets

- 0 .

useful to us in Chapter 7 where we will let the time measure depend on T

= 6t x

for now the reader may wish to think in terms of

w:

P

p-measurable = ( A x P)-measurable.

and

(6.2.4) DEFINITION: Let

U x R.

be a b o u n d e d h y p e r f i n i t e m e a s u r e o n

T

A function X

:

[O.l] x R

+

M

is

r-almost progressiuely

m e a s u r a b l e i f t h e r e is a p r o g r e s s i u e l y m e a s u r a b l e p r o c e s s

Y

such

T

-1 (r.0). s

0

= X(r,w)

Y(r.0)

that

W e say

X

that

is

p(r.0)

r-almost

preuisibly

measurable if there t s a preuistbly measurable process s u c h that

X(r.w)

Suppose that

Xr

is

we may

%(r)

take

=

X

[resp.,

X'(r.0)

Z(r.0)

is

a.e.

(A x P) = (6t@6P

= 0

adaptation fails and have

a.e.

0

for

on the null

X = X'

= r

p(r.0)

3(r)]-measurable

s

0

-1

A

a.e.

set o f

A x P.

fU(r)]

a.e.

h(r)

in the case when

T

Z

(r.0).

s-')-measurable r.

r's

and Then where

This means that

we can weaken the hypothesis of the next result to [resp..

=

a.e.

Xr E %(r)

= 6t@6P.

Section

6.2

283

Progressive & Previsible Measurability

(6.2.5) ALMOST PROGRESSIVE AND PREVISIBLE MEASURABILITY THEOREM: Let

be a bounded hyperfinite measure on U x R -1 denote the completed [Borel[O,l]

T

p =

and let

T

0

s

x

If

x Loeb(R)]-measure.

J(r)-measurable].

progressively

[resp.,

Y

that is

Y(r.0)

is

r-almos t

preuisibly] x

is

F(r)-measurable

X

[Borel[O,l]

%-adapted [resp..

a.e.

is

then

t-almost

that is. there exists a

[O.l] x R + M

Xr

p-measurable and each section [resp..

:

measurable,

Loeb(R)]-measurable X(r.o) =

9-adapted] and

p(r.o).

PROOF : We shall give the proof only in the previsible case, the similar progressive case can be found in Keisler [1984. Lemma 7.51.

For any

Y E [O.l]

(9) = {(r.u)

Let

%

= (9 E Borel[O,l]

a sigma algebra.

Any

x

R,

: u

-

let

x Loeb(R)

:

Y = (9)). Note that

We will

Y = X

with

f

E

$ = {Y E Weas(p) $

E

1

f

(with

%-measurable

:

Y = (Y)

Y = (Y)}.

to finish our

Y = ((r.0)

the form

p-measurable and satisfy

Let

of

is

Y

a.e. this proves our claim.

We know that sets of are

X

show that

since then there will be an

is

f

is previsible

1

measurable with respect to the completion p),

Y

%-measurable function

as the reader can easily check.

respect to

E Y}.

o for some (r.0)

claim.

We Let

:

X(r.0)

B C M.

for any Bore1 wish

Y

E $.

to

show

By

E B}

that

(5.4.3).

284

s

-1 ( 9 ) is

T

r-measurable on

Chauter

6: Hvuerfinite Evolution

x R.

we may pick sequences of

so

internal sets satisfying:

%,I

>

1-

uCx1 = uo

if

x

u

"[dm

For each

n

W

in

.;1

define

where we understand

0. Let

d = U Il 89:

and

m n 9 = U Il 9.; m n

(t,w) E dm

When

m

for some

(t,u) E Il d ; , n where r = st(t).

is fixed, w

-

u

i f and only if

Y = (9).

Since

this means

d

so

and

E s-l(Y)

r[d U % ] = 1

and

( ~ ( 8 )=) s ( % ) .

showing that

Y

This is because

and

s(d) so

(from

corresponds a

E Y E

9)

in fact,

(s(d)) = s(d)

completes the proof by

is sandwiched between sets of

s ( d ) = U Il s ( d z )

tn

s(%).

Lemma '(5.4.7)

m n inclusion is easy to verify. n

% 5 T x n\s-'(Y),

and similarly for

Suppose that for some

such that

tn =: r

and

9.

m.

(tn.o) E I$,

3.

The

E

to each that

n

(r.0) E U

is,

285

6.2 Progressive & Previsible Measurabilitv

Section

Then for each finite

s(d:).

m n [tn-l/n]

n

there exists

~~

w

such that

n

[tn-l/n]

on

extend the sequence

(tn.on) E dm.

and

= o

(tn,wn)

internally using saturation ( 0 . 4 )

Ctp-l/PI and obtain

t

Z

P for some infinite

r,

and

(tp,op)E dm

p

We may

o

Ctp-l/p1 = o

P

(by the internal definition principle and

Robinson's Sequential Lemma). For any finite n , (t,.o) E d: [t -l/n] [ tp-l/n] because o = w , so (r.w) = (st(t ).o) E s ( d ) . P P The reader should note the following consequence of the last result.

If

adapted, then

X

P-continuous

X

is

is

[Borel x Meas(P)]-measurable

r-measurable

bounded hyperfinite

and

H x R.

on

'A

adapted

X = YU

euery

Hence for each

there exists a progressive [resp.. previsible]

R

for

and

YR

with

~(r.w).

a.e.

(6.2.6) PROPOSITION:

X

If

[O.l]

:

x

R

+ tti

is

6tB6P-almost

progres-

sively measurable and a . s . has decent p a t h s , then there is

Y

a progressiuely

measurable

process

paths such that

'X and

are indistinguishable.

Y

w i t h a . s . decent

PROOF : Let

A C R

and x

be a progressive process such that

( A x P)(r,o).

a.s.

S

Z

A

and

with

Xu

and

=

{Sm.Sm."' 0 1

1-sn m

There is a countable dense set

P[A] = 1

such that

is a decent path for

For each

sm

m .sm} n

< -1

m'

X(r.o) = Z(r,w)

in

E S Let

IN.

u

X(s.u)

in

S C [O.l]

= Z(s.u)

on

A.

choose a finite increasing sequence with

0

Zm(r.o)

<

s:

<

k.

equal

0 C sk,"-st Z(sm.o) k

<

-1-,

where

Chapter

286

sk = min{s

h

>

sh m

:

h ” r = 1. Since

r).

z(s;,o)

6: Hyperfinite Evolution

>

1

if

>

r

or

sz

~ ( 1 . w ) if

Zm is

is progressively measurable, each

Z(s;)

Borel x Loeb-measurable.

Define

Y(r,o) = lim sup Z,(r.o). m-W

u E A.

If

since

= Z(s.u)

X(s.u)

lim X(s,u) = X(r,u)

and

s lr

= lim sup Zm(r.u). m-W

Y

we see that

has the same paths as

X

A.

on

For every

%(r)-measurable

is

Y(r.0)

Z,(r.-)

m.

1 %(r+;)-measurable.

is

hence

by (6.1.5).

(6.2.7) PROPOSITION:

X

If paths, and

[O.l] x R

:

X

if

is

X(0)

if

X

a.s.

bt@tiP-almost

and

Y

Y

has

left-continuous

progressiuely

J(0)-measurable.

is

preutstble process s u c h that

*M

w i t h a.s.

then

measurable there

is

left-continuous

a

paths

a r e indtstingutshable.

PROOF :

Z

Let

( A x P)(r,o).

A E n

= 1

P[A]

such that

is left-continuous for

For each

IN.

in

m

u

<

1 r- ;;;}.

or

so

m

if

= Z(s,u)

X(s.u)

a.s.

[O.l]

and a

on

S x A

A.

in

choose a finite increasing sequence

Sm = { s ~ . s ~ . * - * . sE~ }S with 0 < s: 1-s; < ;1;. Let Zm(r.w) = Z(sm.w) k sh m

S

There is a countable dense set

with

Xu

and

Z(r,w) = X(r,w)

b e progressive and satisfy

r

<

0

sm.

<

Each

--, 1

0

<

where

Zm

is

sk+l-sk m m

< I m

sk = max{s: m h

and :

(Borel x Loeb)-

Section

measurable.

287

Proaressive & Previsible Measurabilitv

6.2

Y(0.o) = X(0.o)

Define

r

and i f

>

0.

Y(r,o) = lim sup Z,(r,w). m-)m

Y

so

If r

>

(Bore1 x Loeb)-measurable.

is

0.

u E

A,

X(s.u)

since

lim X(s,u)

= X(r,u)

s tr

X

same paths a s Each

on

Z,(r.*)

measurable,

so

hypothesis,

X(0)

is

s E S.

for

= lim sup Zm(r.u). m*

so

Y

when

has the

A. 1 S(r- -)-measurable m

is

Y(r)

= Z(s,u)

is

9(r)-measurable

fU(0)-measurable

and

and

hence

when

r

Y(0)

>

= X(0).

J(r)0.

By

288

(6.3)

Nonanticipating Processes

If an internal process only anticipates a sample sequence w

by an infinitesimal time into the future, then its projection

will be progressively measurable.

Y(t)

then

L,

for

t

t = 0

>

L.

= X(t--L)

X

If

anticipates by a time

is a strictly nonanticipating process

but not all the way to zero.

Separate treatment of

is incorporated in our next definition.

(6.3.1) DEFINITION: An

internal

X

function

defined

U x R

on

nonanticipating i f there i s an infinitesimal that f o r each

in

t

U

with

X(t.w)

t 2

L ,

With

6"

if

6

>

t

6

fixed

and this

>

0

t

wt = u ,

"X

Recall

L .

X

then

wt = u is

an

is nonanticipating

imply that internal

X(t,w) = X(t,u).

property

"Nonanticipating" as we have defined i t is external: infinitesimal

such

= X(t.u).

Strictly speaking. we should say after

if

L

is

is nonanticipating after

the definition of Lebesgue

of

X.

'for some

L'.

lifting from

(5.4.2).

Here is the result you've been anticipating:

(6.3.2) NONANTICIPATING LIFTING THEOREM:

A

stochastic

bt@bP-almost

process

progressiuely

x

: [O,l] x

measurable

has a nonanticipating Lebesgue lifting.

R

i f and

---f

M

only

is

if

X

Section

289

6.3 Nonanticipating Processes

This

is

an

important

result

because

shows

it

that

progressive measurability with respect to the filtration we have defined corresponds to a simple property of internal processes. Nevertheless, we shall not give the proof which can be found in Keisler [1984, Theorem 7.61.

We shall describe the main lemma

from Keisler's proof and then give the decent path versions of the result.

(6.3.3) DEFINITION:

If

A

is a n

internal

subset

R

of

and

t E

T,

define

P[

T h e u n i f o r m internal conditional probability only d e f i n e d o n internal sets.

P[

of

In

P[

P[

is denoted

general,

a

is

T h e h y p e r f i n i t e extension

lot].

P-measurable

set

might

not

be

t

Iw ]-measurable. If

P[A

Iwt]

lwt]

A

is

P-measurable

and

A'

is

internal

with

v A'] = 0, then

P[A' lat] = Z[P(u)

: u E

= ZIIA.(U)P(U)

A ' & u t = ot]/P[ot]

: Ut =

GJt]/P["t]

= EIIA, jut]

where

IA,

is

the

indicator

of

A'

and

the

internal

ChaDter

290

6: Hvperfinite Evolution

conditional expectation was defined in (1.5.7)with in this case.

P[A' lot]

null

sets

= Cot]

We know from (1.5.9-10) that

EIIA,lat]

so

p(o)

lifts

almost equals

matter

we

so

EIIAl.U(t)].

EIIAI.U(t)].

must

Unfortunately, the

distinguish

between

these

two

things.

(6.3.4) PROPOSITION: Let A

E

t E

Meas(P),

u.

If

A C R

P-almost all

then f o r

P-measurable,

is w.

A

ts

P[

[at]-

measurable. PROOF :

Wu

R =

We express

as a hyperfinite product,

( ( w o , ~ ~ ~ . w t ) ; ( w t + s t . ~ * * , w l ) in )

associating the pair

uniform measures on each factor,

P[

lot]

as in

p

and

R.

The

can be related to

u.

by taking sections in the product, for

A

thought of

wUCO.tI x WT(t.11

t

Aa

t

= {(At+gt.*~-.hl): A t = w ) .

t

Hence,

P I A l o t ] = u[A"

p-almost every is defined.

at

If

1.

Now, by Keisler's

Fubini Theorem,

produces a measurable section. s o R1

PIAlot]

is an internal first factor of

small

Section

6.3

Nonanticipating Processes

29 1

p-measure where the sections are bad, then the same

R.

P-measure and lies in

R 1 x RT(t'll

This proves that

has

PIA)ot]

[PI.

exists a.s.

(6.3.5) DEFINITION:

A E R,

For a n y

A'

the d e r i u e d set

is d e f i n e d by

A ' = {o E A : (Vt E T)PIA]ot] = 1).

Recall defined.

A

that i f

is Loeb.

then

PIAlwt]

is always

Keisler's main tool in the proof of nonanticipating

lifting is the

(6.3.6) DERIVED SET LEMMA:

P[A]

= 1.

Moreover,

tf

A

internal

@

If

ts

E A'

P[A']

then

with

also

= 1

tnternal,

P[@]

and then

A" = A'.

there

is

an

= 1.

For proof, see Keisler [1984, Derived Set Lemma 6.61 and (6.5.16). S-Integrable Lebesgue Lifting Theorem (5.4.5) has a

The

nonanticipating version, Theorem 7.8 of Keisler [1984].

We

prove the decent path form in (6.3.7-8)(c).

(6.3.7)NONANTICIPATING DECENT PATH PROJECTION THEOREM: Suppose

X

:

T

x

R

* + IR

the

tnternal

a.s. h a s a

nonantictpattng

decent path sample.

process Then:

Chapter

292

(a)

The

decent

path

6: Hvperfinite Evolution

j7

projection

indistinguishable

from

is

progressively

a

measurable process. (b)

If each section

Xt

is

%

S-integrable, then

is indistinguishable from a uniformly integrabLe progressive process. (c)

If

llXllp

%

then

is

p 2 1,

6t86P-S-integrable for

Lp(AxP).

€

PROOF : (a)

R.

A.

X

Let

have a m 2 1

For each

t i = nAt.

is a

in

for some

*IN.

in

,

k X(t,.o)

-

st X(1.o)

Y

measurable, s o

is also.

1

<

k

<

m, k

X(tm)

satisfying

r

<

tm

,

<

r

<

,

r = 1

0

<

Y(r,o) = lim inf Y (r.0). m m

and let

k.

E

there - k

X(--)

Define

1 st X(t,.o)

{

=

I and each

n

AO.

except on the null set

Ym(r.w)

At-sample except for a Loeb null set

Each

Xo

If

1 , 1

Ym

is a

is

<

k

<

m

(Bore1xLoeb)-

At-decent path, then

stkXo = Yo. If

m

u.

o

such that

t-r

then

ot = ut

> -1

for some

t

Ym(r,w) = z(X(tm,o)) k

>>

r.

hence for all

where

tm k

<

t.

m' so

X(t,.ko)

= Y(r.u)

so

(b) {st Xt Lemma

k = X(t,.u)

Y

Z

Ym(r.u).

T}

(1.6.2).

that

Y(r.o)

is progressive.

Suppose that each

: t €

'This shows

Xt

is uniformly Hence

each

is

S-integrable.

The family

integrable by an argument m'

is a uniformly

like

integrable

Section

6.3 Nonanticipating Processes

Y

process and by Fatou's lemma, (c)

X

By (5.4.6).

P

(5.4.5),

293

is also (compare to (5.3.26)). so

by

E LP(hxP).

(6.3.8) NONANTICIPATING DECENT PATH LIFTING THEOREM:

X

Suppose a stochastic process

:

%-adapted and a.s. has decent paths. (a)

2,

is a Lebesgue lifting of

X

-

IR

is

Then:

has a nonanticipating decent path lifting,

that Y

[O.l] x R

:

is,

T

x

R

there

infinitesimal

an

internal

nonanticipating

process

after

At

At-decent path sample f o r some

that a.s. has a

in

At

path projection

X.

is

* + IR,

1

such that the decent from

is indistinguishable

X

In particular,

U

is indistinguishable f r o m

a progressiuely measurable process.

(b)

If

X

is uniformly integrable, the

Y

(a) may be chosen with each section,

in part

Yt.

S-

integrable. (c)

If

llXll E Lp(XxP)

in p a r t

p 2 1.

f o r real

(a) may be chosen s o

that

then the

Y

llYllp

is

At@6P-S-integrable.

PROOF : (a)

Let

W

:

T

x

R + *IR

be

a

D-lifting

of

X,

m 2 1 in IN and 1 < k < m k use (6.1.7) to choose Zk an d(t )-measurable random variable m m a.s. and tm k Z k while st[W(tm)]k = X(--) k such that Z i 2 X ( z ) m a. s . (increasing t k i f necessary). Define m'

appealing to (5.3.23).

For each

-.

294

Chapter

(we may take is

:t

= 1

nonanticipating

U[ti,ti+')

with

Z,(l.o)

and

after

&<

tm'

ti+l-tkm

<

6: Hvperfinite Evolution

= Z:(o)).

constant

1.

The process on

the

while

The last statement is internal and holds for every finite hence there is an infinite The process paths because (b)

If

Y(t.w)

W X

m,

for which the statement holds.

m

= Z,(t.u)

m'

intervals

for such an

m.

It has decent

does. is uniformly integrable, we may choose

W

and

ZE S-integrable and conclude moreover that

in the internal statement above.

This proves the theorem (b).

Alternately, see the proof of (5.3.25). (c)

for

m

(5.4.6) so

If

in

Y

*IN

is as in part (a), define truncations

and similarly for

and (5.4.5) we know

there I s an infinite

m

lYmlP

X

when is an

which makes

m

is finite.

By

S-Cauchy sequence,

lYmlP

S-integrable

Section

6.3

295

Nonanticipating Processes

[see the proof of (1.4.8)].

This shows (c).

(6.3.9) REMARK: The

first

projecting

X

:

part

theorems

[O.l] x R

+

M

of

the

can be

Y

and

S-dense internal subset o f space entity.

M,

nonanticipating

proved

T x R

:

*M.

(with +

lifting

little

K

and for

change)

K

where

is an

a complete separable metric

The integrable parts require linear structure on

M

but note that the integrable

= IRd

case follows from

these results one coordinate at a time.

(6.3.10) NONANTICIPATING DOMINATED LEBESGUE LIFTING THEOREM:

X

Suppose progressively

X(r,w)

<

c(w)

:

[O,l] x R

measurable a.s.

[hxP]

IR

is

stochastic

an

6tQ6P-almost

If

process.

f o r some function

c

then there is a nonanticipating Lebesgue lifting

:

Y

R + IR, of

X

satisfying

max Y(t.w)

c(w)

a.s.

P.

t

The complete proof

of

this result

is in Keisler

[1984,

Theorem 7.91. but we shall start i t off for contrast with the next exercise.

Let

= inf[b

For each

0.

X(r.w)

<

E IR : lX(r,w)l

<

b(w)

and

a.s.

h

b a.s. h(r)].

296

ChaDter

{w

b

so

is

Also,

b(w)

:

<

b(o)

a} = {w : X(r,w)

P-measurable.

<

c(w)

6: HvDerfinite Evolution

a

(Use the ordinary Fubini Theorem.)

a.s. by the ordinary Fubini Theorem.

can work with the measurable function a(o).

a.s.},

b(w)

and a

S o we

P-lifting

The rest of the proof can be found in Keisler [1984].

(6.3.11) EXERCISE: State and liftings.

prove

X

Assume

the analog is

a.s. has decent paths

function

(6.3.8)

for decent

path

6t636P-alrnost progressively measurable,

and

corresponding

of

to

X(r,w) b(w)

<_

a.s.

c(w)

in

the

What is the

discussion

after

(6.3.8 ) ?

(6.3.12) EXERCISE: (a)

Show that a stochastic process

X

:

[O.l]

%-adapted and a.s. has nondecreasing paths in

if

only

Y

:

T x R

there

*R

is

and {O.l}.

Y

nonanticipating

that a.s. has a

indistinguishable from (b)

a

X

R

+

R

is

D[O.l]

i f and

increasing

internal

D-sample and

X

is

1.

Prove part (a) in the special setting where both

X

are restricted to have values in the two point space

297

(6.4) Measurable and Internal Stopping

Here

is

the point

of

Exercise

If

(6.3.12).

nonanticipating, increasing and takes only the values

Y

is

0

and

then

1,

= min{t

T ( W )

has

the property

that

Y(t,o) = 1 = Y(t.u), interchanging

if

so

Y(t,o) = 1)

:

= t

T ( W ) T ( U )

and

u

H

E

<

t

shows

o

satisfies the next definition with

t

ut = o ,

and

then

while a symmetric argument = t.

T ( U )

This means

T

A t = 6t.

In numbers (5.3.8-11) we developed some special properties of

particular

stopping

stopping

with

an

approximate

At that time we did not need to know that they

Poisson process. were

times associated

times

in

the

technical

sense

of

the

next

definition but they are examples that we have used. If

At

is a positive time

set of multiples of

HA = {t

H

we shall denote the

by

At

E

A t E H,

kAt for some k

:

E

*IN} U (1).

(6.4.1) DEFINITION:

Let

At E

function

T

ut = o t ,

then

If

T

is a

:

-

T.

A

R

HA

T ( U )

At-stopping

such

that

time is an internal if

T ( W )

= t

and

= t.

At-stopping time, then the internal process

6: Hvperfinite Evolution

Chapter

298

is increasing, takes values in

Y(t,o) = Y(t,u),

that

Y

is.

u t = ot

then

nonanticipating.

To

and i f

(0.1) is

reiterate, the remarks before and after the definition give a correspondence

between

stopping

times

and

increasing

nonanticipating indicator processes.

If t

2

Z(t)

is simply an internal nonanticipating process for

then functions like

L.

= min{t

T'(o)

could fail to be

bt-stopping times.

bt-stopping time. that

>

%(O) 3 8(0).

~ ( o ) ] is a

If

X

:

However,

T

=

T

I

V

L

is a

This basic annoyance is caused by the fact

If

At

-

>

6t.

then

At-stopping time and if

[O,l]

2 a}

: Z(t.o)

x R

(0.1)

is

At

a(o) = min[t %

0 , st(a)

%-adapted

E

PA

: t

= St(T).

(not just a.s.)

and increasing. then letting

(6.4.2) DEFINITION:

A function time i f f o r each

p

: R -+ [O.l]

r.

If we suppose that

{o : p ( o )

p

is an

t s caLled

r}

E

an

%-stopping

%(r).

%-stopping time ahd define

Section

6.4

{0. ,

=

X'(r,w)

then

X'

and

p(u)

if

p(o)

>

if

P(O)

I r

is increasing and when

<

or

r

= X'(r.u)

or

then

<

p(o)

299

Measurable and Internal Stopping

X

>

p(w)

and

r

right continuous.

p(u)

and

s

X'

Since

either both

>

r.

X'(s,w) = 1

If

%-adapted. for all such

s

5 u

w

r

p(w)

so

X'(r,o)

for all

I r.

I r

p(o)

s

>

X'

so

r.

is

has decent paths, (6.3.12)(a)

says i t is indistinguishable from an increasing progressively

X".

measurable process

Let

{

X(r,o) =

The T

%-stopping

= p

time

0 ,

if

X"(r.w)

1 < 5

,

if

X"(r.o)

1 1 5

= inf[r

T(O)

: X(r,o)

1 11

satisfies

a.s.

(6.4.3) THE STOPPING TIME LIFTING LEMMA:

If

: R +

T

p ( o ) = st[~(w)]

%-stopping

is an

is

time

there is a

vt,

U

p

:

a

At-stopping

%-stopping

R

[O,l],

time.

time.

then

For

each

and each infinitesimal

vt-stopping time

such that

T

s~(T) = p

a. s .

PROOF : Let p(w)

5 r

wt = u t p(u)

<

T ,

be

then

T(W)

for some r.

thus

a

{w

t

At-stopping = s

>>

r

: p(w)

time and

for some and

<

s $ , r

us = u s

r} E %(r).

so

p = s~(T).

If T(U)

w

= s

then

u

5

If

r

and

300

If = I

6: Hvperfinite Evolution

Chapter

p

{p(o)
is an

(r,w)

progressive, has values (after

0

s~(T')

in

the discussion

increasing paths

Y

Let

vt-stopping

~ ' ( w ) = min[t time

For almost all

D

in

X

Then

and

takes only

with a n

At-sample.

8.8.

u(o) = inf{r E ~ 0 . 1 1: I(r.o) = 1)

time

where

above.

by

E HA : Y(t.o) = I ] .

defining

yo

o,

is

the

be an increasing nonanticipating

X

lifting of

At)

$-stopping

as

1.

and

X(r,w)

5-stopping time, define the process

equals the path

equals

We make this a

E

ug:

Xu,

SO

= min[t

T(O)

The

t

2

T ' ( w ) ] .

St(T)

= p

a.s.

A good way to think of a stopping time is 'a pruning of the The schematic variation on Figure (6.1.1) is

tree of samples'. as follows.

"

\ \

T

= 6t T

= 26t T

/

T

= 36t = 36t

"

\

\ ~ = 6 t

u . . 0

,

I

*.*

6t 26t 36t Figure (6.4.4)

Y

If

X(r) = S-lim Y(t) t lr

is

a 8 . 5 . .

decent

path

lifting

of

X.

then

so a simple argument shows that there

Section

is

X(r)

an

301

Measurable and Internal Stopping

6.4

such

s 1 r

= st Y(t)

that

whenever

and

t 1 s

a.s., in particular,

st Y(s)

t

2

= X(st(s)).

s,

Our

next result from Hoover & Perkins [1983] extends this to random times.

(6.4.5)THE PATH STOPPING LEMMA: (a)

Let p

:

Y

be a decent path lifting of

Q

has

[O,l]

infinitesimal time

path

and a null

T

whenever

w C

whenever

A.

then

1.

If

for

such

~ ( w Z ) p(o)

and

0

Y(T(u)+L,u)

Z

<

Y

some

At-stopping

E Q

A

set

then

w C A

<

sample

then there is a

At,

and let

%-stopping time.

At-decent

a

T(W)+L

be an

X

L

that and

= kAt Z 0,

X(p(o).w),

in

particular, = X(p)

st[Y(~)]

(b)

Suppose that

while

the hypotheses of (a)

in addition to

X

is

Y

a.s.

%-adapted and

is also nonanticipating after

Then there is a

b

Z

a

time o f the conclusion

so of

that the

At.

At-stopping

(a) has the form:

302

CharJter 6: Hyperfinite Evolution

PROOF : Part (a). Z

Let

l i f t the random variable

st u = p

At-stopping time such that a.s. have a

mp 2 p

IN

in

a.s. where

For every

At-sample.

Let

X(p(w).w).

IN

in

p

be a

u

Y

A t makes

there is an

such that the following set contains every finite

n 2 mp:

{n

*IN

E:

:

1n. E

P[l

&

i]< L}. P

2

lY(u+t)-Zl

max

1 ;;
Since the set is internal i t contains an infinite countable intersection

ll *IN[mp.np]

n

and the

P

contains an infinite

n

by

P saturation.

Let A

Let Z(w)

*

and

= u+

T

be

lim st[

the

Y

X(p(o),w),

n' union

of

the

doesn't have a

max IY(u+t)-Zl] 1 1 ;
At-sample,

If

# 0.

null

o 4

A,

sets

where

*

T(O)

p(w)

Y(T(o)+r.@)

P z Z(w)

z X(p(o).w)

and

p(o),

T(W)

so

this proves part (a).

Part (b). Define

At-stopping times

T,,,(o)

so

rl 2

T~

2

= min[t

If

*-*.

E

Y

U*

:

IY(t.o)

has a

I

2 a+

1 --I.

At-decent path on

w

and

= p

for

N

Y(-.o) = X(*.o).

then we will show that

such random samples there is an

r.

w. p(o)

lim st mTo see this, suppose p ( w )

<

r

<<

t

such that

T~

<<

t.

IX(r.w)l

>

Then

.;

a+ 1

6.4

Section

Measurable and Internal Stopping

This means that for all large enough ~

~

= Y(p)

lim Y(T,) m

a.s. l p and moreover

m,

303

p(o)

T~

<

At-stopping time such that

= X(p)

(both a.s., using part

p E IN m > m

st(a)

there is a finite

m

P

= p

(a)).

>

p

so

a.s.

We need to 'internalize' these limits as follows. be a

t.

and

Let

CJ

st[Y(a)]

Then for every finite such that for all finite

P'

P[~T,-cJJ

> 1 P

or ~Y(T~)-Y(cJ)~> -3 1 P

Since the definition of

{T~)

< -. 1 P

is internal, there exists an

infinite

n satisfying the probability above. P to pick an infinite n E n[m n 1. This T~ P P' P 1 At-stopping time with b = a+ n'

Use saturation is the claimed

-

( 6 . 4 . 6 ) EXERCISE:

Show that i f

p

is only

an internal (nonstopping)

T

P-measurable we may still select so

that the remaining conclusions

in (6.3.14)are true, in particular.

(6.4.7)

st Y ( T ) = X(st(s)).

EXAMPLE:

This example shows one kind of difference between and

st X ( T )

It is based on a remark in Hoover 81 Perkins

%(st(T)).

[1983] and an example in Lindstrom [1980].

On the

*finite

jump function is a.s. (4.1.5)

j :

set

W + *IR

finite but

not

W

(where

R = W lr )

choose an internal

with a symmetric distribution which S-integrable.

to obtain a distribution:

For

example, apply

304

Chapter

Notice that

x

P[lJl

=

m]

= 0, but

6: Hvperfinite Evolution

E[lJl]

=

m.

Define a process

by

0

where

at

is a positive infinitesimal.

L

The infinitesimal jump at 1 t = - + Gt. The functions

are

1 t = 2

announces the finite jump

2

Gt-stopping times for each

N

m

and

1 and 0 for r < -, we see 2 2 that X(st(T,)) = X(l) with the same distribution as j while 1 =: 0 and 1 P[st X(T~) = O] + 1 as m + m , since X(z) T~ = 5,

Since

X(r)

= j(ul12)

N

when

Ijl

N

<

m.

for

1 r 1 -

N

305

(6.5)

Martingales The notion

obtained by

*martingale

of

*-transform.

on

evolution

our

scheme

is

This makes well known facts about

finite martingales available for the study of hypermartingales.

(6.5.1) DEFINITIONS: An

*martingale

* d

IR

X : T x R +

internal after

At

if

s

<

Hi

in

t

called

is

and

a

R

in

w

imply

I

E[X(t)

A nonanticipattng

*submartingale s

<

t

because t

X(t.w)

.

w

and

=

X

internal

:

*

U x R + IR

*supermartingale)

(resp.,

*martingale

A

wt = u

Ti

in

wS] = X(s.0).

R

in

i s called a

after

if

At

impLy

is automatically nonanticipating after

I

ECX(1)

I

wt] = E[X(l)

Also, we could simply say

X

ut] = X(t.u)

is a

At

when

*martingale

after

if

At

I

E[X(l)

for

in

t

Ti

and

w

ot] = X(t.w)

in

R.

since a simple

* finite

calculation yields the defining property above from this.

If is

<

M

:

H x R

*convex,

then

1CjP(Xj).

so

4 *IRd

X

is a

= q(M)

is

*martingale and q * submartingale: a

:*Rd **IR q(Xcjxj)

ChaDter

306

If

X

* submartingale and q is increasing and *convex, * submartingale. = q(X) is also a *martingale is given by a sum of important kind of

is a

Y

then

An

* independent

mean zero terms,

X(t,o) = H[f(s,ws)

f

where each

6: HvDerfinite Evolution

:

T

in

s

x

T.

W

+

*IR

I

s

satisfies

see (4.3.1).

t. step 6t]

H[f(s.w)

:

: 0

f(s,w)

<

s

*martingale

is a

does not depend on

on

t

o ,

Our theorems.

next

for

s.

It is

Example (4.3.3)

by a ‘drift’ term, that is,

related

the Poisson process

to

(5.3.17).

does not lead to a

if those increments are anticipating.

not

= 0

*martingale

Notice that the full generality of ( 4 . 3 . 4 )

*martingale

W]

E

t , step bt]

a central example leading to Brownian motion. only differs from a

w

Anderson’s random walk

B(t.o) = H [ e p ( w s )

is such an example where

<

0

:

(We condition

X(tl).**-,X(tn).) two

results

are

*-transforms

of

well

known

Section

307

6.5 Martingales

*WARTINGALE

(6.5.2) THE

If

X

:

T x R

then f o r any

>

x

+

WAXIKAL INEQUALITY-

*IR

At,

after

0

>

P[ max X(t) At
(a)

*s u b m a r t i n g a l e

is a

x]

I

1 ;EIX(1))

and

If x

M

>

:

0.

T x R

* *IRd

is

*m a r t i n g a l e

a

At and

after

then

PROOF :

See Breiman Cl96S. 5.131 or Doob [1953] or Helms & Loeb [1980].

IMl

Note

* submartingale

is a

-

by remarks above.

(6.5.3) THE UPCROSSING LEMWA:

X

Let Let

b pa(w)

X(At.o), [a.b]

:

T

x R

equal

*IR

be a

the number

of

*s u b m a r t i n g a l e ttmes

X(At+6t.w).**-,X(l.w) f r o m left to right.

that

crosses

after

At.

the sequence the

interval

Then

PROOF :

See Breiman Cl968. 5.171 or Doob [1953].

Our

next

definition

*martingales

of

result,

(6.5.6).

At-martingales.

are not

too

is

a

It

preliminary shows

ill-behaved, but

that

for

the

paths

of

first we give an

308

Chapter

6:

Hvuerfinite Evolution

example that shows that i t is the best we can hope for.

(6.5.4) EXAMPLE:

p

Let

:

W

+

{-l,l}

be

+1

on one half of

on the other half, as in (4.3.2).

N

is a nasty martingale with

(6.5.5)-it

W

and

-1

Define

the paths as shown in Figure

splits a jump in infinitesimal time.

p = -1

4

p = -1

4

1. I 6 t

0

1

2 2 Figure (6.5.5)

(6.5.6)PROPOSITION:

M

Suppose

:

T

E[IM(1)1]

A t Z 0 and

x

"IR~

R

has a

M(l)

is l i m i t e d .

t h e r e is a n L n f i n i t e s i m a L

vt

*m a r t i n g a l e

is a

in

T

E FL1(R).

s u c h that

after Then

M

a.s.

vt-decent path sample.

PROOF : By (6.5.2). of the proof

we

max IM(t)l is finite a.s., so for the rest O
Section

6.5

309

Martinzales

We use (6.5.3) to show that both

S-lim M(t)

S-lim M(t)

and

t tr

t lr

exist and are finite for every that

the

limits exist

r

It suffices to show

a.s.

Mj,

for each coordinate

I

1

j

I

d.

M.

of

Let

be

X(t)

S-lim X(t.o)

one

coordinate

If

H(t).

of

either

S-lim X(t,o) fail to exist for some r in t lr 1 2 C0.11 then there are sequences t,(w) and t,(w) that tend 1 r and S-lim X(t,(o).w) = S-lim inf X(t,o) monotonically to m 2 < S-lim sup X(t.w) = S-lim X(tm(w).o). As a result there must m be a rational interval [a.b] that X(t.w) crosses an infinite b b number of times, or pa(o) is infinite (in *IN) for pa as in or

tt’

(6.5.3). and

Since

U[{Pa

b

=:

m}

E [ P : ] :

a

<

b in Q]

is a

>

P[tl{P:

is finite, by (6.5.3).

= 0

m}]

P-null set.

As a result of the last two paragraphs, except for a null set all

the

(5.3.25)

M

coordinate has a

limits exist

and

are

finite, s o

vt-decent path sample.

(6.5.7) PROPOSITION: Let and let

X(U) x(T)

X T t S

E

after

SL’(R). A t and

be a positiue be a

*submartingale

At-stopping time mith

S-integrabLe.

X(u)

In particular. if

M(1)

E SL1(R),

then

w

after T

<

u

At Z 0

<

E SLl(R),

is a

M(T)

1.

If

then

*martingale

E SL1(R).

by

310

Chapter

6: Hvperfinite Evolution

PROOF : Let

ET[X(u)]

definition,

X(T)

is

2 X(T(U).W)

E,[X(u)](o) k = 6.

infinite and

I ~ ~ ( ~ By~ the 1 . submartingale

= ECX(u)

2 0.

Let

h

be positive

The following calculation shows that

S-integrable.

For the second inequality, break the sum into a sum over classes

[u~'~)].

For the second to last use the

The

last

part

follows

X = lMl 1

maximal

5

inequality, (6.5.2) E,[X(u)](w)

*submartingale

*martingale

max X(t.o). o5t5u from the first

by

taking

the

~ ( 1 )E FL~(R).

or

0.

(6.5.8) REMARK:

If

E[lMI2]

H E 0.

is an then

L2-finite I

indicator function of

is

*martingale.

S-integrable.

A C R.

then

Suppose

IA

is the

Section

311

6.5 Martingales

*Cauchy

by the

the left hand side is also. just shown, implies A t 2 6t

When

P[A]

inequality, s o i f

S-integrability for

E

II

:

S-AC. which we have

The condition

(3k

P.

U. let

belongs to

UA = {t

is infinitesimal, then

= k-At]}

*lN)[t

E

U (1)

(6.5.9) DEFINITION:

We

shall

say

M:Uxn+*Rd infinitesimal

that

is

an

a

A t 1 6t

internal

At-martingale

U

in

M

if

At-decent path sample and if wheneuer UA,

to

M(t)

is

I

for

is a

correspond

s

a.s.

has

<

belong

t

a

S-integrable.

*martingale

some

<

some

u s ] = M(s.u)

Our results above show that if

M

0

for

then

E[M(t)

and

process

after

infinitesimal under

M(l)

vt =: 0. then At

projection

in

U.

to

the

is

I

S-integrable and

is a The

At-martingale At-martingales

uniformly

martingales with respect to the progressive

integrable

filtration.

The

last phrase is such a mouthful, that we introduce a term for it.

312

Chapter

6:

Hvperfinite Evolution

(6.5.10) DEFINITION:

A M

:

-

uniformly

[O,l] x R

integrable

IRd

progressiue

(6.5.11) THE

If

M

is called a hypermartingale if

a.s. has decent paths and for each

I

E[M(s)

process

<

s

in

a.s.

P.

r

%(I-)] = M(r)

[O.l],

At-HARTINGALE PROJECTION THEOREM:

M

:

II x R + *IRd

is a

M,

decent path projection.

At-martingale. then its

is a hypermartingaLe.

PROOF : By (6.3.7)(b) uniformly

I

E[M(l)

we see that

integrable

ot]

= S-lim M(t) t lr

progressive

E[%(1)

lifts = %(r).

P

I

is indistinguishable from a process.

By

and

S-lim E[M(1) t lr

A(t)].

(1.5.10).

We could finish by using (6.1.5)(e)

I

ot] and

the reverse martingale convergence theorem from Doob [1953. p.

3281, but here is a direct proof.

If

A ' E %(r).

= E[%(r)IA,].

with %(r)

t l =: r

= st M(t2)

we

wish

to

show

that

E[%(l)IA,]

Let

A be an internal event determined at

and

P[A

a.s.

v A'] = 0. Take

EIM(t)IA]

=

Let

t2 Z r

t = max(tl.t2).

1 M(t,o)6P[ot] CotlGA

tl,

be such that

Section

where

6.5

the first

sum

is over a selection of

A.

classes contained in nonanticipating Z

EIM(l)IA]

313

Martinvales

the equivalence

M(t)

which is well-defined since

E[8(r)IA,]

A E .U(t).

and

is

z EIM(t)IA]

E[8(l)IA,].

Z

(6.5.12) DEFINITION:

M

If we

N

say

provided

:

II x R

there

N

that

[O,l] x R

:

is a h y p e r m a r t i n g a l e .

+ “ I R ~is a m a r t i n g a l e

is

is a

IRd

+

a positiue

then

M

lifting o f

At E T

infinitesimal

so

At-martingale w h o s e decent p a t h p r o j e c t i o n ,

N

N,

M.

is i n d i s t i n g u i s h a b l e f r o m

(6.5.13) THE HYPERMARTINGALE LIFTING THEOREH: Every

N.

hypermartingale

Mp(l)

If

has

is i n t e g r a b l e . f o r

NP(t)

so that

M

a martingale

lifting

p 2 1 , w e may c h o s e

N

is S - i n t e g r a b l e .

PROOF : Let

N(1)

be an

T

has a

Np( 1)

I

E[N(l)

at]

lifts

We know that

ot].

I

E[M(l)

NP(t)

(Integrability of

when

.U(t)].

At

N in

t = kAt

follows from (6.5.7) when

is S-integrable. ) One way to prove that

is to use

M(r)

N

is a

At-martingale lifting of

M

the reverse martingale convergence theorem of Doob

[1953. p . 3281. and

(resp. an

At-decent path sample for some infintesimal

and that

(1.5.10).

I

= E[N(l)

N(t)

SLP-lifting) and define a.s.

M(l)

S-integrable lifting of

= E[M(l)

Since

I

%(r)]

%(r)

= n[l(t) a.s.

: t

we get

>> E(r)

N(t)

r]. =

N(r)

fi(r),

a.s. for

Chapter

314

all rational

6: Hvperfinite Evolution

This implies indistinguishability since both

r.

have decent paths.

N

Another way to prove that

M

of

then

I

are

a.s.

4 %(1-)

l(t,)]

every

9(l)-measurable.

%(1-)

I

= M(1-)

a.s..

a.s.

(Note

%(r)]

t = 1

by

At.) tmT1

sequence

Also, both

Exercise

Since gives

and

M(1-)

%(1-)

shows

(6.1.11)

that

2 9(1), so ordinary martingale convergence, Doob [1953,

3191

M(1-)

= E[M(l-)

%(r)]

a. s. ,

= R(l-)

S-lim N(t) t Tr E[M(l)

I

E[E(l-)

At-sample could fall short of

that the

p.

At-martingale lifting

is to show that the left limits,

because

X4(tm)

is a

=

see

or

8(1-)

integrable

the

a.s.

proof In

martingale

of

Helms

fact, we

convergence

8

Loeb

[1980].

only

need

the

theorem.

shows

uniformly

Helms

€k

Loeb's

[1980] proof of the special case (as well as their proof of the

full theorem) is based on a result like (6.5.14) below.

-

A third way to complete the argument is as follows. know

g(r)

a.s. as

%(1-)

limits; in fact, for a.s.

We

claim that

rfl.

t = 1

on the

E[M(l)

I

since decent paths have left D-sample,

9(1)] =

claim establishes the result, because if

and the paths of

M

and

N

We

8(1-). r < 1

are decent.

st N(t)

= i(1-)

Proof of

this

is rational,

The claim above is

6.5 Martinaales

Section

315

If

established as follows.

A

there is an internal

N(l)

(6.1.12). We know

The next

E

result

E J(1)

A

A(t)

t

1

P[A v A ‘ ] = 0

by

then for each

such that

M(1).

ntegrable lifting of

is an

(mentioned above)

is included because

so

it

seems potentially useful.

(6.5.14)ALMOST EVERYWHERE CONVERGENCE LEMMA:

Zm

Let

W i t h o u t Loss o f g e n e r a l i t y . m e may a s s u m e that

uariables.

{Zm

:

m

E

be a c o u n t a b l e sequence o f measurable random

‘‘w)

is

subscripted

{Xk

:

the

projection

elements

k E *IN}.

of

an

of

the

finitely

internal

sequence

T h e following a r e equiualent:

o

(a)

jim

(b)

For s o m e i n f i n i t e

+

”

a.s. n

IX

max Iz = k<.j
a n d all i n f i n i t e

o

k

<

n.

n.s.

PROOF : There

is no

because each

Zm

loss of generality

in

could be lifted to

Xm

the

internal

sequence

and then the sequence

could be extended using countable comprehension. Let a

>>

0.

nm& then

= {u : Ixm(o) I

>

a}.

If

Zm

+

o

a.s.

and

Chapter

316

where

the

intersection

S-lim P[ U R:] h m>h mp(e)

p

= 0

and

union

or, for each

P[

such that

]:R

U

6:

Hvperf ini te Evolution

are p

1 < F,

(I

IN.

over

Thus

there exists

in

for each finite

so

n,

m>m

P

n

Ri]

P [ , U

1

< F.

Since the last statement is internal i t must

P also hold for some infinite p E aIN.

E

E

For any

E

internal

Let

n E fl{[mp(E),np(e)]

:

which is nonempty by saturation ( 0 . 4 ) . n 0 and infinite k , P[ U R;] Z 0. Therefore the j=k

OQ+)

>>

np(e).

n *IN.

set

{E

E

* R+

:

P[ max

IX I

> €3 <

contains all

E )

k<j
E

and (b) holds for a particular infinitesimal

from

6

the set. If

(b) holds, then whenever

internal set k(9).

{k

P[ max I X j l > kljln n P[ U ]:R < 9. m>k(8)

E *IN

Therefore

E

:

>>

€1 < so

9

and

0

>>

0

the

contains a finite

9)

P[fl

R:]

U

= 0

and

k m>k

final ly ,

The last null set is precisely the set where

X

j

-/-+0, s o (b)

implies (a).

(6.5.15) EXERCISE: Let =

P[A

I

A E R ot].

be

Show that

an

M

internal is a

set

and

*martingale.

define

M(t,w)

Section

6.5

317

Martingales

(6.5.16) EXERCISE: Prove

the Derived Set Lemma

pick

an

P[A]

If

hints of Perkins.

= 1,

Rm 2 AC

internal

(6.3.6)

with

the following

for each finite

such

<

P[Rm]

that

m , we may 1 3 );( . Use

(6.5.15) and (6.5.2) to show that

The Borel-Cantelli Lemma (cf. Breiman [1968]) says that if 2

<

P[A,]

U Ak] = 0. (In other words m k>m "infinitely often" is zero.) P[n

then

m,

Am

probability of

Am =

We let

{w

I

max P[Rm

:

wt]

> -1-}

the

and apply the Borel-

t

Cantelli Also,

R,

let

I

P[R,

to obtain a Loeb null

lemma

wt]

provided

= fl

R m.

is always w

Q Ro

2 AC

R,

so

defined.

(because

and being a Loeb

Show

I

max P[R,

ot]

that 1

--

<

t

large

m).

Use this to conclude that

Ro 2 ( f l U Ak). m k>m

set

->

A'

I

P[R,

w

t

set

]=

0

for sufficiently

A\Ro.

(6.5.17) PROPOSITION:

M

Let

be

At-stopptng time. also a

a

At-martingale

The process

and

N(t,o)

let

= M(t A

be a

T

T(o).w)

is

At-martingale.

PROOF : Exercise with these hints: two cases,

T ( W )

<

t

path for the sample S-integrable?

and w.

T(O)

Compute

>

t.

what about

If N

E[N(t+at)

M

has an

along

o?

I

ot]

in

S-decent Is

N(l)

ChaDter

318

6:

Doob’s inequality is a well-known

*martingales

obtain for

by transfer.

the standard proof yields extra

Hvperfinite Evolution

result which we could However, a variation on

S-integrability

information

which we develop now.

(6.5.18) DEFINITION:

0

<

T.

At E

xA (t.w)

*IR

X:TxR--,

Let

D e f i n e the

= max[lX(s.o)l

be

internal

At-maximaL f u n c t i o n o f

:

o <

s

<

t. s E

and

X

H~], f o r

Let

by

t E

uA

where

TA = {t

T : (3k

E

E

*a)[t

= kAt])

U (1).

Here is a useful basic fact about maximal functions.

(6.5.19) LEMMA: For e u e r y p o s i t i u e

aP[XA(t)

>

a]

* s u b m a r t i n g a l e X,

<

1.

E[X(t)I

{XA( t 1>a)

PROOF : Whenever

is a

6t-stopping time, the reader should

E[X(t)

I

T]

when

E TA : X(t)

>

a] A 1.

T

verify that

X(T)

Let

T(W)

= min[t

<

T(W)

so

<

t

Section

6.5 Martingales

aP[~
319

H[X(~(o),o)6P(w)

I

I B[E[X(t.o)

>

aP[XA(t)

a

>

if either

a]

<

T

T ( W )

s(o)]6P(o)

H[X(t.W)6P(W) Finally, XA(t)

:

<

Z[X(t.w)6P(o)

:

T ( W )

t

or

:

<

<

t]

:

T(O)

<

t]

t].

XA (t)

=

X(t)

>

a.

so

XA (t.w) > a].

0

This proves the lemma.

The standard form of Doob’s inequality is obtained from the lemma as follows. positive

Let

M

* submartingale.

be a

*martingale,

Write

the result of

]MI

so

is a

the previous

lemma as

and integrate both sides with respect to the measure on

[O,m),

obtaining

by Fubini’s Theorem.

Holder’s inequality gives

paP-’da

Chapter

320

6:

Hvperfinite Evolution

This proves the first part of the next result.

(6.5.20) DOOB’S INEQUALITY: Let vt

<

M

be a

*martingale

A t E T, A t 2 0. Then for

If

IM(t)lP

IMA(t)

Ip

is

i s also

vt,

after t E

S-integrable.

1

<

p

<

m

and

HA,

M(t)

E

SLP(n)

then

S-tntegrable.

PROOF : Let

b

>

0

be infinite and compare

] da.

Use Fubini’s Theorem and Holder’s inequality as above to obtain

S-integrability of

because

of

Lemma

infinite constant

lM(t)

1’

(6.5.19)

b = a.

means that

applied

to

lMl = X

with

the

321

(6.6) Predictable Processes

of "predictable"

S(tandardizab1e)-notions

The internal and a r e a s follows.

(6.6.1) DEFINITION:

For each

L

>

internal.

r-predictable O,u

t E

function

Y,

let

I

E T : s

G

for all

if

:

t €

T

t-r].

R + *M

x

Y.

called

is

whenever

two samples

satisfy

An internal set process t E

and

= 0 V max[s

[t-r]

An

0

T.

Iq((t.w)

ts.

the sections denote

1-Pred

1-predictable t f the indtcator

t s

9(

the

or

in other

words, t f

are determtned at

qt

internal algebra o f

for all Let

[t-11.

1-predictable

sets.

An tnternal set is called ftnttely predictable if tt is the

e-predtctable for some external

algebra

>>

0.

Let

conststtng

of

e

F-Pred

all

denote fintteLy

predictable tnternaL sets. Ftnally. we say a subset

IE

T x R

t s predictable

if tt belongs to the smallest stgma algebra contatntng the ftnttely

predictable

predictable sets by

sets. Pred.

We

denote

the

famtly

of

ChaDter 6: HvDerfinite Evolution

322

The

0-predictable

sets are

nonanticipating for all time.

When

<

0

L

the

sets which

are

not just after some infinitesimal

t,

we have the inclusions

0.

%

simply

F-Pred C r-Pred

C

0-Pred.

(6.6.2) NOTATION: Let

3.8

the collection o f a l l

denote

C0.13 x R

of

subset o f

[O,l]

K

the f o r m and

A

is

3.8

adapted or previsible.

where

K

is a c o m p a c t

is a n i n t e r n a l s e t d e t e r m i n e d

r = min K.

before the instant

Every set in

A

x

subsets o f

A

E

J(r).

Bore1 x Loeb

The family

3.8

measurable and

8-

is a semicompact paving

in the sense of (2.2.2).

If

S

is a family of sets. we denote the smallest sigma

algebra containing

F

by

I(%).

finitely predictable sets and

Here is the connection between

3-8.

(6.6.3) LEMMA:

If 9( E

91

F-Pred.

in f a c t ,

is

a

finitely

predictable

internal

set,

t h e n i t s ( h a l f ) s t a n d a r d p a r t is p r e u i s i b l e .

s(91) E 2(3-8).

PROOF : The proof is a refinement of the proof of (5.4.1). Suppose that

91

is

€-predictable.

Let

{(am,bm)}

be an enumeration

of all standard open rational endpoint intervals of length less

Section

than

Define a sequence of internal sets by

E.

91,

= {a E

Since

b m -am

before

am.

<

R

: (t,w) E: 'II

t E

€-predictable.

* (am,bm)}. 'IIm

is determined

l-8-sets by the remarks above.

( r . 0 ) E s(91)

and

(r.0) E [O.am] x R

then either

r Q (am.bm)

then

st(t.o) = ( r . 0 ) I 'I,.

is

for some

Each of the sets

Suppose that

o Q 91,.

'II

and

E

is a finite union of

of

323

Predictable Processes

6.6

and

Hence,

(r.0) E [O.a,]

some

If

t % r

satisfies

w

an element

which would make x R

w E qm.

( r . 0 ) E [am,l] x 'II.

or

because

(t.a) E 91

If

is fixed.

m

We claim

or

(r.0)

E

[bm.l] x I'I;.

To show the opposite inclusion of our claim, suppose that (r.0) E

4".

Choose

r.

decreasing to

a

(r.0) E '?lm

Since

subsequence. we must have tm

* (am.bm)

such

sub-subsequence {t,}

so

subsequence

and

intervals

(t,,W)

ltm-rl

E

91.

(tn.a) E 'II the lemma.

m

<

n.

and

(t,,o) st(t

n

E 9(

Now

is decreasing.

) = r.

and

It,-rl

<

for the

so there is a

to an internal sequence and choose an infinite

for all

(am.bm)

r E (a,.bm)

(r.0) E (am.bm) x 91,.

that

that

of

choose Extend n

ltm-l-rl.

a

this

so that Then

This proves the claim and hence

324

Chapter 6:

Hvperfinite Evolution

(6.6.4) PROPOSITION:

W E

Z(1.9))

if and onLy if

s

-1

(W) E Pred.

PROOF : A proof very similar to (5.4.8) works here.

algebra of

I(%-9))sets

I x A

sets of the form or closed) and

instant of

I.

A

T(a+ ,;1

n {"

U(a- ,;1

m

in

where

I

is an interval (either open

is internal and determined before the left

U

A

consisting of finite disjoint unions of

Since

-1 (IxA) = s

and

1 b- ;)

x

1 b+ m -)

x A,

is determined finitely before

Pred.

sets.

Consider the

I

A ,

a

= (a,b)

I = [a.b]

in

T,

these sets are

the sigma algebra generated by finitely predictable

The Monotone Class Lemma completes

the proof

of

the

forward implication, because the monotone class of this algebra has

s

-1

(I) E Pred.

The proof of the converse is nearly the same as (5.4.8) except now we may derive s -1 (W) and s -1 ( W c ) from the Souslin operation (2.2.1) on finitely predictable sets so

and

9

o In

and

$,I,.

Section

Thus, by 1y E

325

Predictable Processes

6.6

the

Separation

(2.2.3)

Theorem

and

Lemma

(6.6.3).

Z(Jy.9).

Recall the Souslin operation from ( 2 . 2 . 1 ) . X.’3-Souslin

set is

if i t can be derived from

We say that a sets by the

Jy-’3

Souslin operation.

(6.6.5) LEMMA: 91 C T x R

If s(91)

’3-adaptation o f

i s internal, then the

Jy*’3-Souslin; in f a c t ,

is

:

’3(91) = ((r.0)

3(t.u)

E

91 with t

%

r and u

;a}

E I(%-9).

PROOF : The

set

containing

J(91)

is simply

the smallest

9-adapted

set

~(91).

Define a decreasing sequence of internal sets by

: 3u E R

Each

is finitely predictable, so by Lemma (6.6.3) each

s(91,)

91m

with ( t . u )

E

91

and

u [t-l/m]

E Z(X-’3).

A simple extension of the proof of Lemma whenever

is

m’

s(nrm) = n s ( 9 1 , ) . = 9(9)

for the

2

%a(%).

a

decreasing

(2.2.4)

sequence

of

shows that

internal

thus i t is sufficient to prove that

sets

s(n91,)

’Ilm’s defined above.

%Lm 2 91

Each ns(91,)

= Jt-l/m]

1.

q m = {(t,a)

and

E(X.’3)-sets

Conversely, suppose

(r.a

are E

9-adapted.

s(nrm).

so

Then there

Chauter 6: Hvperfinite Evolution

326

is a u

m

t so

that

sequence n

such that for every

r

N

and

E 91

(t,u,)

u

(t,w)

m,

[t-l/ml

€ 91,,

or there is a

= w[t-l/ml.

Extend the

m

to an internal sequence and select an infinite

{u,}

such that

E J(91).

(t,un)

This proves the lemma.

Recall from section (2.2) that we call a set a Henson set

if

it

can

be

derived

from

internal

sets

with

the

Souslin

operation.

(6.6.6) LEMMA:

If s(Y)

Y

i s a Henson set and

ts

s ( f )

9 - a d a p t e d , then

%-%-Sousltn.

ts

PROOF : f

Let

be generated by Souslin's operation on internal

sets,

where

we

may

assume

each

intersection

is

decreasing

(if

n necessary, replace ' u In

).

9

fl

by

m=l

Our lemma reduces to

ulm

the claim

%(9uln) denotes the

where

previous lemma which says words,

the

Souslin

%adaptation

9(9uln) is

operation

of

~ ( 9 ~ 1as~ in ) the

%*%-analytic.

on

l-9-Souslin

In other

sets

gives

l-9-Souslin. Suppose

that

)

(r.0) € fl 9(9

so

that

for every

0 1 .

there exist

(tn.un)

9uln

with

tn

N

r

and

u

-

n r

w.

n

We may

Section

{(tn,un),Suln}

extend

whenever

m

<

to an internal sequence satisfying

Y u l m 2 YDln.

n.

there is an infinite (tn.un)

(r.un)

327

6.6 Predictable Processes

E

s(U).

E

for

9(J I m

n

have assumed that

s(f)

satisfying these conditions.

all

Finally,

Since the extension is internal

finite

m

( r , w ) E s(U)

is

and

tn

because

N

Such a so

r.

and we

un r * o

%adapted.

(6.6.7) THEOREH:

A

I g [O,l] x R

set

[Borel x'Loeb]-measurabLe

IE

1(1*9)

if

and

and only t f

is

preuisible.

that

is,

J-adapted. if and onLy if -1 s (I) E Pred. its inuerse

standard part is predtctabLe. PROOF : Assume that and

s-'(IC)

I

are

is previsible.

Loeb(HxR)-measurable

By ( 5 . 4 . 8 ) .

both

s

-1

(I)

and hence analytic over

s(s-'(I)) = I and internal sets. By Lemma (6.6.6). -1 c s(s (I ) ) = ' I are both l-8-Souslin sets. Therefore the the

Separation Theorem (2.2.3) says

I

If

E

8(l*J).

I E X(l.9).

i t is previsible because every

1.9

set

is. Equivalence of the last two conditions was proved in Lemma (6.6.4).

Every previsible process is is

p =

(T

If

Z

B E [O.l]

0

s-l)-measurable is a

and therefore

r-almost previsible.

p-measurable process and there is a Borel set

such that

and such that

Borel x Loeb-measurable. hence

p[BxR]

Z(r) = 0.

is

%(r)-measurable

then the process

unless

X ( r ) = Z(r)

r E

B

for

328

ChaDter 6 :

r 4 B.

= 0

X(r)

r E B

for

is

p-measurable and

In this case (6.2.6) says that

X

the result above implies that

X

r-predictable

If

r-lifting.

Hvperfinite Evolution

vi

is

%adapted.

w-almost previsible and

Z

and therefore

= u

has an

is the path measure of a

differential process from Chapter 7 and i f

b

is a point of

b = 0).

a.s. continuity of that process (e.g..

then

B

= {b}

satisfies the conditions above.

(6.6.8) THE PREDICTABLE LIFTING THEOREM: Let and

x

:

be a n y b o u n d e d h y p e r f i n i t e m e a s u r e o n

T

let

be

L

[0.1] x R

a n internal

*M

is

inftni tesimal,

X

:

st Y(t.o)

has a n

a normed space and

Y

2 0.

If

s u c h that

# X(s.t(t),w)l

r-predictabLe

X

L

r - a l m o s t p r e u i s i b L e . t h e n t h e r e is

L-predictable process

w{(t.w)

that i s ,

an

Y x R

is b o u n d e d ,

= 0.

r-lifting.

Y

If

M

is

may b e c h o s e n w i t h

the same bound.

PROOF : Since process

W

X

is

r-almost previsible, there is a previsible

such that

W

(We may choose previsible if

W

Pred-measurable.

bounded if is.)

X

is. for example;

By Theorem (6.6.7).

W(st(t).o)

W A b

is

is

The sigma algebras generated respectively by

Section

6.6

finitely

329

Predictable Processes

predictable,

r-predictable.

0-predictable

and

internal sets satisfy the inclusions:

Pred

C_

Therefore

Z(L-Pred)

Z(0-Pred)

W(st(t),o)

and (1.3.12),

Y

C_

W

o

is

0 2

Loeb(Uxf2).

Z(L-Pred)-measurable

has a n

s

C_

L

so by

(1.1.8)

Y.

(Also,

r-predictable lifting

W

may be chosen with the same bound a s

if

2 0

W

is bounded.)

This completes the proof. Consider

Bore1 x Loeb-sets of the forms

(0) x A .

A

E

J(0) t l Loeb(f2)

or A E %(rl)

x A,

(rl.r2]

We shall call these sets

n

Loeb(f2).

basic preuisible sets.

Because of the

open left end on the intervals, i t is easy to see that they are 9-adapted.

O n the other hand, i f

then

A

This

implies

E

generated

%(rl

by

-

1 -) m

that

A

is internal and in

for sufficiently large finite

each

Z(JI*8)-set

the basic previsible

is

in

sets.

the

m.

important,

but

requirement and more.

now

we

relax

the

By Theorem

Loeb

so that

sigma algebra

basic previsible sets generate all previsible sets. very

9(rl),

(6.6 7).

This is not measurabi ity

330

Chapter 6:

Hyperfinite Evolution

(6.6.9)DEFINITIONS: W e c a l l sets o f the f o l l o w i n g f o r m s

(0) x A.

A E A(0)

for

(the

P-completion o f

!lJ(O)),

i n the

P-completion o f

%(rl).

or

(rl.r2]

x

A,

A

for

basic almost-preuisible g e n e r a t e d by

sets.

Sets in the sigma algebra

the basic almost-preuisible

sets are called

almost-preuisible sets.

A bounded hyperfinite measure called

P-continuous

i f urheneuer

T[U

The

x A]

total path variation

r

on

P[A] = 0.

U x R

is

then

= 0.

measures

of

Chapter

7 are all

P-continuous.

(6.1.6) tells us that basic almost-previsible

Proposition sets are

r-almost

previsible

A E %(rl).

T.

For example, i f

R1

determined at a time

tl

for every

P-continuous

then there is an internal set rl

such that

P[A v R1] = 0.

The set

(rl,r2]

x Q1

is a countable union of

and

r

0

s -1 [(rl,r2]

measure

x (A v

R,)]

= 0.

3-J-sets

Section

6.6

331

Predictable Processes

We may extend this observation a s follows.

(6.6.10) PROPOSITION:

W

If to

the

is a n a l m o s t p r e u i s i b L e set. that i s , b e L o n g s

sigma

preuisible

algebra

sets and

hyperfinite measure

generated

if

is a

7

T

on

by

R.

x

the

basic

almost

P-continuous

W

then

bounded r-aLmost

is

previsible.

PROOF : Disjoint finite unions of basic almost previsible sets form a n algebra. for example.

Thus we

may

apply

the Monotone Class Lemma

with this algebra.

(3.3.4)

T h e remarks before the statement show that

If

the algebra has this property.

Wm

is a monotone sequence

of almost previsible sets such that for every

Ym

previsible sets (resp..

nWm)

and i f

W = Ulm

is

beginning

such that

7

0

7

s -1 [Wm v Ym] = 0,

r-almost previsible because

nWm)

(resp..

there exist

U = UUm

and

r[Ulm

then

UWm

v Y,]

= 0

(resp..

flym),

then r [ W v U]

Therefore r-almost class

<

r[Ulm v Y ,]

the collection of previsible for each

containing

previsible sets.

the

almost

= 0.

previsible

P-continuous

algebra

generated

This proves the result.

7

by

sets

which a r e

is a monotone basic

almost-

ChaDter 6:

332

HvDerfinite Evolution

The almost-previsible sets are very nearly the sets which are "predictable with respect to the usual augmentation of in the sense of Dellacherie & Meyer [1978].

%''

They would also

permit basic sets of the form

(0) x A ,

for

A

in the completion of

We say that the measure T

n]

s-~[(O) x

o

= 0.

%(O).

is continuous at

T

zero

if

The total path variation measures from

Chapter 7 are continuous at zero.

(6.6.11) PROPOSITION:

K

If

[O,l] x R + M

:

the usual augmentation o f continuous

G

:

[O.l]

zero,

at x

R

T 0

M

-+

s

is preutsible with respect to %

then

and

7

there

is

P-continuous and

extsts

a

preuisible

such that

-1 ((r.0)

: K(r,w)

#

G(r,o)) = 0.

PROOF :

Let

a E M

H(r.o) = K(r.w)

and define

T

We

may

apply

r

if

o

>

H(r.o) = a.

0. Then

s -1 [K(r.w)

(6.6.10)

and

#

H

H.

r = 0.

and

is almost previsible and

H(r.o)l

the

approximations to obtain a previsible equal to

if

= 0.

usual G

simple

function

which is

r-almost

Section

333

6.6 Predictable Processes

(6.6.12) DEFINITION:

A

basic

almost-previsible

process

is

bounded

a

process of the form

m

<

O = r

where

<

r2

***

<

rm

<

r m + l = 1.

is

h0

a(0).

measurable with respect to the completion of

and D(rj)

is measurable with respect to the completion of

hj for

j = l,***.m

(and all

h.'s are bounded). J An almost preuisible process is a process

measurable with

resepct

to

the almost

that

preuisible

is

sigma

algebra generated by the basic almost-preuisible sets.

Notice that we have extended our use of the term "basic." The indicator function of a basic almost-previsible s e t basic

almost-previsible

process,

but

so

is

any

is a

linear

combination of disjoint indicator functions.

(6.6.13) LEMMA: Let measure on (bounded)

be any

T

T x R basic

r-predicttble

P-continuous bounded hyperfinite

and let

L

satisfy

almost-prevtsible

n-lifting of the form

process

01

H

L

>

0. has

A an

334

ChaDter 6:

-

j = 1.-** ,m,

for tj rj internal functions gj

where

Hvperfinite Evolution

tm+l -

1,

and

haue the same bound a s

the

H.

PROOF : By

Lemma

measurable sj

Z

rj,

(6.1.7)

there

functions,

gj(o),

j = 1 . - - * .m,

for

exist

bounded

determined

0

J

times

s

<

this, for each in t erna 1 sk = min[t

(t,w) E qk will do.) to

k

such that

IN

in

qk E st-'(a)

x R

because

-1

R

x

(r.)

J

a E CO.11. n[[s,t]

with

x

r[qk]

for some

(If

[sk.tk] x R

The set

st-l(a) x R ,

01.

st

sk

N

is

T-

There exist

R]

p.

Z

use inner-measurability

E U : (t.o) E duk for some

0'

t

j' for any

(a) x R]

s Z t

a ( t.

s

. i

measurable to show how to select

-1

J

0 =

times

h (w)] = 0.

We use the fact that each of these sets

p = r[st

at

.U(s.)-

satisfying

P[g.(w)

Let

internal

qk

01

and

>

to find an

P -

1 s*

t k = max[t

is empty, any

To see

Let E U :

t k Z sk Z a

is a larger inner approximation -1 tk Z a since qk st (a) x R.

This means that

r"s,.t,l

Extend

the sequence

{sk,tk}

x

n1 >

P

1 - E.

to an internal sequence and use

Section

335

6.6 Predictable Processes

Robinson's

Sequential

Lemma

and

the

Internal

Definition

Principle to select an infinite n s o that sn Z tn 1 n[[sk,tk] x R] > p - i; f o r all k < n. Let s = min[sk t = max[tk

and

:

k

n].

Now, having chosen sj z rj,

so

gl

imally

gj's

is determined at if

k

<

n]

R] z p.

x

determined at

to = 0.

let

:

and

a

Then

r[[s.t]

and

Z

Choose

[tl-r].

necessary, s o

that

tl

>

J

n

tl

so = 0 tl z sl.

but

sl+t,

Also increase r[(st-'(O)

with

s.

infinites-

[tl.l])

x

R] = 0.

The latter is possible by the remarks of the previous paragraph. Choose each t

>

t

s +L.

j

tJ

5 j tm+l = 1.

j = l.**-.m

for

sJ

and

n

w[(st-l(r,)

r[( ( tj. tj+ll v st-l(rj.rj+l

(0) if

G

(tj.tj+l ]

j = 0).

x

[t,,l])

R] = 0.

Let

j = O.l.--*.m.

Then for each

(replacing

in this manner, that is,

by

[O,tl]

1)

x

n] = 0

and replacing

This means that for these

j

I s

(rj*rj+ll and

gj

by

with

as above,

r[st

by the

G(t,w)

P-continuity of

f

H(st(t).w)]

= 0

7.

The next result is also helpful in stochastic integration.

336

ChaDter 6:

HvDerfinite Evolution

(6.6.14) LEMMA:

V

S u p p o s e that preuisible

processes

If

space.

processes

V

with

ualues

of

a

in

bounded almost-

separable

normed

c o n t a i n s the basic almost-preuisible

and

convergence,

is a u e c t o r s p a c e

is

then

closed

V

under

contains

bounded all

pointwise

bounded

almost-

p r e u i s i b l e processes w i t h u a l u e s i n that space.

PROOF : We will show that for any almost previsible set vector

b

in the range space, the function

1y

bI#(r,w)

and any

E 1.

This

proves the lemma because all bounded measurable functions are bounded pointwise limits of sums of these "simple" functions. Let

b

be an arbitrary but fixed range vector and consider

the collection of sets

Z(b) = {H-l(b)

Every basic function

:

H E V

&

H takes at most the values 0

almost-previsible

bIA(w)I(q,s,(r)

set

is

belongs

in

to

Z(b)

V.

V

is a vector

because if

space.

Hi1(b)

Finally,

Z(b)

because

Finite

unions of basic almost-previsible sets belong to

& b)

Z(b)

disjoint because

is a monotone class lim Hm

is either increasing or decreasing,

takes only the values

0

and

b

and belongs to

Monotone Class Lemma (3.3.4) shows that sigma algebra of almost-previsible

the

Z(b)

V.

The

is the whole

sets for each vector

This proves the lemma as remarked above, since all bounded

b.

Section

6.6

functions

337

Predictable Processes

are

limits

of

"simple"

functions

(partition

the

to extend

the

range).

(6.6.15) REMARKS ON EXTENSION TO Only minor

technical changes are required C0.m) x R

results of this section to (5.5.4).

C0.m):

in the framework of

One change which we mention explicitly is this.

In

order for finite disjoint unions of basic almost-previsible sets to form an algebra we must also include sets of the form

(r.m)

x A,

for

A

in the

P-completion of

Also, a b a s i c a l m o s t - p r e u i s t b l e p r o c e s s o n

C0.m)

O(r).

is one of the

form

-

m- 1

where

0 = rl

<

r2

<

0 . .

<

to the completion of

9(0)

to the completion of

%(rj)

rm, and for

ho hj

is measurable with respect is measurable with respect

j = l.***,m.

One would also expect the measures

T

to satisfy

r { O x R } z r{T x R } ,

as the path variation measures of Chapter 7 will: however, this is not required f o r the results we have stated. meaningless. not false.)

(They become

338

ChaDter 6:

HvDerfinite Evolution

Extension of some of the progressive notions to

[O,m)

more technical and is outlined in detail in the next section.

is

339

(6.7) Beyond In

[O.l]

section

with Localization

(5.5) we

indicated

analysis of paths of processes on

how

[O,l]

the

infinitesimal

extends to paths on

We will refer to the same internal time scheme (5.5.4)

[O,m).

in this section.

The definitions and results of this chapter

carry over to this setting with little formal change other replacing the condition times adding 'when r = 1

plays in

t

[O.l].

r E [O,l]

with

r

E

[O.m)

than

and some-

is finite' or ignoring the special r o l e Some of the results can be proved the

same way or by a change of scale, while most require one more countable sequence in a saturation

argument.

This would have

cluttered our proofs. This

section

is

only

intended

to

ingredients needed for this extension.

highlight

the

new

It does not give many

detai Is. Our primary interest here is the extension of our treatment of martingales.

The uniform integrability assumption we made on

[O,l]

strong for

is

too

C0.m).

We

could

localize our

martingale with a deterministic sequence of times, but use of random times is important and requires the additional technical details that we wish to outline.

Randomly localized martingales

are the main topic of this section.

The proof of the lifting

theorem (6.7.5) is quite tehcnical. but we omit i t anyway.

The

extra stopping time in the coarser sample theorem (6.7.6)is a special feature of hyperfinite local martingales.

Chapter 6:

340

Hvperfinite Evolution

(6.7.1) NONANTICIPATING DECENT PATH LIFTING THEOREM FOR

X

Suppose a stochastic process 6t@6P-almost paths.

progressiuely

R

x

: C0.m)

C0.m):

* IR

is

measurable and a.s. has decent

Then:

X

has a nonanticipating decent path

that i s , there is an infinitesimal and an internal process

Y

is nonanticipating a f t e r

:

T

x

lifting,

II

in

At

R + *IR

that

and a.s. has a

At

At-decent path sample whose decent path N

Y.

projection,

If

X

r

ea ch be

I

is indistinguishable f r o m

[O,r] x R

is uniformly integrable for then the

E C0.m).

chosen s o

section

Yt

that is

X.

Y

for each

o f part (a) may finite

the

t,

S-integrable.

The proof follows the steps in the proof of (6.3.8). F i r s t one extends

the results used

and (6.1.7).

[O.l]

in

(6.2.7). (5.3.23)

that proof,

[O.m]

These extensions are made by rescaling

to

and taking a l i m i t .

PROOF SKETCH: Referring know

by

the

to the

steps of

extension

of

the proof

(6.2.7) that we may assume

in case

0

<

(b))

k ,( inL,

variable

X

of with use

Z i Z

(make

sample

W 6t.

a.s.

where

has

a

S-integrable at each finite

t

Next, by

For each

the extension of

X($)

X

W

decent paths and is progressive. DC0.m)-lifting

of (6.3.8). first we

(6.1.7)

Zm k

(5.5.12) take

m

>

1

in

IN

and

to choose a random

is determined at

k z k

tm

m

Section

6.7

Beyond

0 5 k 5 m2

for in

case

rO.11

k 2 W(tm) k Zm

and

Define

(b)).

341

with Localization

a.s.

Zm(t.u)

k S-integrable Zm

(make as

before

except

take

n

d

m Zm(t.w) = Zm

for

ing after

constant on

t1

m

t 2 m.

This gives us a "Zm nonanticipatT[t,. k tk+l ) , with 2 2 m < tk+l-tk m m < 2 m

and

k k P[ max 21W(tm)-Z(tm)l

1 1 (resp.. E > --I < ;

Oikim

Zm

We may extend the saturation.

Choose

an

internal probabilistic

Y = Zm

is

our

t o

an internal family of processes by

infinite

m

inequalities on the

ti

Finally we may

At = tL

nonanticipating after

satisfying

formula in quotes above.

nonanticipating

take

in case (b))."

At.

decent

Z

keep

path

the whole The process

lifting.

The

W

decent because

is.

so that Z has a At-sample m This sketches the proof of (6.7.1).

FILTRATIONS: The relations of Definition (6.1.4) and the filtrations and

d

scheme

have

the same formal definition on our

(5.5.4).

Similarly,

the

definition

larger time

(6.4.2)

of

an

r

is

%-stopping time carries over with the only change that any instant o f

[O,m)

rather than just

P A = {t E T

is infinitesimal and

: t =

stopping time is an internal function whenever

T(O)

= t

just as in (6.4.1).

and

t

vt = w ,

When

[O.l].

kAt. k E *IN}. T

then

:

R

+

T ( U )

%

TA

= t.

At E

a

T

At-

such that formally

The Stopping Lemma (6.4.3) carries over to

our larger time scheme as well since (6.7.1) is just what is needed to extend the proof of (6.4.3) to this setting. why

we

chose

(6.7.1)

to

illustrate

the

simple

(This is

localization

342

Chapter 6:

Hvperfinite Evolution

technique.) The

Definitions

(6.2.1)

of

extend via the same formulas to

adapted,

progressive,

etc.

C0.m).

(6.7.2) DEFINITION:

A decent path process a

M

x R +

: LO,..)

IRd

is called

(d-dimensional) local hypermartingaLe provided

is

progressiuely

{p,}

sequence

sequence for

f o r each

m.

M

r

<

and

%-stopping

of

means

p,

is

times f o r

M.

5 pm+l,

a

reducing

A reducing

lim p = m

a.s. and

00

is a uniformly integrable

I

ECM(s A p,)

= M(r A p,)

%(r)]

P

a.s.

s.

Again, the definition of

%-martingale

extension of (6.5.10) to the case only

there

A p , )

= M(r

Mm(r)

%-martingale. for

measurabLe

M

that

r

<

s

is just the formal

in

C0.m)

rather than

Uniform integrability was defined in (5.3.25) and

[O.l].

studied from

the internal

point of view as early a s section

(1.6).

(6.7.3) DEFINITION:

An internal process (d-dimensional) tesimal

At E T

At-local

M

:

Y

x R +

martingale

proutded that

M

*Rd

is called a

some

for

is a

infini-

*martingale.

that

i s , provided

E[M(t)

M

has a

At-decent

I

us] = M(s.u),

A t 5 s 5 t,

for

path sample a.s. and

M

has a

At-

Section

reducing

rO.11

Bevond

6.7

sequence

We

{T~}.

external) sequence

{ T ~ }

(a)

T

(b)

The

<-

~

,$ m

T~~

that

say

of internal

M

At-reducing sequence for

is a

343

with Localization

a

(countable,

At-stopping times

provided:

and a.s.

lim m*

in ) =

m.

S~(T

At-maximal function

M A (t) = max[IM(s)l

:

At

<

<

s

s E UA],

t,

t E

HA

satisfies:

,$ m

M A (r,-At)

for all

with

w

>

T,,,(w)

(paths are bounded by

m

At.

before

rm)

and

MA (

T ~ ) is

S-integrable (so

(c)

M(t

A

T

~

is also

)

The decent path projection

P(st

Tm)

= st[M(Tm)]

%

S-integrable).

satisfies

a.s

The lifting theorem (6.7.5) contains the additional facts about

turning

reducing

mere

sequence

uniform

into

the

integrability technically

of

useful

the

standard

bounds

on

a

At-reducing sequence without loss of generality. Example (6.4.7) shows why we add the last requirement that 'the standard part of the localization equals the localization of the standard part'. because

X

This is an essential extra technicality

of (6.4.7) is a

*martingale

with a decent path

344

Chapter 6:

%

sample while

is not a local martingale.

The purpose martingales

Hvuerfinite Evolution

of

and

this

local

section

is

to

show

hypermartingales

that

are

At-local

corresponding

internal and measurable notions.

(6.7.4) LOCAL MARTINGALE PROJECTION THEOREM:

M

Let

At-local martingale.

be a

The decent path

N

M

projection

is a local hypermartingale.

PROOF : The

standard

sequence

of

=

lim p,

m

in

= st

times

T

form

m'

satisfying a.s.

increasing

an

m,

p,

Since each

a.s. is

T~

we can rescale and apply (6.5.11). that is. for

lN.

Ml;l(t)

let

lim Ml;l(t)

projection,

p,

= st M(T~)

%(p,)

m+l

bounded by each

%-stopping and

(0

parts,

= M(t(m+l)

= %'(r)

A T ~ ) . The decent path

is a

(uniformly

integrable)

t lr

hypermartingale on

C0.11.

Hence

A p , )

%(r

= lim M(t t lr

A T ~ )is N

a uniformly integrable martingale on N

= M(p,).

a.s.. for all

r 2 m.

%

[O.m+l].

Since

M(r

A ),p

is a local hypermartingale.

(6.7.5) LOCAL HARTINGALE LIFTING THEOREM:

Let

M

be a local hypermartingale. A t E H.

infinitesimal whose decent from

M.

and

path projection,

Such an

N

a

Then there is an

At-local

8,

is

martingale

N

indistinguishable

is called a local martingale lifting

M.

of

PROOF : This proof is quite technical and hence omitted, see Hoover &

Perklns [1983].

Section

6.7 Bevond

rO.11

with Localization

345

In order to keep a path sample of one process comparable to another, we frequently want to take a *coarser sample'.

When we

do this with local martingales we must also modify the reducing

sequence.

M

stop

M

and

m

S-integrability we may need to

at a time not in the coarser time sample, for example,

in case T

In order to maintain

jumps an infinite amount before

after one of the times

6t

T ~ + A ~ .We need

to do

this on a set of

infinitesimal probability (a.s. is not enough). We

call

vt-sample

the

M

of

vM(t)

writing

N

internal and

for

from

later abuse

the

next

the notation

theorem

the

slightly by

t E Uv.

N(t+vt)-N(t).

(6.7.6)COARSER SAMPLING LEMMA FOR LOCAL MARTINGALES: Suppose vt E

+ Ug

time

M

that

is a

is LnfinitesimaL. T

such

-

N(t)

that

martingaLe. whiLe

T

=

At-Local

martingaLe

Then there is a = M(t

A

T )

and

vt-stopping

is a

vt-Local

a.s.

03

PROOF : Let

T~

be a

At-reducing sequence for

(6.4.5) (details omitted),

for each finite

finite sequence of

Mv(r:-vt)

$, m ,

vt-stopping satisfies: n < m -n = T T and T m m m

= st M(T:

A T ~ )=

%(ym)

S-integrable,

and

-

= st

M(7,)

a.s.

n,

M.

By Lemma

an increasing

the maximal function

Since

a.s.,

E(T; M(T~)

A T ~ )

is

ChaDter 6: Hvperfinite Evolution

346

n

We may extend seiect

an

m

{T~.{T~ :

infinite

<

n

n}}

to an internal sequence and

satisfying

the

probability inequalities above, so that the satisfy

T~

<

T

vt-stopping times Since

T

sequence since

M(T,)

>

~

+

{T:

k

for ~ :

m

<

<

n),

n.

and

{T:

for finite : m € IN}

is

k

reduces

S-integrable.

and

M(t

and

At-stopping times each

is increasing. N

T~

expectation

N

sequence Let

=

of

T = T

a.s.

n.

The

rk +

m,

A T)

by the formulas above

T

347

CHAPTER 7:

STOCHASTIC INTEGRATION

In this chapter we study pathwise integration with respect to a process that is a sum of a martingale and a process of bounded variation.

The infinitesimal analysis of the latter is

similar to section (2.3) and analogous to the classical analysis of Lebesgue-Stieltjes path-integrals.

The new feature of this

approach is that infinitesimal Stieltjes sums also work in the general case.

(7.1) Pathrise Stieltjes Sums In section (2.3) we showed how to represent every Borel measure on

[O.l]

by choosing an internal measure on

T.

In

section ( 4 . 1 ) we saw how an internal measure can arise first and how to make a standard Borel measure from it. the classical Stieltjes measure

dF

equals

We also saw that dF

0

st-',

where

u

F = S-lim F(s)

and

dF

is the hyperfinite projection measure.

s lr

We saw a hint of some problems with jumps of (where

F

was increasing and finite).

F

in Chapter 4

In section (5.3) we

resolved similar problems for more general processes by taking At-decent path samples where time increment. chapter.

At

was a coarser infinitesimal

We will use the same basic approach in this

In this section the first step is to show how to

sample a process simultaneously with its pathwise variation.

We

begin by fixing some basic notation that we shall use for the rest of the chapter.

Chapter 7: Stochastic Integration

348

(7.1.1) NOTATION:

$2.

T h e infinitesimal time a x i s ,

U.

P

on

and the u n i f o r m probability

the sample space,

R

are the same as

i n Chapters 5 and 6 except that now w e let infinitesimal element o f

U.

(For example, if

smallest positive element of and

U.

H.

w e might haue

-.

a larger increment.)

H6 =

{t E

1

At = n

6t

For any

denote any

n!

i s the

6t =

2 [i]

6t. A t

in

let

U

: t =

k6t. k E *IN}

U (1)

and U A = {t E U : t = kAt, k E *IN}

If

g : U + *lRd

is internal

(d

denote the formard differences o f and

At.

U (1).

finite) let

g

corresponding to

6t

Also let

16t 16gl . t

6Var g(t) =

for

t E

= )[l6g(s)I

: 6t

<

s

<

t , s E US]

AVar g(t) = )[lAg(u)I

: At

<

u

<

t. u E HA]

and

denote the uariations to time

t.

where

of

1-1

g

in steps o f

denotes the

6t

or

d-dimensional

At

up

7.1

Section

eucLidean norm f : H

349

Pathwise Stielties Intecrrals

* d IR

on

.

FCnaLLy.

* * L ~ ~ ( R ~ . I Rt ~s )a]n

if

f : U + *IR

[or

internat function. Let

and

1 f(u)Ag(u) t

ltfAg = S

,

for

+

s.t E PA

u=s step A t

[ W h e n the uaLues o f

f(u)

are

linear maps.

means the map evaluated at the uector

Our

convention

variations at

after

the

sums

6g(u).]

defining

the

internal

is to make i t compatible with our

At

Bt-decent path samples especially in the case of

processes

whose

liftings are only nonanticipating

6t. Our (artificial

a right-most

this

instant

=

max[U6\{1}] In

start

or

6t

definition of progressive

to

f(u)bg(u)

case

6t = [ l - ~ ]

T

< we

1.

D-space convenience-) convention of having r = 1

causes us an extra headache when

(We could ignore this problem on

take

6g(T) = [g(l)-g(~)]

and

[O.m).) interpret

i f necessary and also let

with a similar convention for

uA*

The last convention will allow us to place a final jump at

ChaDter 7: Stochastic Intearation

350

r = 1

on our internal paths and account for the corresponding

X(l)

We simply l i f t

measure.

Suppose

that

g : T + *IR

we

whose

begin

against

of

Z( )6g

the

st g = 0

so

that

is O.K.

variation

works

too

6t-variation

sampling

along

)dh

when

then

isn't,

so

h = st g.

On

Ag = 0

and

the

then

the standard part f(kAt) = 0

then

is zero.

coarser

The standard

f(k6t) = (-1) k -1.

A t = 26t.

so the standard part

Suppose

the variation

s(

is infinite and

if

We

sampling always works; perhaps

If

well.

Zl6fI

'Borelable'. but

is too simple.

A t = 26t.

Coarser

function

is limited.

Bt16g[ = 2t.

but

does not properly represent let

internal

represents the integral of

is zero, while

function

the other hand, i f we

even

but

an

B116g(s)l.

Zf(s)6g(s)

d(st(g)).

g(k6t) = (-l)k6t. part

with

6t-variation.

would like to say that st(f)

separately.

and

it

the

is not

Af = 0

The following results show how

infinitesimal

time

axes

works

for

Stieltjes integration.

(7.1.2) PROPOSITION:

If and

if

almost

var

X

var

<

R + *Rd

tn

T6

path has

has a

then

such that the

has a of

6t-decent

projection ftnite

a.s..

03,

projectton

Z.

x

surely

(X,AVar X) path

H

its decent

g(m.0)

A t 2 6t

:

At-decent AVar X

i[ : [O.l] x

classtcal

there

path sample

is

an

n

+

IR d

uariatton, tnfinttestmal

(d+l)-dtaenstonal

process

path sample and the decent is

indistinguishable

from

351

7.1 Pathwise Stielties Intearals

Section

Recall that the classical variation of a path

0

r

to

is defined to be the

sup

of all sums

over the set of all finite partitions of

[O.r].

Finiteness of

this sup is equivalent to saying that each component. the vector

2

from

%(a,")

2,.

of

is the difference of two increasing functions.

We c a n supplement (7.1.2) with the hypothesis in the next result.

(7.1.3) PROPOSITION:

X

If

U

:

R

*Rd

W a r X(l)

6t-vartatton,

T.

x

6t

in

Us

such that

and

the projectton

E 0

aLmost

sureLy

of

tndtsttngutshable from

that

(2.

Limtted

Q.s., f o r some LnftnttestmaL

then there ts a n LnftnttestmaL

(X. AVar X)

has

A t 2 6t

in

has a

At-decent path sampLe

sample

of

var

(X. AVar X)

is

2).

PROOF : First we shall prove that the hypothesis of (7.1.3) implies

X

that

has a

At-sample and

x"

has bounded variation.

Then

we shall prove (7.1.2).

A C R

Suppose o E A.

6Var Xo(l)

measures o n

1;

by

is measurable, I s finite.

For each

P[A] = 1. o E R

and whenever define internal

ChaDter 7: Stochastic Integration

352

= 6X(t.o)

u (t) w

+

where

a

+

= (6X,(t,w)

u;(t)

= (6X;(t.w)

= max(a.0)

S

whenever

+

u,(t)

+ ....,6Xd(t,w)) .*...6X,(t,o))

a- = -[min(a.O)].

and

:U

is an internal subset of

We

know

u = +.

and

that

-

or

blank, then

Therefore

w E

whenever

A,

the

P"u =

formulas

d-tuples of Bore1 measures on

define

r

For

of Chapter 2 can be used

the machinery

€

(0.1)

and

finitely decreasing to

pz[O.r]

any

u 0

( u = +,-)

st-1

0

[O.l].

countable

to see that

+ -

pw = j ~ ~ - p ~ .

Let

sequence

r.

= S-lim u~[T6[0.tm]]. m*

(I

= +,-,blank.

and for any sequence strictly finitely increasing to

r,

u = +.-.blank.

= S-lim u~[Ua[0.tm]].

p:[O,r)

strictly

tm

m*

This shows us that

S-lim X(t.o) = pw[0,r],

that the

S-limit

t lr

as of

t

increases to w,[O.r]

increasing S-limits on

is

r the

functions.

A

pw[O.r)

equals

difference

of

Existence of

implies that

X

and that each component

has a

two

right

continuous

increasing and decreasing At-sample whose

7.1

Section

2,

projection, process,

353

Pathwise Stielties Integrals

is

indistinguishable

(5.3.25).

Lemma

This

shows

from

that

a

decent

path

the hypothesis

of

(7.1.3) implies that of (7.1.2). but our sampling convention at At y[O]

means

that

# 0.

example, let at

-

may not equal

Notice that close jumps of

so that we may need

cancel

-1

%(O) = st X(At)

6t.

u+

to choose

w.

-

and

u

At

even

larger.

if

can also For

-21 + 6t and u - be unit mass + - = 0. The proof of y = y -y

be unit mass at

for each

st X(6t)

+ u

Then

(7.1.2) given next completes this part of the argument.

PROOF OF (7.1.2): Suppose

At

>

6t

%(*,w) = stk X(*.w) q.r E [O.l]

= r

and

is infinitesimal

and B

>>

0.

var z(1.0) there exist

and

<

w

OD.

q = ro

i s such that

Then

for

< rl <

* * *

every

'

rm

such that

There are also times X(tj.w)

Z

%(rj.w)

and

s.tj.t E TA X(t,o)

Hence for each infinitesimal

var

-

Z

At

such that

P(r.w),

>

6t,

1 1 ~ x 1a.s. t

var %(q)

so

S

X(s.0)

z %(q,u),

Chapter 7: Stochastic Integration

354

Next we find one infinitesimal time sample satisfying the

V(t)

opposite inequality. Let know

S-lim(X.V) = ( 2 , var 2)

number 0

<

j .( m .

For this

A 1) Z

X(jAt

whenever

At,

-

g(i)

Thus the internal set of

and

T6

in

At Z m

s.t

g.

We

A 1)

V(jAt

such that for Z'

var

%(i) a.s.

.:1

€

At's

2 6t

in

T6

such that

):11

>

t

IAXII

P[max(IV(t)-V(s)-l

var

a.s.. so for every finite natural

there exists

m,

6t-lifting of

be a

:

s.t

E

At]

<

At

S

contains an infinitesimal.

(X(t)sZA:lAXl)

a

Such an infinitesimal

At-lifting of

(2,

var

At

makes

g).

(7.1.4) DEFINITIONS:

If

U

:

T x R

+

(U, 6Var U)

that

*Rd

has a

is an internal process such

6t-decent path sample with

projection indistinguishable from say

U

has

S-bounded

:

[O.l] x R + Rd

(c,

var

6t-variation or

c),

then we 6t-bounded

variation. If

W

variation and

U

has

a.s. has bounded classical.

S-bounded

6t-variation with the

Section

7.1

projection that

U

355

Pathwise Stielties Integrals

fi

is a

When

indistinguishabLe

W,

from

then we scy

6t-bounded variation Lifting o f

U

has

S-bounded

6t-variation. T6

internal. pathwise measures o n

W. we

define

by the weight functions

T6 x R .

as weLL as a measure o n

6u(t.w) = 6pw(t)6P(o).

T

Extend these measures to either all of 6pw(t) = 0

by taking

or all of

T

x R

t Q T6.

if

The measures we have just introduced play a role in showing the connection between internal summation and classical pathwise integration.

The hyperfinite measures

variation measures of the paths of so

that both

f(w) = p,[lT]

pw

<

1

u

and

S-integrable

with

<

fi, 1.

K

0

0

while

% C T x R.

respect

to

denotes the section,

=

{t

E T

:

are the total

p,

is normalized

This makes

weaker conditions would suffice for this).

If

st-l

(t,w) 6 % } .

P

the (of

function course,

Chapter 7: Stochastic Integration

356

(7.1.5) THE ITERATED INTEGRATION LEMMA FOR PATH MEASURES: 6 p : Y x R + *[O.l]

Let For

each

the weight

o

T.

measure o n

be an internal function.

function

Suppose that the function

P.

is S-integrable w i t h respect to the

weight

defines a

6vw(t)

Let

f(w) = vw[U]

be given by

u

6 ~ ( t , o ) = 6po(t)6P(w).

function

The

hyperfinite extension measures satisfy: (a)

(b)

(c)

If

Y

is

Loeb(R)-measurable

If

Loeb(T x Q ) .

E

lr

is

then the function and

a-measurable. then for a.a.

is

p,-measurable.

If

X

for almost all

is

pw(Ww)

[--,-I

: 'U x R

o.

pw(Qo)

Xu

is is

o , lro

P-measurable and

u-integrable. then p,-integrable

.

and

.

E[ Jxo ( t dw, ( t ) 1 = JX ( t o )du ( t o ) . PROOF : (a)

If

91

<

and

PWC91,1

u[91]

= E[~,(91~)].

Monotone

E[lim

V"Cr1. The

Class

p,(91:)]

hypothesis that

Yo

is internal, then

Lemma by

p,[T]

the

is internal for each

w

ECV"(*,)l.

so

=

Moreover.

uC*l

of

follows

rest

(3.3.4). Dominated

(a)

because Convergence

is P-S-integrable.

easily

from

the

l i m ECvw(9:)1

=

Theorem 'and

the

Section

(b)

If

internal

'21

W

is

u-measurable, then by (1.2.13) there is an

such that

v '213 = 0.

u[#

N.

contained in a Loeb null set and since

is

a.s.

Since

= 0

p,[N,]

is

'21

a.s.

has measure zero a.s.

Y,

Ww

we see that

0,

= p,[SCw]

p,[W,]

v

v

W

Therefore

By part (a)

W,

is complete,

p,

Using (1.2.13) for these a.s. and

357

Pathwise Stielties InteFrals

7.1

P

is

p -measurable 0

is complete,

p , [ W , ]

P-measurable i f we take any value for the null set of

0's

where i t may fail to be defined (for example, we may take the outer

measure

Fw[Ww]).

= ~ [ " u ] = E[p,(91,)]

u[W]

Finally.

= EC~,(W,)I.

(c)

If

X 1 0 is

sequence

of

simple

convergence

we

functions

know

JXkdu

Sk

JXkdu = E [ Xwdpw].

S"

for

positive

X = X+-X-

with

by

X.

By

By

part

(b)

monotone

SXidpw] = E[[Xdp,].

integrable

be a monotone

Xk

1 JXdu.

Again

= E[lim

[ X,dpw] lim E

{Xk}

u-integrable. let

Finally, we

and apply the positive part to

X+

and

we

know

convergence

Thus part

functions.

monotone

(c) holds may

X-

write

in order

to finish part (c).

(7.1.6) THE STIELTJES DIFFERENTIAL LIFTING LEWMA:

Let

W

:

bounded vartatton. ltfttng

(a)

U.

Rd

[O.l] x R

If

U

Then

W

a.s. have decent paths o f has a

&it-bounded vartatton

t s such a ltfttng, then for a.a.

the Borel measures

I

w

= 6Uw o st-'

equal the

Lebesgue-Stieltjes measures generated by

(b)

the Ic

W

total 0

( c ) n,(O)

st-1: = 0.

uartatton

measure,

ldW,l

a,

dWw : equals

ChaDter 7: Stochastic Integration

358

PROOF :

U

Let

be a

to obtain a Let

A

At-decent path lifting of

>

6t

so

At

that

U

be the null set where

We know that i f

o Q A.

Apply (7.1.2)

S-bounded 6t-variation. U] # [W, var W].

stk[U.6Var then

= S-lim U(t)-U(6t) t lr

ru[O.r]

has

W.

= Wo(r)-Wo(0)

= dWw[O.r]

and Ir I[o.r] o

so

= S-lim 6Var U(t) t lr

= var W(r)

(a) and (b) hold. = W(0)

lim W(r) r 10

Since

uo(0) = 0.

a.s..

This proves the

1emma. Next we deal with the measures from (7.1.4) that we are most interested in for stochastic integration.

(7.1.7) DEFINITION"

H

Let

G : T x R

[O.l] x R

:

+

*IR

-4

IR

be a function.

such that f o r a.a.

o.

An internal

the hyperfinite

measure :

~"{t

i s

called a

In

in

properties on

to

compute

(7.3). G

we

= 0

# H(st(t).w)}

6U-path lifting o f

order

summation

st G(t.w)

H.

martingale will

need

and hence also on

H.

integrals to

require

by

internal additional

Section

(7.1.8) THE

H

U :

:

H

[o.il

x R

R

x

H

-,

H

measurable, then if

-

bU-PATH LIFTING LEMMA:

Let

rf

359

Pathwise Stielties Integrals

7.1

*Rd IR

S-bounded

is

has a

Gt-variation.

[Borel[O.l]

GU-path Lifting

b.

i s bounded b y

bounded b y

have

we may

x

G. G

choose

Meas(P)]Moreover, so

it

is

b.

PROOF :

K

Let

be

indistinguishable K(st(t).o)

bounded

Let

H

from

(see

(5.4.10)).

G be a

u-lifting of

u-lifting, see (1.3.9)).

function

By

(Loeb x Loeb)-measurable

is

u-measurable.

(Bore1 x Loeb)-measurable

a

(5.4.9).

and

K(st(t).w)

hence

(resp. a

By the Iterated Integration

Lemma (7.1.5).

Except for a null set K

0

A C R.

is a simple multiple of

for a.a.

po

is limited so that

on the Loeb sets of

H.

Hence

w.

~

Finally,

GVar U(1.w)

~

K(st(t),w)

lemma is proved.

:{

stt G(t.o)

= H(st(t),o)

# K(st(t),o)l

for all

= 0.

t.

8.5.

w,

so

our

ChaDter 7: Stochastic Intepration

360

( 7 . 1 . 9 ) THEOREW:

Let measurable

H

x

: [O.l]

R

+

and bounded

lR

by

be b.

x Meas(P)]-

[Borel[O.l]

W

Let

x R +

[O,l]

:

IR

d

be a process wtth a.a. decent paths o f bounded uartatton. U : T x Q

Let

W

of

H

and let

-+

*Rd II

G :

also bounded b y

be a

6t-bounded uariatton lifting

x R -+

*R

be a

6U-path ltfttng o f S(t.o) =

b. T h e n the tnternal process

t

G(s.o)6U(s.w)

ts a

6t-bounded

uartatton

Lifting o f

16, the pathlvtse classical tntegral

I(r.o)

=

s:

€I-dW.

PROOF : First we show that b

so

<<

S G

denote a bound for

S

and

6t-decent path sample.

H.

only has finite jumps where

right-S-continuous where By

has a

(7.1.6).

for

measure generated by

6Var U(t)

8.8.

dWw.

o.

6Uo

If

t.t

6Var U(t)

+ At

€

Let

Us.

does and

S

is

is. 0

st-'

= u

0

is the Bore1

By change of variables,

We apply a similar argument to

ZG-6U. Thfs proves the theorem.

7.1

Section

361

Pathwise Stieltjes Integrals

(7.1.10) DEFINITION: We

caLL

the

G

function

the

of

next

theorem

a

H.

6U-summable path Lifting o f the process

(7.1.11) THE STIELTJES SUHHABLE LIFTING THEOREH:

U

Let

H

is

be a

[Borel[O,l]

H

K

0

has

a

-

-S-integrable for a.a.

w,

a.s.

G

6U-Lifting

If

and i f

x Weas(P)]-measurable

~~IHol=ldW,l < then

W.

6t-bounded variation Lifting o f

such

Go

that

is

In this case

o.

t-6t S(t,w) =

1 G(s.o)6U(s.o)

S-6 t is a

6t-bounded variation Lifting o f

I(r.o) =

J)(q.u)dW(q.o).

PROOF : Let

G'

be any

6U-lifting of

H. We will show that all G'

sufficiently small infinite truncations of S-integrable.

are pathwise

Of course. infinite truncations of a lifting are

also liftings. Define m .

Hm =

{H . -m

for finite

m

,

H > m

IHI < H

<

m m

and

Gm =

{ G' -m

-m

in the case of

H

and all

m

. . .

G > m IGl G

for

<

<

m.

-m

G'.

For a.a.

Chapter 7: Stochastic Intevration

362

w,

{Hm}

for every

L 1 (var W)-norm.

is a Cauchy sequence in the r

in ,'Q

there is a finite

m(r)

>

1

so

that

such that if

k 2 m(a).

Thus, the internal set

contains an infinite

n = n(e).

By saturation the countable

intersection n*m[m(r) .n(r)l contains an infinite n. We claim n that G = G is our summable lifting for H. This follows from the definition of standard

n

because

P[ZlG-Gkl16Ul

> €1 <

r

f o r all

r.

By the bounded lifting theorem (7.1.9) above, for each N

finite

We

m

we know

also know

stkSm

-B

stkZG6U

S,(t.o)

Im

JHdW

= Im(r,w)

where

in probability in

in probability, hence

By the bounded case ( 7 . 1 . 8 ) . that the decent path projection of

D[O,l]

stk I G 6 U = JHdW

for each finite

m

and a.s.

we know

7.1

Section

363

Pathwise Stieltles Inteprals

are indistinguishable.

IGn-Gkl 1

Since

I IGnI-IGkl 1 ,

the same

convergence estimates show that the decent path projection of

are indistinguishable. lifting of

Hence

ItG6U

is a

6t-bounded variation

S'HdW.

(7.1.12) EXAWPLE: Consider

the process

J(t.o)

of

(5.3.8). (4.3.3). and

N

(0.3.6) whose decent path projection process.

one on a

pat)

is a classical Poisson

We wish to calculate

as an example.

6J(t.w)

J

Since

J

is finite and increasing by jumps of

6t-sample, i t has is a function of

and

since

only

6J-liftings. we see that

S-bounded

o t+6t the

alone

times

J(t+6t.o)

6t-variation.

(=1

Since

with probability jumps

count

when

it

ts a

6J-Ltfttng o f

for

N

J(r.o). too.)

(This depends on our right continuous path convention The important fact that we are trying to illustrate is

that the lifting the "coin, "

J(t+Bt,o)

must anttctpate the next

o t+6t. Now we compute

toss

364

ChaDter 7: Stochastic Integration

t-6t

1J(s+6t,w)6J(s,o)

+ 2.1 +

= 1.1

+

0 . 0

J(~~.u)-l

s=6t

where

is the time of the last jump of

T~

time

J(*.o)

at or before

t.

Notice obtained

that

if

we

want

lift

to

from projecting Anderson's

the Brownian

infinitesimal

any infinitesimal time advance or delay can be

B

6J-lifting because the paths of

B(t,w)

are

random walk,

tolerated in a

S-continuous.

6J-lifting of

is a nonanticipating

%

motion

%.

Hence,

Path liftings

need to be done more carefully when the differential process is a martingale

with paths of

infinite variation as we shall see

below. Here

is

structure

to

a

result

that

* finite

our

adds

the

stochastic

representation

of

evolution

Stieltjes

path

integrals.

(7.1.13) THE NONANTICIPATING STIELTJES LIFTING THEOREM: Let

W

: [0,1] x

R

4

IRd

be a progressiue

with a.a. decent paths of bounded variation. a.s.

Let

H

:

[O.l]

x R +R

process

var W(l)

<

03

be a preutsible process

mt th

There is an tnftnttesiaal tit-bounded variation Lifting after

6t

and

H

has a

>

6t

U

0

such that

W

has a

mhich i s nonanttcipating

0-predictable

6U-suamable path

Section

365

7.1 Pathwise Stielties InteErals

lifting

G.

ItG*61J

These Ltftings make

6t-bounded uariation Lifting of

Q

which is nonanticipating after

JrH*dW

6t.

PROOF : Apply

the

Nonanticipating

W

(6.3.8) to

obtaining a n

nonanticipating a f t e r some infinitesimal

6t-variation.

(7.1.4)

H

U

process

which

U

U

to find a n

has

S-bounded

G.

U

with

to

obtain

a

similarly

u

of

bounded

In the integrable case, apply the truncation

the proof

which a r e called

U

W.

is bounded we apply (6.6.8) with the measure

0-predictable

is

At-decent path sample for

that

so

Theorem

T h e coarser sample still has decent paths, so

associated

argument of

Lifting

Next. apply (7.1.2) to

At.

is our lifting of

If

Path

internal

and has a

At

6t E U A

infinitesimal

Decent

(7.1.11) to a sequence of processes

of

0-predictable.

This proves the theorem.

(7.1.14) SUMMARY: There a r e two main

ideas in this section.

T h e first

is

that coarse enough time samples of a process whose standard part has

finite

classical

interchanged

with

variation

have

the standard part

a

variation

(in

that

D[O.l].)

can be

The second

idea is that Iterated Integration allows us to connect a.s. path approximation

to

internal

sums.

The

following

two

exercises

test your understanding of the second idea on internal summands which need not be liftings of any standard process.

Showing

366

Chapter 7: Stochastic Integration

this

internal

stability,

separate

development of more general sections.

from

lifting,

integrals easier

makes

the

in the following

The internal sums also have "standard" applications.

(7.1.15) EXERCISE: U

Let

Gl(t.w)

and

and for

8.8.

have

6t-bounded

G2(t.w)

variation.

that

b E 0,

are internal, bounded by

w ~

T h e n f o r a.a.

~

:{

Stt

Gl(t.w)

f

= 0.

st G ( t . w ) } 2

w

t-6t (Gl(s,o)-G2(s,o))6U(s.o)l

max[I'

Suppose

: 6t

<

t

<

11

Z

0.

s=6t In other words, both

summands give nearly

the same Stieltjes

Bt-bounded variation.

Suppose that

sums.

(7.1.16) EXERCISE: Let

U

internal

is

have and

has

a

limited

bound.

Then

G

S(t.o) =

t

G-6U

a.s. only jumps where

U

does.

'6 t

Hence

S(t)

has a decent path sample.

See the proof of

(7.3.8) i f you have trouble formulating the jump condition. is easy to see that

S(t)

need not have

It

6t-bounded variation.

367

(7.2)

Quadratic Variation of Martingales One of the main ingredients in martingale

integration is

the quadratic variation process associated with a martingale.

It is used in estimates similar to the classical domination of a signed measure by its total variation in the previous section, but there are also surprises.

Consider these curious heuristic

formulas for one-dimensional Brownian motion:

(db) 2 = dt

&

dbdt = 0 ,

so

d(f(b))

= f'(b)db

1 + 5 f"(b)(db)2

= f'(b)db

1 + 5 f"(b)dt.

+

* * -

for example. d(b

No

doubt

our

reader

2

) = 2bdb + dt.

will

see

some

tempting

analogies

for

Andersonn's infinitesimal random walk, for example,

6(B2)

= (6B+B)2 - B2 = (6B)2 + 2B6B = ( ~ +f2B6B i)~ = 2B6B + 6 t . *

Such calculations are made

precise

transformation formula given below. the generalized study of the

(db)2

by

the

(generalized) I t o

In this section we take up term.

Brownian motion is

an important test case. The following is an extension of (7.1.1) where

AM, etc. are defined ( f o r the internal functions

6M

t + M(t,w)).

and

Chauter 7: Stochastic Integration

368

(7.2.1) NOTATION: Let

6t

and

At

*lRd

are internal processes with values in denotes

I f

be t i m e i n c r e m e n t s .

the e u c l i d e a n inner

product

M

and

and if *Eld,

on

N

(x.y)

we define

the joint q u a d r a t i c v a r i a t i o n processes f o r the respective time increments by:

+ )[(6M(s.o).6N(s.w))

:

0

<

s

<

t, s E

f o r 6t

<

H6]

t E

Us,

and

We

also

define

maximal

functions

for

the

respective

increments by:

6

M (t.o) = max[IM(s.o)l

: 0

<

s

<

t, s E

Us]. for

6t

<

t E

At

<

t E UA.

T6,

and

M A (t,o) = rnax[lM(s.o)l

:

0

<

s

<

t, s E

U,], for

7.2 Quadratic Variation of Martingales

Section

All

the

little

details

variation are important.

in

our

We include

369

definition M(6t)

of

quadratic

in the quadratic

variation. while no such term was needed

in first variation.

This term corresponds to the standard term

%(O)

convention of starting

M

If (d = 1)

and

= B(t,w)

because of our

6t-decent path samples at

6t.

is Anderson’s infinitesimal random walk

is as in (5.2.3). then

6t

=: t =

[6B,6B](t,o)

:

0

<

s

<

t , step at]

It is well known that the paths of classical Brownian motions such as

are nowhere differentiable (see one of the

g(r,o)

books by Breiman, Doob or Loeve from the references). various

classical

formulations

of

the

idea

increments of Brownian motion tend toward We would

that

infinitesimal?

when

At

finite

could be given.

like to turn the question around and ask:

[AB,AB](t,w)

In fact,

is much larger than

6t,

What is but still

This is answered by Lemma (7.2.10).

We begin with some simple, but illustrative calculations.

(7.2.2) EXERCISE

(Cauchy’s inequality for quadratic variation):

For interna

mensional processes

M

and

N.

I [6M, 6N] (t HINT:

Apply the

* transform

with

components

6Mi(s).

t

>

s E

r;.

of Cauchy’s inequality to vectors 6Ni(s)

for

l < i < d

and

Chapter 7: Stochastic Intearation

370

The next result frequently allows us to focus our attention on single martingales, yet conclude results about pairs.

(7.2.3) EXERCISE (Polarization identities)

For internal

HINT:

M

d-dimensional processes

* d IR

Sum the corresponding identity of

and

N.

.

(7.2.4) EXERCISE: Let be a

T

be a one-dimensional

M

6t-stopping

time.

6t-martingale and let

Show

that

E[M2(~(~),o)]

= E{[~M.~M](T(w).w)}.

General martingales require some coarser time sampling just as in the last section.

N6

makes

# NA

and

The nasty martingale

then is, M

of (6.5.4)

A main result about

[6N,6N] # [AN,AN].

quadratic variation says that i f

N(t)

(M,N)

is a

6t-martingale.

6t-sampling also works for the quadratic variation, that

[6M,6N] and

N;

has a

Bt-decent path sample for the same

moreover,

infinitesimal

At

in

[6M,sN]

T6.

Z

[AM,AN]

6t

as

a.s. for any coarser

The path property

(7.4.9) using estimates for stochastic integrals.

is proved

in

Stability for

bigger increments is Theorem (7.2.10).

We had a hard time deciding what

level of generality to

Section

7.2

Quadratic Variation of Martingales

present in hyperfinite stochastic integration.

L2

the

case on

371

We shall present

[O,l] in section (7.3).

This reduces the

that would be required for a treatment of the

technicalities

full local theory .

We do offer notes on the extension to local

martingales in sections (7.4) and (7.6) (which our reader may ignore).

L2

Lindstrom [1980] treats local

our outline

toward Hoover 81 Perkins [1983] more

is directed

general theory.

martingales, but

Our reader must consult their paper for more

details of the local case. in the local case,

This section is not very technical

we also give

so

the local results.

reader may ignore the statements such as "t is limited in she is only

interested

[O,l].

in

Some stopping

The

T" i f

times are

needed anyway, so this should cause no trouble. Definition martingale" function.

(6.7.3)

is

automatically

set

has

a

up

so

that

a

"

6t - 1 oca 1

locally-S-integrable

maximal

The lifting theorem (6.7.5) shows that there is no

l o s s in generality with this definition, or, to put i t another

way, the maximal always

locally

functions of integrable

sequence in general).

(but

standard

local martingales are

localizing

with

a

different

Our next result gives integrability of

the quadratic variation in both the local and "global" cases.

(7.2.5) THEOREM:

M

Let

and Let

p 2 1,

T

be a be a

6t-stopping

(M6(~))p

([~M.~M](T))~'~ is

d-dimensional

is is

S-integrable

time.

and

only

after

6t

Then for each finite

S-integrable

S-integrable. if

*m a r t i n g a l e if

and

only

In p a r t i c u l a r . if

[6M,6M](1)

if

M2(1) is

Chapter 7: Stochastic Integration

372

S-integrable. 6t-reducing

If

M

is a

sequence

6t-local martingale with the

S-integrable for each

[6M,6M]1’2(~m)

then

{ T ~ } ,

is

m.

PROOF : p = 1

The last remark follows from the first part with simply applying Both (1.4.17)

(6.7.3)(b).

implications and

given next.

by

of

the

first

part

the Burkholder-Davis-Gundy Assuming that

(M6(~))p

are

proved

inequalities

(resp.

using (7.2.6)

[6#,6M](~)~’~)

is

S-integrable. there is a convex increasing internal function satisfying the conditions of (1.4.17) with (resp. x

[~M.~M](T)~’~).

* C0.a).

E

rk(2x)

<

for all

= (M6(7(w)))’

f(w)

rk(x) = @(x’).

The internal function rk(0) = 0

is convex, increasing, has

krk(x),

x

CJ

and satisfies +1 k = 4’ . The

where

E *[O,m),

inequality (7.2.6) completes the proof of the first assertion (because one implication in (1.4.17) does not require part (b) as noted in its proof). The

S-integrable

M2(1)

is

Doob

inequality

S-integrable. if

and

(6.5.20) shows

only

that

[M6(1)I2

if

is

S-integrable and the first part of the result connects this with quadratic variation using

T

5

1.

(7.2.6) THE BURKHOLDER-DAVIS-GUNDY For every standard real

exist every t E Us :

This completes the proof.

INEQUALITIES: k

standard real constants d-dimensional

-

*martingale

>

and

0

c,C

M

>

0

d

E

such

and every

N,

there

that f o r 6t

and

and for every internal convex increasing function

*[o.-)

*~ 0 . m )

satisfying

Section

7.2

373

Quadratic Variation of Martinaales

q(0) = 0

9(2x)

and

k*(x)

x

f o r all

E *[O,m)

the following inequalities hold:

PROOF : This

result

finite case

of

follows by the

taking

extension o f )

d-dimensional

[1972] Theorem 1 . 1 .

Davis-Gundy's

* transform

the

of

(the

Burkholder-

While this is a cornerstone

of our theory, we shall not give a proof since i t is a "wellknown standard result."

(7.2.7) PATHWISE PROJECTION OF For

process

any

internal

[6M,6M](t)

of squares.

process

each

t

The (local)

is finite when

r E [O,m)

M,

the

quadratic variation

is increasing for all

o

S-integrability of

(7.2.5) means that except for whenever

[6M.6M]:

o

[6M,6M](t,o) o

e A.

since i t is a sum

[6M.6M]

proved in

in a single null set is also finite.

A,

Hence for

the left and right limits along

S-lim[6M.6M](t) t tr

= inf{st[6M,6M](t)},

S-lim[6M.6M](t) t lr

= sup{st[6M,6M](t)}.

t E Us.

tZr

and

both exist in

IR.

t E

Us

tZr

It follows, (5.3.25).

that

[6M,6M]

has a

Chapter 7: Stochastic Integration

374

At-decent path sample for some infinitesimal A t actually has a

[6M,6M]

as the process M.

in

T6,

but

6t-decent path sample for the same

The proof that

has a

[6M,bM]

6t

6t-decent

path sample, Theorem (7.4.9). uses the machinery we develop for stochastic integration.

We believe that there should be simple

direct proofs of this basic fact, but do not know any. We abuse notation and define a pathwise projected process using the extended standard part:

By the preceding remarks a process with paths

[P.%](r,w)

is indistinguishable from

Dt0.m).

The abuse of notation is

in

justified by (7.2.11) The following is a key technical lemma that tells, us some information about paths of quadratic variation processes.

(7.2.8) LEHMA:

Let

M

suppose that part,

;(a) =

be a u

d-dimensional

6t-stopping time whose standard

is a u(u).

6t-local martingale and

satisfies

S-lim M(t) t lo

= st[M(u)]

<

and

m

a.s.

Then

[ % , 1 ] ( ;=)

st{[6M,6M](u)}

a.s.

PROOF : We will show that for any infinitesimal

At

in

H6,

7.2 Quadratic Variation of Martingales

Section

[GM,6M](o+At)

[6M,6M](a)

Z

a.s

The dependence of the exceptional null set on "a.s.")

does not matter.

375

(in the

At

The external almost sure statement

means that the internal probability

-

P{[6M,6M](a+At)

holds

for

all

At

st[6M,GM](a+At) decreases

tends to

finitely

subsequence, but

finite

in a

to zero.

Thus

S-integrable and

a

the

<

SL2,

so

maximal function.

Y6.

Hence

an a.s.

At

convergent

is increasing, the

= st[bM,dM](a)

a.s. M2(1)

is

(6.5.20) has an

SL2

In this case, the martingale

1.

t l a < t

9

{ M(t)-#(a)

=

of

lemma in the case where

M(o+At)-M(a) is also

i t has

[%,%](a)

0

N(t)

B

in probability as

since quadratic variation

shall prove

<

B }

interval

st[6M,6M](a)

whole limit converges a.s. and We

>

[6M,6M](o)

by Doob's

,

a

,

t

2

<

o+At

a+At

inequality

That maximal function

N 6 (1) = max[lM(t)-M(a)l

:

u

<

t

<

o+At]

t

is

infinitesimal

st[M(a)].

yields

by

the

hypothesis

S-integrability means

expected value.

N

a.s.

it

has

S-lim M(t) = t la infinitesimal square that

Finally, applying the BDG inequality (7.2.6) to

376

Chapter 7: Stochastic Integration

CE{[~M,~M](U+A~)-[~M,~M](CJ)}

<

12}

E{maxlM(t)-#(a)

z 0.

t

This proves

the

lemma in the global

SL2[0.1]

case.

(The

reader can easily prove the local case by introducing a reducing sequence.

Moreover, the bounded integrability of the reducing

sequence is all that is needed, not the fact that 6t-decent path sample.

M

has a

This is helpful in Exercise (7.4.4).)

(7.2.9) COROLLARY: If

M

is a

is

[6M.6M]

d-dimensional

6t-local martingale, then

t = 0, a.s.

S-continuous at

PROOF : a(o) = 6t

The stopping time a.s. s o the lemma yields

satisfies

S-continuity of

G(0) = st M(6t)

[6M,6M]

at zero.

(7.2.10) THE QUADRATIC VARIATION LEMMA: Let

M

{tj

: j E

*IN.

of

Ui

with

be a 0 to

<

n}

tl

<

j

<

d-dimensional is any 0 . -

<

tn,

6t-martingale.

I f

S-dense internal subset then

PROOF : The components of a martingale

d

[6M.6M](t)

=

are also martingales and

1 [6Mi.6Mi](t).

Section

7.2

Quadratic Variation of Martingales

377

If we prove the lemma f o r one-dimensional martingales, i t follows for

d-dimensional ones by summing components.

shall assume that

is a one-dimensional

M

Hence we

6t-local martingale

for the rest of the proof.

M

Since S-continuous

l2

IM(tl)

has a

at

zero

a.s. a.s.

IM(6t)I2

6t-decent path along

Corollary

Ti

sample,

so

9

(7.2.9)

Hence, we may a s well assume that

that we only have

to compare

[6M,6M](to) to = 6t

the difference between

large and small increment5 beginning at the same time. a useful

formula

for

comparing

large

and

small

-

M(tO)

= 1[6M(s)

: to

<

s

<

t l , step 6tl.

so

2

=1

+ 2

11 6M(r)6M(s) s>r s-6 t

=I Hence,

t -6t 1

= [6M,6M](tl)

+ 2

1

5=t0

so

summing Here is

increments,

starting with the first large one:

M(tl)

2

=: IM(t0)l

IWt)I2

shows that

is right

it

(M(s)-M(to))6M(s).

Chanter 7: Stochastic Integration

378

In general,

where

(using

o r , letting

our

[s]

convention

= max[t

. t

j .

j

on

<

s,

0

<

j

<

The same sum formula may be used to define summand

t

in

Us

yielding a

n].

N(t)

*martingale

for any upper along

Ui.

By

direct calculation the quadratic variation

t

= 4 )lM(s)-M([s])

[&N,6N](t)

l2

6M(s)

12.

To conclude the proof we need a reduc ng sequence even when

M2

is

S-integrable on

Stopping Lemma ( 6 . 4 . 5 ) 6

M (Tm-6t) $, m.

i(st

T

[O.l].

In this case, apply the Path

to obtain stopping times ~

=) st(M(~,)]

and

T~

such that

~ , f l . In general, if

Section

Quadratic Variation of Martinaales

7.2

{ T ~ } is a

6t-reducing sequence for

1 [6N,6NI2(rm)

<

M

M 6 (rm)

with

-

1 5m[6M.6M] 2 (

379

5

then

m,

T ~ ) .

1 By

(7.2.5) and

[6N.6Nl2(~,)

(1.4.14).

Burkholder-Davis-Gundy's

is

S-integrable.

(7.2.6) inequality and (1.4.13) tell us

that E(max[lN(t)l

: t

I

T~])

1 -

<

2

CE([GN.GN]

<

u

-21

(T,,,)),

CE(st[6NS6N]

C.

for a standard positive constant

(T,,,))

W e will show that

1 E(stC6N.6NI

and

IN(t)(

therefore

Z

0

2

for

( T ~ ) )= 0

finite

t

8.5.

proving

our

lemma.

For each

n

in

IN

1 -

< $ E ( S ~ [ ~ M . ~ M ] ~ ( T ~ )+) BmE(st{I[

: s E

A:(")]}

where

A",")

= { s E Ui : s

<

T ~ ( o& ) IM(s)-M([s])l

> f},

1 -

2) .

380

Chapter 7: Stochastic Integration

estimating ( s

I

T

m

).

,$ 2m

IM(s)-M([s])l

where

is

it

large

It is sufficient to prove that : s E A”,(w)}

E(st{2[lBM(s)12

for every po

A:

on

m,n

= uo = Bt

E

IN.

Define

1 -

2) = 0

6t-stopping

times

as

follows:

and

th

p i = i-

timelM(t)-M(t-Cit)l

>

1

<

n].

and

u

If

M

then

has a

i

= min[t

: tj

j

2 pi,

Bt-decent path for the sample

>

lM(s)-M([s])1

$

and

s

,$ m.

a finite amount infinitely near

Us

path along

number

M(*.w) of

(5.3.4)(c).

w

and

Therefore

U6

[s]

and

s.

could only have jumped by more than s

,$ m

Thus for almost all

A”,(o).

varies

but since i t has a decent

s,

between

times before

s E

M(*.o)

i t must have jumped by an amount

one single time in hand,

0 5 j

by w.

(the A:(o)

2 > 1 at > - n n On the other 1 n

a finite

C0.m)-version

of)

is contained in the

countable (external) union

Hence, by of (7.2.8).

S-integrability and (7.2.6) applied a s in the proof

7.2

Section

Quadratic Variation of Martingales

E(stJP[ 16M(s)

1'

: s E

1

c

I

2

38 1

A:(o)])

st E(max[lM(t)-M(p,Ar,)I

i €IN

Pi A

Tm

<

t

:

<

ai A

T

~

t. E T,])

= 0.

We get zero because

M6(

T ~ )

I

max[ IM(t)-M(piATm)

whenever

M

has

happens a.s. Since

T~

is

: pi A

T,,,

6t-decent paths

This proves that

1

S-integrable while

a.s. as

m +

m,

5 t

<

for

the sample

ui A

max[IN(t)l

: t

T

I

~ Z]

0

which

o,

Tm] Z

0

a.s.

this concludes the proof of the

1 emma.

The primary consequence of this lemma i s the fact that the quadratic

variation

independent of

of

a

standard

the lifting and

local

hypermartingale

the infinitesimal

increment

is

in

particular (that is, once the increment is coarse enough to make the paths of

Fix an

A

r

decent). E

[O.m)

and i f

an increasing sequence. define

0 = ro

<

rl

<

0 . -

<

rk = r

is

Chapter 7: Stochastic Integration

382

(7.2.11) COROLLARY:

%

Let

be a local hypermartingale and let

%.

6t-local martingale lifting o f

S(%,{rj})

converges to

[%.fi](r)

P

standard quadratic variation o f of

lifting

and

indistinguishability)

we

r E C0.m).

tends to zero.

The

does not depend o n the

denote

decent

be a

in probability as the

maxlrj-rj-ll,

mesh o f the sequence,

choice

For each

M

path

the

unique

standard

(up

process

t,o

by

[ii.G](r).

PROOF : Choose any

[%,i](r) 6t = t

such that

= st[6M.6M](t) (7.2.10).

Lemma

H

t E

<

tl

<

a.s.

whenever -**

<

Let

the

tk = t

= st M(t)

G(r)

in

m

mesh

of

a

By

sequence

>>

such that

em

0

max(t -t I < e m , the probability above holds. .i j-1 0 = r < rl < * * - < rk = r be a standard sequence in

whenever Let

choose

be finite.

* finite

is infinitesimal,

This is an internal statement s o there is an

CO.-)

IN

a.s. and

with t

j'

maxlr -r

J

0

<

j

<

k

j-1

I <<

Let

em.

to = 6t.

so

p(rj)

= st M(t

)

. i

a.s.

tk = t

and

Section

7.2

The sequence

Quadratic Variation of Martingales

383

satisfies the internal probability above. s o

{tj}

standard parts yield

proving the corollary. Two

M

6t-martingales

N

and

distinct infinitely close times. M+N

is not a

could have finite jumps at

This would mean that the sum

6t-martingale because the paths no longer have

separated jumps.

M+N

Of course

is a

*martingale,

coarser infinitesimal time sample increment make

M.N

M+N

and

all

If we start with

%

we may apply the

%

lifting theorem to

(%,%). If

(%.N)

lifting of

(i.5).

at all),

so

may define

M+N

is a

is a

[g,%](r,u)

the

and

2d-dimensional

2d-dimensional

M

then

would

At-martingales.

standard local hypermartingales martingale

116

in

At

some

so

N

and

martingale

6t-local martingale

must jump together (or not

6t-local martingale.

In this case we

= S-lim[6#,6N](t.o) tl r

a.s. and use the

polarization identities (7.2.3) and (7.2.11) to see that this is independent

of

the

indistinguishability.

choice

of

the

lifted

pair

This shows how to extend

up

(7.2.11) to

pairs and the following extends (7.2.6) to pairs.

(7.2.12) LEMWA: Let

(%.H)

marttngaLe wtth the

If %(;)

17

is a

= st M(a)

be

LocaL

6t-Local marttngale Lifting

Gt-stopptng a.s..

2d-dimensionaL

a

ttme satisfying

then

[%.HI(;)

G <

to

OJ

= st[BM,6N](o)

hyper-

(M,N). a.s. and a.s.

Chapter 7: Stochastic Integration

384

PROOF : By the preceding remarks and ( 7 . 2 . 8 ) we know that

[1.8](;)

= S-lim[6M.6N](o+At)

a.s.

At10

[i,%](;)

and that

>

0

for

We simply apply the

* finite

[6M,6N](a)

Z

Z

0

infinitesimal

* transform

a.s.

of Cauchy's

dimensional vectors with components

I [ 6M.6N] (u+A t)-[

that

: 1

<

i

[6M,6M](o+At)

S-integrability of finite a.s.

inequality to the

6Mi(t).6Ni(t).

6%.6N] ( a )I =

=I)[6Mi(t)6Ni(t)

know

At

'E6.

in

[6M,GN](u+At)

We

= st[GM,6M](u+At)

Hence i t suffices to show that for every

a.s. At

= st[61,6M](u)

[6N,6N]

<

d.

Z

<

<

o+At, t E

[6M,aM](a)

a.s.

(I

makes

This proves the lemma.

t

[6N,6N](o+At)

T,]l

and

local

- [6N,6N](o)

385

(7.3) Square Martingale Integrals

i

Let

:

[O,l]

R

x

+

IR

P(0) = 0.

integrable and

be a hypermartingale with

We know from the Martingale Lifting

Theorem (6.5.13) that there exists a

M2(1)

S-integrable and

M(6t)

0

E

whose

s.

1.

with

6t-decent path

By Theorem (7.2.5). we

is S-integrable for all

[6M,6M](t)

M

6t-martingale

a.

projection is indistinguishable from know that

g2(1)

t

<

1.

In this

section we use estimates on the quadratic variation to show that the martingale integral

is

well-defined

as

the

6t-decent

path

projection

of

the

Stieltjes sums t-6t 1s=6t G(s.o)6M(sSo).

S(t.o) =

for a pathwise lifting

G

H.

of

"Well-defined" means this

a. s. does not depend on the choice of the martingale lifting,

M. or the path lifting, G. once

M

is chosen.

construction is the analog of (7.1.4).

Our first

The development runs

parallel to section 7.1. except that we use martingale maximal inequalities

(instead

of

the

triangle

inequality)

and

this

requires that our summands be predictable.

(7.3.1)

DEFINITION: Let

M2(1)

M

be a 6t-martingale w i t h

S-tntegrabte.

p a t h uartatton m e a s u r e

For e a c h ho

on

o E R

U

W(6t)

2 0

a. s. and

deftne a quadrattc

by t h e w e t g h t f u n c t i o n

Chapter 7: Stochastic InteFration

386

D e f i n e a total quadratic variation measure as

on

u

the hyperfinite extension o f the measure

Y x R

with weight

function

du(t.o) = 6Ao(t).6P(o).

Since

E{[6M,6M](l)}

hyperfinite measure, u .

extends to a bounded

is limited, u Since

[6M,dM](l)

P[A]

= 0, then

is P-continuous. i. e., i f

Iterated Integration (7.1.5) applies to

u.

is S-integrable. u u[Y

x

A]

Since

continuous at zero, u is continuous at zero, u[st-'(o)

(7.3.2)

G :

= 0. Also,

%

is right x R]

= 0.

DEFINITION: Let

H

:

T x R

+

*IR

[O.l] x R + IR

ho{t

i s called a

be any function.

such that for almost all

:

st[G(t.o)]

# H(st[t].w)}

2 61 -path lifting o f

An internal

o

= 0

H.

We can prove a path lifting theorem like (7.1.8) for the quadratic path variation measure, but unpredistable integrands give the "wrong" answer, as shown in the following exercise.

387

7.3 MartinFale Integrals

Section

(7.3.3) EXERCISE:

B(t,o)

Let

be Anderson's infinitesimal random

w a l k associated w i t h

6t

as above in (5.2.3).

Define

2 2B6B( o) = )[2B(s,o)[B(s+6t,o)-B(s,o)]

:

0

I

s

<

Ir]

t, s E

Show that

Z ~ Z B ~=BB2(t1-t pt2B6B = B2(t)+t and

2 St2B6B = B (t).

(HINT:

Write

B2(t)

Show that

Pt

Show

when

K(t.o)

that

as a double sum and compare.)

a n d St are not

= ZB(t.w).

= B(t.o)+B(t+6t.o)

*martingales.

H ( r ) = 2g(r).

K(t.w) are

then all

= ZB(t+6t.o)

.-.

the

functions

and

6B"-path ltftings o f

The exercise above shows that

Pt is.

but

K(t.o)

H.

6M2-path lifting alone is

not enough to make infinitesimal Stieltjes sums independent of the

infinitesimal

differences

in

liftings.

Moreover,

the

388

Chapter 7: Stochastic Intearation

internal sum

is infinite a. sgn[aB(t)]

for all noninfinitesimal

s.

depends precisely

on

t.

but

w t+6t,

The function

is internal and

bounded.

(7.3.4) DEFINITION: G : H x R + *IR

An internal. process if

G

is 0-predictable

S-tntegrabLe

is

and the function

with respect

6M2-summabLe IG(t,w)I2

is

to the hyperftnite measure 6u =

generated by the weight function

u

16MI2-6P.

This summability condition is equivalent to the condition

by the Iterated Integration Lemma (7.1.5). Our next result is part of a closure law f o r stochastic Stieltjes sums.

(It lacks the decent path property.)

understood that the martingale

M

It is

is a s above.

(7.3.5) PROPOSITION: G

Suppose

ts

6 M2-summabLe

1

(where

M2(1)

is

t-6t

S-integrable).

*marttngaLe

N(t)

Then

=

G(s)BM(s)

s=6t

after

6t

with

N 2 (1)

S-LntegrabLe

is

a

Section

7.3

389

Martingale Intecrals

PROOF : Since

G

is nonanticipating after

E[6Nlwt] = G(t)E[6Mlw

Moving

st

t

6t.

] = 0.

inside always produces the inequalities:

st[I B G26u] = st[E{[6N,6N](l)}] w t

>

E{st[6N,6NI(l)}

- s > E{

=

The

two extremes of

u-S-integrable. [6N.&iN]

st G2d(A,)}

[ st G 2du

by (7.1.5).

these inequalities agree because

Hence

st E{[6N,6N](l)}

G2

= E{st[6N,6N](l)},

is so

is S-integrable and (7.2.5) completes the proof.

Our next result says nearly the same sums.

u-equivalent

Again,

M

summands pathwise give

is a s above.

(7.3.6) PROPOSITION: Suppose

G1

U{(t,w)

*

and

:

G2

I

2 6M -summable and

st Gl(t.w) # st G2(t.w)}

Then the marttngale tnftnttely close to N2(t)

E{

are

= 0.

t

Nl(t) = B G1(s)6M(s) = 2 t G2(s)6M(s), in f a c t ,

max Et[Gl(s)-G2(s)]6M(s) 6t
I 2}

0.

t S

ChaDter 7: Stochastic Integration

390

PROOF : Since GIG2

IG1I2 and 2 2 max[G1.GU], while

<

IG2I2

are

u-S-integrable. so is 2 st GlG2 = st G1 u-a.e. Therefore,

N

Applying the BDG Inequality (7.2.6) to the martingale

=

N 1-N 2

yields

E{[

max IBt[G1-G2]6Mll2} 6t
In particular, the whole path of path of N2

<

CE{BlG1-G2 216M12}

N1

0.

Z

is inf nitely close to the

a.s.

(7.3.7) PROPOSITION: Suppose

B

is

6 -mar ingal

wtth

M2(1)

F.G

a n d the sequence

{Fm}

are all

S-tntegrable.

If

2 6 1 -summabLe.

st F m

IFm!

<

IGl,

tends to

st F

in

u-measure

and

then

tends to zero in probabiltty.

PROOF : Use comprehension from section ( 0 . 4 ) to extend the sequence {F,}

to an internal sequence satisfying

IFm(

<

IGl

for all

Section

m E

7.3 Martingale Integrals

*# .

Use the Internal Definition Principle (in the style of

Chapter 1) to pick an infinite is

391

Fn

infinite, then

Z

n1

F

n 5 n1

such that whenever

u-a.e.

Now

use

the preceding

proposition to show that

max IPtFn6M

E{[

- ZtF6M1I2}

0.

Z

6t
This means that for every standard positive

I(€) = {m E

*IN

<

I m

contains a finite

k

I

the internal set

E

>

n1 3 P{maxlZt[F-Fk]6MI

B }

<

8 )

This proves the claim.

m(e).

Up to this point we have not assumed that of a standard function of any kind.

G

is a lifting

However, we have not been

able to make any conclusions about the paths of The weakest pathwise result is taken up next.

= PtG.6M.

N(t)

In section (7.6)

we describe Hoover 8 Perkins’ [1983] better martingale lifting that makes 2 6M -summable standard

N

have

internal

6t-decent function-not

paths just

when

G

is

the lifting of

any some

H.

(7.3.8) THEOREM: Let

:

6t-martingale a. s.

[O.l] x R

M

Suppose

preuisible

or

with that

even

+

be

the projection

S-integrable a n d

H : [O,l] x R + IR

only

usual agumentatton of

IRd

M 2 (1)

5

preuisible

with

of

M(6t)

is respect

the Z

0

u-almost to

the

a n d that the pathwise S t i e l t j e s

Chapter 7: Stochastic Integration

392

integrals satisfy

<

E[lL H2(r.w)d[T,~](r,o)]

H

Then

G

:

U

x R

has

+*R.

2

6M -summable

a

Such a

G

m.

2 6M - p a t h

lifting

Pt G(s)6M(s)

makes

haue a

Gt-decent p a t h s a m p l e , so t h e s q u a r e s u m m a b l e m a r t i n g a l e

may the

be d e f i n e d independent

of

the p a r t i c u l a r

lifting as

6t-decent path projection o f

t-6t

2

G(s.o)6M(s.o)

s=6 t

PROOF : First, as long as

N(t)

= ZtG-6M

has a

sample, Proposition (7.3.6) shows that any two G's

have

the

same decent path projection.

remaining claims are 6M-sums have a

that such a

G

PROOF OF LIFTING: Change of variables gives us

=

2

6 1 -summable

Hence

exists and

Gt-decent path sample.

H2(r.o)d[~.$](r.w)

at-decent path

H2(st(t).u)dAa(t)

the

two

that

its

Section

for

393

7.3 Martingale Integrals

all

such

o

that

is

[%,%](1)

finite.

Iterated

Integration gives

= E{b2(st(

E[1H2d[%.%]}

E

t) ,w)dXu( t)}

1

<

H2(st(t).o)du

m.

We know by (6.6.8) or (6.6.11) that there is a

0-predictable

F

such that

: st[F(t.o)]

u{(t.o)

For each

m E N,

= 0

# H(st[t].o)}

the truncations of

F

and

H

at

m

satisfy

Thus

by

infinite

Robinson’s

Sequential

Lemma,

8 2 F26u z

st Fndu 2

for

sufficiently

small

n.

=

The function

G =

F n

H2 st du.

is our square-summable predictable path

ChaDter 7: Stochastic Intewration

394

1 i f t ing .

PROOF OF OF 6t-DECENT PATHS:

N(t)

We say that

M(t) w

N(t+At)

N(t)

@

t , t+At E

N(t)

H6

implies

M(t+At)

N(t)

6 M2-path lifting of

M(t)

only jumps where

@

Z

if

0, then

M(t).

M(t)

does and

has a

also has a 6t-decent path

H

We begin by showing that when

a bounded

= 0, such that

0 C At

and

M(t)

a. s. only jumps where

6t-decent path sample, then sample.

n, PCA]

A

if there is a null set

E A , then whenever

If

a. s. only jumps at the same times as

is bounded and

H. then

= ZtG.6M

N(t)

G

is

a.

s.

does.

H

First suppose that

is bounded and

u-equivalent

to a

basic almost previsible process o f the form in (6.6.13) and that

.-.m is the

6M2-path

0-predictable

(6.6.13).

Since

u

is continuous at zero, we may assume that

go = 0. The functions = ZtG*6M

lifting guaranteed by Lemma

gj

are bounded,

that

N(t)

M(t)

has a 6t-decent path sample at

Case 1:

N(t+At)

t

<

t

- N(t)

<

only jumps where

t+At, for

j

tj

w

lgj(w)I

M

<

b.

does we assume that

and consider two cases.

a s above in

G.

= = [N(t+At)

- N(t

+6t)]

j

To show

+ [N(tj+6t)

- N(t)]

7.3

Section

395

Martingale Inteerals

M

A t most one of the terms with a difference in

M

noninfinitesimal because and

has a decent path at

Thus, i f

are bounded.

‘5-1

- M(tj)] and g [M(t+At) j noninfinitesimal. Therefore, M(t+At)

<

i:

t+At

N(t+At)

so i f

*

N(t+At)

These

t

for

j’

- N(t)

N(t)

H

N

Proposition

(7.3.6) shows

N(t)

exactly one

- M(t)]

is

G

above

- M(t)].

M

a .s. only jumps where

is the particular

makes

gj

is a bounded basic almost previsible

G

G

and

M(t).

M(t+At)

two cases show that

process and

does.

as in

j

o

M(t).

= g.(o)[M(t+At) 3

then

does in the case that

lifting

t

N(t).

gj-l[M(tj)

*

t

*

N(t+At)

of the terms

Case 2:

can be

that

lifting obtained in (6.6.13). any

= BtG-6M

other

2

6M -path

bounded

also a. s. only jump where

M

The next step of the proof uses Lemma (6.6.14) to show

that every bounded almost previsible process produces nice path sums.

V

Let

be

processes, H.

N(t)

have

shown

set

of

all

bounded

that have a bounded

= BtG*6M

that

processes

the

that

2 6 M -path

a. s. only jumps where

V

contains

all

in the paragraphs above.

basic The

set

almost

previsible

lifting, G. does.

M(t)

almost

V

such We

previsible is a vector

space, because i f two stochastic sums only jump where

M

does,

CharJter 7: Stochastic Integration

396

so

Proposition (7.3.7) and Lemma

does a linear combination.

(6.6.14)

I

show that

contains all bounded almost previsible

processes.

Hm be the finite truncations of an

Finally, let

H.

integrable unbounded

N,(t)

G

lGml

2

= ZtGm6M

IGl a.

and

6M -summable

Gm + G

M

only jump

s.

Gm

in does.

Us.

A

Let

Hm

and

u-measure.

The

In particular, Nm 6t

Proposition (7.3.7) says that a subsequence of the

N

6M -path

lift

a. s. has a 6t-decent path sample for the same

tends uniformly to

2

2

is a

H. then the finite truncations

lifting of satisfy

If

as Nm

M.

a. s.

in the space of internal functions on

be the countable union of null sets where some 6t-decent path or the subsequence of

does not have a

not tend uniformly to

If

N.

z ~ + ~ ~ *G z- ~~G M- ~ M .

m.

H ~ + ~m ~ G . L ~ ~MG ; ~ M ,

since we may make the uniform error between less than one third of

the difference.

M

a. s. only jumps where

does

and

o Q A

then for sufficiently large finite

Nm

Nm

does, s o does

Gm -sums and Since each LtG*6M.

G-sums ZtG;6M

This proves

half of the main result of the section. Since we have actually shown more than that

ZtG-6M has a

6t-decent path sample, we have the following corollary t o our proof.

(7.3.9) THEOREM: If

M

i s an

S-integrabLe and summabLe

S-conttnuous

M(6t)

6M2 -path

process. then

=: 0

Lifting

ZtG-6M is also

6t-martingaLe luith and i f

of

a

G

standard

S-continuous.

is

a

M2(1) 6 M 2-

preuistble

Section

7.3

397

Martinnale Integrals

PROOF : Since the proof of (7.3.8) shows that

M

where

M

does and

stochastic sum is

HtG-6M

does not jump by

only jumps

S-continuity, the

S-continuous.

The remaining half

of

the well-definedness

question is:

"What i f we take a different martingale lifting?"

(7.3.10) THEOREM: Suppose integrable

%

that

%(O) = 0.

and

6t-martingale lifting of

N

is a

Suppose

% with

At-martingale

S-integrable.

G2(1)

is a hypermartingale with that

M2(1)

F

is

%

with N2(1) 2 6M -suminable 6M2-

is a

path lifting of a standard almost preuisible process

G

while Then

2 6N -suminable

is a

the

At-decent

path

indistinguishable from the

a

S-integrable and

lifting of

Suppose that

M

H.

6N2-path Lifting of

H.

LtG-6N

is

projection

of

6t-decent path projection of

PF-~M. PROOF : First suppose that process and (6.6.12).

F In

and both

H

is a bounded basic almost previsible

G

have

cases

the

the form of decent

path

the

lifting of

projections

are

indistinguishable from

m

1 hj(.o)*[%(r

j + l)

-

%(rj)].

j=1

Proposition (7.3.6) shows that

F

and

G

can be any bounded

398

Chapter 7: Stochastic Integration

H

path-liftings of Next

we

and still have the above projection.

apply

previsible

processes

previsible

H

and

G

V

V

such that if

is a bounded

projections of set

(6.6.13).

Let

consist F

of

subset

all

6N -lifting of

H.

of

almost

bounded

almost

2 6 M -lifting of

is a bounded

2

LSF*6M and

the

H

then the decent path

ZtG*6N are indistinguishable.

The

is a vector space containing the basic almost previsible

processes by the remarks above.

The space

V

is closed under

bounded pointwise convergence by Proposition (7.3.7).

Thus

V

contains all bounded almost previsible processes. Finally, a truncation argument similar to the last part of the proof of Theorem (7.3.8) shows that every square summable integrand produces indistinguishable stochastic sums.

(7.3.11) EXERCISE:

Show that alL the p r o o f s in this sectton actually apply to the case of take

G :

d-dimensional Linear functtonals.

H

x

R

+

IRd

That ts, ure may

and tnterpret our stochastic Stteltjes

sums as sums of tnner products,

The meaning of lifting is clear.

A

6M"-summable vector

valued function is one for which the square of the vector norm, IG(s.0)

12.

is

u-S-integrable.

399

(7.4)

Toward Local Martingale Integrals To extend the treatment of martingale integrals from the

square

summable martingales

local martingales,

we

of

may

use

Theorem (6.7.5) and the special (6.7.3)

the

the Local

stability

to standard

Martingale

Lifting

Gt-reducing sequence

together with Theorem (7.2.5).

infinitesimal

last section

results

{T~}

of

This gives us the basic

without

use

of

Iterated

these

stability

b

Integration

(7.1.5).

This

section

proves

results, but does not give a general standard lifting theorem. (Sections

(7.6) and

(7.7)

sketch the proofs of

the powerful

lifting theorems of Hoover & Perkins [1983].) We do show how these simple extensions of section (7.3) may be applied to prove that paths of the quadratic variation are decent.

This is the main application of the section.

For this section we work in the evolution scheme of (5.5) and (6.7).

(7.4.1) DEFINITION: Let

M

Gt-reducing

be a

d-dimensional 6t-local martingale with

sequence

{T~}

and

M(Gt)

Z 0

a.s.

The

quadratic path uariation measure is defined by the weight function

The hyperfinite extension measure, in fact,

k

E IN.

[GM,6M](k)

Xu

may now be an unbounded

may not be

S-integrable even for

Proof of lifting theorems require that we control the

Chapter

400

growth of version

6hW's

of

so

7:

Stochastic Integration

that the total measure

6ho*6P.

The

simple procedure

6u

is a bounded

that we

used

in

Definition (7.1.4) will not work. s o we postpone this problem to section (7.7). We do know from the definition of

6t-reducing sequence

that

M 6 ( T ~ ) is

S-integrable for each

m

By (7.2.5) we also know that

-/[~M,~M](T,)

is

S-integrable for each

in.

Despite the technical problem with boundedness of the total quadratic variation measure, the notion of path-lifting remains the same.

(7.4.2) DEFINITION:

Let

M

be a

internal fucntion G function H

hu{t

6t-local martingale as above. is a

An

2 6M -path lifting of a standard

prouided

: st[G(t.w)]

#

H(st[t],o)}

= 0

a.s.

W .

Since we do not yet have an analog of the measure

u

from

the last section, we define local summability with the standard part.

Section

401

7.4: Toward Local Martingale Integrals

(7.4.3) DEFINITION:

M

Let sequence

be a

6t-local martingale with and

{T~}

0-predictable process prouided

that

there

Bt-stopping times

<

M(6t} G

0

Z

a.s.

is called Locally is

an

increasing

6t-reducing An internal. 2 6M -summable sequence

of

{urn} satisfying:

(a)

am

(b)

st[M(om)]

(c)

S-lim urn = m

Tm

= %(S~[CJ,])

a

.

~

a.s.

r

Our first result should be a "closure law" for stochastic sums.

Unfortunately, we cannot conclude that internal sums have

Gt-decent

paths

section (7.6).

without

the

stronger

martingale

However, we will now show that

has all the other properties of a

lifting

N(t)

of

= Zt G-6M

Bt-local martingale.

The maximal function

N 6 (urn) is

S-integrable

by (7.2.5) because we have assumed by (d) that

d[BN,6N](um)

Without knowing that can still show that

is

N

S-integrable.

has a

6t-decent path sample, we

Chapter

402

7: Stochastic Integration

= S-lim N(t) t lom

st[N(o,)]

a.s.

First we show that

= [a,a](st

st[6N.6N](am)

a.s.

am)

We have the inequality:

am) - st[6N,6N](am)

[%.%](st

= S-

- [6N,6N](am)

lim[6N,6N](am+At) A t 10

<

S- lim I [ I G ( S ) ~ ~ ~ S M ( S: )am ~~ At10

= lim At10

s

<

s

<

um+At]

st IG(s) I2dXw(s).

{am<s
The final integral in this inequality tends to zero because we have assumed that

(7.4.4)

%(st

am) = s t H(am)

a.s.

EXERCISE: If

N

is a

*martingale

6t-stopping time such that

[fi,a](st

after

N6(a)

is

a) = S-lim[6N.6N](a+At),

6t

and

a.s..

then = S-lim N(o+At)

At10

is a

S-integrable and

At10

st[N(o)]

a

a.s.

403

7.4: Toward Local Martinpale Integrals

Section

HINT: This is a "converse" to Lemma (7.2.8).

The ideas in that

proof can be used here. This exercise together with the preceding remarks mean that if we show that

N

has a

6t-decent path sample, then

N

is a

6t-local martingale. The

next

result

independent of

implies

choice o f

the

that

stochastic

lifting, but

sums

applies

to

are more

general internal summands.

(7.4.5) PROPOSITION:

M

Let

be

GI

s u p p o s e that

ho{t

that

a

€ 0

and

6t-local

G2

martingale

a r e locally

: st Gl(t.w)

# st G2(t.o)]

T h e n except for a single null s e t o f t E

O ' S

as aboue and 2 6 M -surnmabLe a n d = 0

a.s.

o.

f o r all f t n t t e

T6' It G1(s,o)6M(s,o)

%

Bt G2(s.w)6M

S,W).

PROOF :

If

and

u:

conditions for

2 am

GI

are the stopping times in the summability 1 2 let u = u A am. The am and G2, m m

satisfy the summability conditions for both Let

N

*martingales.j

We

can

see

(t)

= B

t

Gj(s.o)6M(s.o).

for

G1

and

G2.

j = 1.2.

define

The BDG Inequality (7.2.6) says that

that

the right hand

side of

this

inequality is

404

ChaDter

infinitesimal

1)

by

showing

7: Stochastic Intearation

that

the

internal

function

U

mlG1-G21216M12 Summability

S-integrable. Nj(um) 6

are

is P-S-integrable. of

means

Gj

that

[6Nj,6Nj](um)

is

By (7.2.5) this means that the maximal functions P-S-integrable.

Therefore,

max

<

IH(G1-G2)6MI

6 t t
<

N:(u~)+N~(u~)

see that

is

S-integrable.

)umlG1-G21216M12

is

Applying (7.2.5) again. we

P-S-integrable, s o

Finally, condition (d) of the summability hypothesis says that a.s.

a,

= 0.

Hence the right side of the BDG Inequality is infinitesimal. This concludes the proof, since

u

m

+

a.s.

We also have a convergence in measure result similar to the

last section.

Section

7.4:

Toward Local Martingale Integrals

405

(7.4.6) PROPOSITION:

M

Let

that

the

be a

6t-Local martingale a s a b o v e .

F,G

functions

6M2-suminable

locally stopping

all

and each standard

e

>

mith

sequence respect

<

IGI

{Fk}

are

the

same

to

and f o r each

m

0.

u,

then for each finite

- ItF6M]

m a x [XtFk6M

st

the

IF,]

If

{am}.

times

and

Suppose

+0

6t
in probability.

PROOF :

IFk}

Extend

IFkI

<

IGI.

to

For each

and an infinite

k

for every

an

m E IN

between

m

and

E

E Q+

sequence

satisfying

there is a finite

n1

such that

n2

n1

sufficinetly small infinite

all standard

internal

and

E .

and

k

n2.

By

saturation, all

satisfy these inequalities for

Hence, we may apply (7.4.5) to prove

our claim (along the lines of the analogous result from the last section). We

shall not

following

prove

a general

lifting

result may be proved along

theorem, but

the same

lines as

the the

406

Chapter

decent path part

of (7.3.8).

This

7:

Stochastic Integration

in turn has a n interesting

application.

(7.4.7) PROPOSITION: Let that

G

M

be a

6t-local martingale a s above. Suppose is a locally 6 M2-summable process w h i c h lifts a

H, that i s ,

standard preuisible process

ho{t

:

E 0

st[G(t,w)]

# H(st[t],o)}

= 0

a.s.

0 .

T h e n the stochastic Stieltjes sum

a.s. has a

6t-decent path sample

This "half" of the standard stochastic integrat on the0 r em

for local martingales can be applied to the interna

quadratic

variation process as follows.

( 7 . 4 . 8 ) LEHHA

If

the

pair

L = (M.N)

6t-local martingale. then

N

is

2d-dimensional 2 is a locally 6M -suminable a

612 -path lifting o f the left limit process

N

N ( r - ) = S-lim N(t) t tr

= E(r).

7.4:

Section

Toward Local Martingale Integrals

407

PROOF :

Let

be a

T~

6

N (Tm-6t) .$ m

6t-reducing sequence for

on

{Tm

>

6t)

(M,N).

Since

4/C6M.6M1(Tm)

and

is

S-integrable,

or

N

2 6M -summable up to

is

L = (M,N)

Since

define

stopping

times

16L(t-6t)l

> 71 . ' ' J

st M(a i ) =

%("'3)

i.

for

6N(t-6t) i,j.

J

*

0

6t-decent path sample a.s.

= G(st(t)) = " i c time

finite so

Also,

i.j

by if

a.s.

have

then

(7.2.8). t E t =

U6

in

t

E IN,

if

Similarly, i f we

a.s.

on a decent path, then

Therefore, we

We know that

a

aj

8.5.

i = S-lim [GM,6M](u.+At).

Atlo

has

st M(t)

0, then

aL(t-6t)

rm.

T6

if st [

is

that

i u

j

<<

6M. 6M] finite

m,

(03) and

for some finite

Chapter

408

= S-lim [6M,6M](a!(u)+At) 3

Atlo

= 0

7:

Stochastic Integration

- [dM.bM](~;(u))

a.s.

by the remarks above.

Hence

N(t)

is a

6M2-path lifting of

E(r).

(7.4.9) THEOREM: If

L = (M,N)

martingaLe, then

is

[6M.6N]

2d-dimensionaL

a

has a

6t-local

6t-decent path sampLe.

PROOF : This follows easily from (7.4.8) and (7.4.7) together with the

(*transform of the finite) formula for summation by parts:

[6M,6N](t)

= (M(t).N(t))

- Xt(N,6M)

-

Xt(M,6N).

409

(7.5) Notes on Continuous Hartingales In this section we present a few basic facts about local hypermartingales

with

internal objects.

continuous paths

and

the corresponding

The basic references for this material are

Keisler [1984], Panetta [1978], Lindstrom [1980a] and especially Hoover

&

[1983],

Perkins

section 8 .

part

11,

which has

the

strongest results. The first result says that sampling in order to obtain nice path properties of a lifting is not necessary.

(7.5.1) THEOREM: Let

+.

6t = min T

If

M

w i t h a.s. continuous paths and 6t-local martingale that

the

N

M(0)

with a.s.

continuous

indtsttnguishable f r o m

is a Local hypermartingale

path

= 0.

then there is a

S-continuous paths such

M

is

M.

The next result of Hoover & Perkins [1983] of

fi

projection

says continuity

can be measured by the quadratic variation [in contrast

to the decent path case. see (7.6.2)].

It extends results in

Keisler [1984] and Lindstrom [1980a].

(7.5.2) THEOREM: Let is

M

be a

*martingale

S-continuous and locally

d[6M,6M]

is

and

+.

6t = min T

Then

M

S-integrable i f and only i f

S-continuous and locally

S-tntegrable.

This has the following important consequence.

410

7:

Chauter

Stochastic Intevration

(7.5.3)COROLLARY: Suppose

martingale and process.

M

that G

Then

is

an

S-continuous

6t-local 2 6 M -summable internal

is a locally

ZtG*6M

is an

S-continuous

6t-local

martingale.

We add that

G

need not be

process [compare to (7.3.9).] Keisler's

existence

of

internal

This idea plays a key role in

theorem

equations mentioned below. difference

the lifting of a standard

for

stochastic

G

Since

equations

differential

may be internal, solutions have

standard

parts

which

satisfy an associated stochastic differential equation. The criterion in (7.5.6) for continuity plays a role in constructing strong martingale liftings which satisfy the decent path version of Corollary (7.5.3).

(7.5.4) NOTATION: 6t E 1

Suppose

is a positive infinitesimal and

is nonanticipating after

x

(t) = X(6t)

+

16

(7.5.4)EXAMPLE:

Z

Let

Z[Z(W) cr2

:

w

= Z[Z2(w)

E

:

W

W] :

-

*R

= 0.

w E W]/#[W].

6t.

X

We define

E[6X(s)los]. s=6t step 6t

for

t E Ui.

be an internal function with zero mean, and Define a

1 imi ted

*martingale

variance , by

+

6t = min T.

where

41 1

7.5: Notes on Continuous Martingales

Section

I

1'

E[ l6M(t)

In this case

2 = Z (Ut+6t)6t.

l6M(t)I2

but

wt] = a2*6t. so

(t) = u2*t,

[6M,6M]1 6

while

[6M.6M](t) With

some

itself is usually not s o easily computed. extra

[6M,6M]

estimate

&

integrability, Hoover

Perkins

[1983]

a s follows.

(7.5.5) THEOREH:

IF

M

is

S-integrable and

The

following

a

M~

6t-martingale,

sup[stl6M(t)l] t€0

result

uses

= 0.

is

locally

a.s.. then

[6M.aMl]

to

(t)

check

6

continuity.

(7.5.6) THEOREM: Let

M

be a

6t-local martingale.

If

[6M.6M]1

(t)

6

is a.s. then

M

S-continuous and t f is

S-conttnuous.

sup[stlBM(t)l] tEO

= 0

a.s..

412

(7.6) Stable Hartingale Liftings &

Hoover

[1983]

Perkins

show

that

coarse

enough

6t-martingales have the property that all their Stieltjes sums also have a ingredients Example

6t-decent path sample. of

(7.6.2)

necessary.

their

result

helps

explain

Roughly

internal

the

why

into

hyperfinite

coarser

speaking, the

martingale

martingales.

in

This section outlines the

idea

continuous

Since the internal

time

time

is

to

and

framework. sampling

is

decompose

an

discontinuous

line is discrete

this

means we "take a limit." Let

M

be a

6t-local martingale and let

m

be a natural

We define

number.

,t-6t

and

1

' t-6t

Mm(t)

=

6M(s)I s=6t {IbM(s)l>

The conditioned processes

[see

1 ' m)

(7.5.4)]

and

are each

M(t)

*martingales = M(6t)

and

+ [W,(t)-M

,I6

(t)]

+ [Mm(t)-Mm16(t)].

Section

7.6

m

In the l i m i t as jumps of

we expect

m

the martingale,

Mml6(t)

so

[Mm(t)-Mmlti(t)]

is infinite,

to contain all the

should "tend toward a

is an

using Theorem (7.5.6) above.

the limit condition on

MmIti(t)

m

S-continuous process by

[Mm(t)-Mm16(t)]

This means that

tends to a continuous process as

X

Mm(t)

Hoover & Perkins [1983] show that when

continuous process."

If

413

Stable Martingale Liftinas

m + m.

We shall formulate

more precisely.

is any internal process we denote the infinitesimal

X

oscillation of

OX(t)

up to

by

t

= sup[stlX(u)-X(v)l

:

u

=

v

<

tl.

We want to have

'C6Var

-

Mmlti](t)

for each

limited

t.

happen.

However,

there

A t E Hti

in probability, as

0

Unfortunately, is

always

which makes this happen.

TA

coarser time line

m +

this does

a

coarser

A sample of

a,

not

always

infinitesimal

M

along the

produces only decent path sums.

Recall

the special form of a coarser sample of a local martingale from (6.7.6).

(7.6.1) DEFINITION: Let

M

be

a

tit-martingate.

If

LnftnttestmaZ such that

'[AVar

Mm16](t)

-+

0

A t E H6

t s

an

414

Chapter

in probabiLity f o r aLL Limited

M

At-sample o f

Our next

is a stabLe

example

6t-martingale.

Stochastic Integration

then we say that a

t.

At-LocaL martingaLe.

should help

Here is the outline of what

7:

to clarify

the situation.

M

the example contains:

is a

is an internal bounded predictable summand,

G

1G-6M t

yet

=

N(t)

variation

does

= [6N,aN].

has indecent paths.

not

detect

the

A coarser sample of

M

problem

Moreover, quadratic because

[6M,6M]

is stable.

(7.6.2) EXAMPLE: Let

u

:

W + (0.1)

We may think of rate starting at

Recall

that

our

be an internal function such that

as Bernoulli trials with an infinite success

u

t = 1

by summing “successesn as

infinitesimal

approximation

to

the

process in (0.3.7). (4.3.3) and (5.3.8) had a jump rate

Now we have = b*6t , #W

with

b

Z

1

a-

Poisson a

when

415

7.6 Stable Martingale Liftings

Section

6t-stopping time for the first "success,"

Define a

h n.

k

Then the probability of beginning with at least

"failures"

is

PCT-1

>

k*6t] = (1-p)

PCT-1

>

t]

k

or

T

s

means

t at

"success"

limited multiple of

happens

8

7--1

a

Now we define a

<

and

M(t)

=

1 6M(s), p

(-l)k(l-p) 0

Notice

for

6t-martingale using

(-l)k+l 6M(t-6t.o) =

that

[6M,6M](l+t)

6M = 5 f l + o(6t)

=: t

Ii t es ima 1

a nonin

E 01 = 1 .

t

1,

.

within

t = r c ,

To see this, observe that if

t

t/6 t

a. P[O

for

= (1-p)

for

t

r

T .

>

1.

limited, then

Let

<

if

t = l+k6t

,

if

t = l+k6t =

I

if

t

until

= 0.

where

,

>

M(t,w)

T ( W ) T(O)

.(&I).

the first success,

until the first success,

l+t =

T .

so

However,

7: Stochastic Integration

Chapter

416

we have seen that multiple of

1

Z

T

a.s., since

T

is a.s. a finite

a.Hence

*martingale

Next, we define a

1 G-6M. t

N(t)

where

G

=

is deterministic and bounded,

G(l+k6t)

= (-1)

k

.

This makes

aN(t-6t)

= p-1

(-l)k-l(-l)k(l-p)

We know that

p =

6+

noninfinitesimal limited

o(6t)

r

an

N(T) does not have a

first builds up to infinitely near

r

t = 1.

if

t = l+k6t

if

t =

T ( W T

T(O)

<

= l+k6t

= l + r a

=: 1

T(O)

for some

and

= 0

N(T-6t)

N

s

a.s., hence

N(l)

Therefore.

.

k-l(-uk+l P = P

=

2

Z

r

r-1.

6t-decent path sample because i t

and then jumps down by

1

for times

Section

7.6

Stable Martingale Liftinas

N

The quadratic variation o f

l6Ml = 16"

because

for all

417

M.

is the same as that of

t.

Finally, we compute the decomposition

This decomposition is the same for all values o f

2

that

<

we simply let

so

m

6M( s) I

1

i 3)

0

m = 2.

E

*

IN

such

First,

= l+h*6t

,

if

,

otherwise

s

m

<

T-6t

s o that

EC6M(s)I

5)I

WS]

{laM(s)l<

= (-1)

(s-l)/6t p(1-p)

,

if

then (~-1)/6t 6X(s)

0

hence

P

2

.

for

l < S < T

= {(-l)

otherwise,

1

<

S

<

T.

418

X(t)

z 0

Chapter

7: Stochastic Integration

for all

t

The other part of the decomposition of

a.s

M

satisfies

while (s-l)/6t (p-1)p

This means that the variation in steps of

where

A t E U6

satisfies

At

Z

Mm la

At-sample of

M

s

<

T

otherwise.

satisfies

6t

on a time axis

uA*

0, then

AVar Mm16(~) 1 0

and a

if

,

0

On the other hand, i f we sample

,

a.s

makes i t a stable

At-martingale.

general, such coarser time sampling always works.

(7.6.3) LEHHA:

Let

M

inftnitestmal stable

be a A t E Us

6t-local martingale. so that a

At-local martingale.

At-sample

There is an of

M

is a

In

A

419

7.6 Stable Martingale Liftinas

Section

value

of

that makes

At

the

standard

part

of

the

variation equal the variation of the standard part always exists by results in section 7.1.

Such a

At

satisfies the lemma

above. The point of this lemma together with the Local Martingale Lifting Theorem is that every local hypermartingale has a stable At-local

martingale

lifting,

These

are

the

liftings

that

Hoover 8 Perkins [1983] show make internal Stieltjes sums have decent paths.

(This is why we called them "stable.")

(7.6.4) THE HOOVER-PERKINS THEOREM: If

M

Loca Lg

is a stabLe

At-Local martingale and

AM 2 -summabLe process, then

2

G

is a

G-AN

is a

t

N(t)

=

At-local martingale.

This is the main technical result of the Hoover & Perkins [1983]

for

article.

It is applied to give a new existence theorem

semimartingale

equations.

The

decent

path

property

is

needed to show that the internal sum arising from the solution of a

* finite

difference equation has a standard part.

420

(7.7)

Semimartingale Integrals We want to define integrals

for a wide class of integrands and 'differentials'.

About the A

Z

most general kind of process

that we can use for an Ito-

calculus is defined in (7.7.3).

The term "semimartingale" is

not completely standardized in the literature and our use of i t is relative to our own evolution scheme. gale" to warn you of the latter above.) predominant custom and even

(We used "hypermartinOur usage is close to a

sense of humor couldn't bear

out-

'local-semi-hyper' . . . A semimartingale

Z

Z(r)

where

N

is a process which may be written as

is a martingale, with

variation with

W(0)

+ W(r)

= Z(0) + N(r)

N(0)

= 0

W

and

has bounded

= 0. For the time being let us assume that

N2(1)

r

integrable.

The complete carefully developed theory of sections

(7.1) and

E

[O,l]

(7.3) applies

previsible process. and let a

U

where

Let

the

M

be a lifting of

6t-martingale with

W M

lifting

6t-bounded variation for

as in (7.1). and

6t E TA.

[6MS6M](l)

bounded variation with

Z-integration of

to

be a lifting of

At-martingale

and

var W(l)

we work on

6Var U(l)

N

are both

a bounded

as in (7.2-3)

By first choosing

then choosing

we may suppose that

S-integrable and S-integrable.

U

U

with

M has

is a

6t-

Section

42 1

7.7: Semimartinnale Integrals

We may combine the path variation measure for

M

and

U.

Define the weight functions

and

+ 6hw(t).

6pw(t) = 6Kw(t)

In this case

is

p,[U]

(7.1.5) applies

P-S-integrable, so Iterated Integration

to the hyperfinite measure

given by the

u

internal weight function

6u(t,w) = 6pU*6P(o).

The Predictable Lifting Theorems applied to bounded

u

(6.6.8) and

and a bounded previsible process

0-predictable internal

u{st[G(t,o)]

G

(6.6.11) may be

H.

satisfying

# H(st[t].w)}

= 0.

Since this joint total variation measure

u

measures of sections (7.1) and (7.3).

is both

and

G

dominates the path 6Mz-summable

6U-summable and

is well-defined as

of

yielding a

H(O)*Z(O)

plus the decent path projection

422

There is something extra theory

of

Chapter

7:

to prove

in

(7.1) and

sections

Stochastic Integration

this definition.

The

only

the

(7.3)

shows

that

infinitesimal Stieltjes sums give the same answer for different liftings of N. W

€I.

and

Z

The decomposition of a semimartingale plus bounded

variation

terms is not unique.

into martingale

Z

If

is a

Z(r) = J(r)-Xr

martingale of bounded variation [such as

for a

Poisson process as in (5.3.8), (4.3.3), (0.3.7)] then we may view either

Z

N

W

or

is continuous, then

and then are unique.]

[If

as zero in the decomposition above.

N

and

W

may also be chosen continuous

This means that Stieltjes and martingale

integrals must agree when both are defined since classically one defines integrals against trouble

in

the

classical

dZ

by

JdN

approach

+ JdW.

This causes some

which we

at

least

avoid

conceptually since we use infinitesimal Stieltjes sums for both parts

of

the

26X = 26M + 26U.

lifting,

(We do

still

use

separate estimates for the two terms.) The nonuniqueness

in the decomposition

Z = Z(0) + N + W

causes a far more irritating problem when i t comes to trying to identify a space of

(7.7.1) EXAHPLE: Let half of that

p

W

5. 2 3.

:

W

and

-

dZ-integrable processes.

-1

on one

on the other half (see 4.3.2).

We know

{-l,+l}

+1

be internal and equal

2 E U for each 4,***,e m

infinite natural number such that

finite

{G

:

m, m

<

so

let

n} C T.

n

be an

Define a

Section

6t-martingale by

= 0

M(0)

M(t)

and

.

{ P ( w ~ + 1~ ~ )i~f =

6M(t,o)

Since

423

7.7: Semimartinaale Integrals

2'16M1

.

=: H

m

31 <

t =

where

m-l m '

m < n

otherwise.

we

m,

t

= H 6M,

see

M

that

has

bounded

m

variation. as

The semimartingale

X(t) = 0 +

%(t)

+ 0

or

X(t) = G(t) as

may be decomposed

X(t) = 0 + 0 + %(t).

The

deterministic internal function

is not because

m- 1

m ,

if

t = - m .

0 ,

otherwise

m l n

S-integrable with respect to the first variation z'Gl6MI

= Hn

is infinite.

However,

G

ISMI, is

S-

integrable with respect to the quadratic variation,

and

It

turns out

that semimartingale integrability

defined by saying that a process integrable with

respect

decomposition

H

may

H

is integrable if i t is

For another

to .some decomposition. not

be

is well-

integrable.

it

is

integrable, then i t produces an indistinguishable integral.

In

other words, given another decomposition of

Z.

but

if

either we get

424

ChaDter

the same answer or "infinity."

7:

Stochastic Integration

This is the point of (7.7.10).

Strange, but at least consistent . . . .

A simple special case of

this may be proved as follows. Suppose that and that

(Mj,Uj)

Z(r) = 0 + N1(r) lifts

(N..Wj). J associate total variation measures for infinitesimal increments

6tl

+ W,(r)

+ W2(r)

j = 1,2, as above.

for u1

= 0 + N2(r)

and

and

u2

We

to each lifting

6t2.

If

H

is an

almost previsible process satisfying

Then we can find

0-predictable internal processes

G

j

such

that

This just requires a slightly different truncation argument and the bounded that

H

Predictable Lifting Theorem

(6.6.8).

This shows

has liftings for both decompositions.

The decent path property

of

the infinitesimal Stieltjes

sums can be proved in the same manner as the decent path part of the proof of (7.3.8).

Finally. the indistinguishability can be

proved in the style of the proof of (7.3.10).

In fact, these

two parts can be combined in a single proof using 6.6.14) where we let

W

(6.6.12

-

be the set of bounded almost previsible

Section

7.7:

Semimartingale Intenals

425

processes which have liftings for each decomposition producing 6t -decent path j

sums and

indistinguishable

projections.

The

details are left as an exercise. We hope that the separate treatments of sections (7.1) and

(7.3) are clearer even than a combined summable plus

treatment of

"square

For the

integrable variation" semimartingales.

remainder of this section we simply state the full-blown general results needed to define semimartingale integrals by lifting to infinitesimal Stieltjes sums.

Further details must be found in

Hoover & Perkins [1983].

(7.7.2) SEMIHARTINGALE INTEGRALS: Here

a

is

summary

of

the

classical semimartingale integrals. an

S-semimartingale

Z-integrable

process,

then

If

H

and the

of

effect

is

Z an

this with

section Z(0)

= 0

on is

almost-previsible

decomposition

that makes

H

integrable has a

= 0 + M(t)

+ U(t)

The decent

and

H

6t-semimartingale lifting X(t) has a (6M 2 ,6U)-summable lifting G.

path projection of the

Gt-semimartingale

1 G6X t

Y(t)

=

is what we take as our definition of the process

Ji

H(s)dZ(s)

= y(r).

A slight variation on Theorem (7.3.8) shows that the decent path

Chapter

426

Y

projection exists and moreover that

U6.

does on

well-defined path

7: Stochastic Intezration

a.s. only jumps where

X

Theorem (7.7.10) shows that this definition is up

to

projection

indistinguishability, that

is

the

same

no

matter

lifting, which integrand lifting

G,

is.

the decent

which

decomposition

or which

infinitesimal

time sample (subject to all the "lifting" requirements) we take. We

want

to

set o f

variation

up

var W(r.o)

: C0.m)

any

to

W

the paths of

O'S .

x R

r E

*R

[O.r].

a r

<

definition

of

*IRd

with

that

the

process

is defined f o r a.a. paths. to the setting (5.5.4) with

the only formal change that restrictions at need

4

have finite classical or

C0.m).

We extend the notation (7.1.1)

We

x R

: [O.m)

This means that except for a single

locally bounded variation.

null

W

study processes

bounded

r = 1

variation

on

are dropped. each

interval

m.

(7.7.3) DEFINITIONS: We has

say

that

the internal

T x R

* *Rd

(U.6 x Var U)

has a

process

6t-locally bounded variaiton i f

U

6t-decent path sample and its projection is able f r o m

A

(c,var

ndistinguish-

z).

process

z

: C0.m)

x R +R

is

called

a

d-dimensional semimarttngale i f there exist progressively measurable decent path processes

= W(0)

= 0

such that

N

N

and

- Z(0)

with

N(0)

is a local hypermartingale.

has locally bounded variation. and

Z(r)

W

= N(r)

+ W(r)

W

Section

X

An internal process d-dimensional

M(6t)

U(6t)

f

martingale,

:

f x R

*Rd

-P

is called a

6t-semimartingale for an infinitesimal

there exist

if

427

7.7: Semimartinvale Intevrals

such that

0

%

M

internal processes

U

M

and

U

is a stable

is nonanticipating

after

6t-locally bounded uariation, the pair

with

6t-local and has

6t

(M,U)

6t

has a

6t-

decent path sample a.s.. and

X(t)

for

-

X(6t)

= M(t)

+ U(t)

+

t E H6.

X

An internal process lifting

of

measurable

a

is called a semimartingale

process

Z

Gt-semimartingale for some infinitesimal

2

6t-decent path projection

If

X

is a

X

if

6t

is

and if the

is indistinguishable from

6t-semimartingale.

a

the projection

clearly a semimartingale because the projected pair

Z.

2

(8.c)

is is

the required decomposition.

If

Z

is a semimartingale. then there is a semimartingale

lifting

X

for

Z.

(7.7.4) THE SEMIMARTINGALE LIFTING AND PROJECTION THEOREM:

A process

Z

is a semimartingale if and only tf i t

has a semimartingale lifting

Semimartingales differentials'

are

because

a they

X.

"good" contain

class a

of wide

'stochastic class

of

428

Chapter

7:

Stochastic Intepration

traditionally important processes, are closed under integration by a wide class of integrands. and because they are also closed A

under

change

of

variables

in

the

sense

of

Moreover, there are several technical ways

"Ito's

to say that semi-

martingales are the widest possible class of for

example.

see

Metivier

and

formula."

'differentials',

[1980].

Pellaumail

12.12, or

Williams [1981]. p. 68.

(7.7.5) EXAMPLES We would like to indicate how one goes about verifying that the classical

stationary

independent

Z

semimartingales.

Suppose

process, that is,

Z(0) = 0 and

increment

processes

is such an adapted decent path Z(s)

- Z(r)

is independent of

and its distribution is only a function of

3(r)

Z(r) = %(r)

example, we may have

are

(s-r).

For

where

1 6X t

X(t)

for a

* independent

function of

wt+6t

family

=

{6X(t)

: t €

Y}

as in (5.3.21) or (4.3.6).

need not be integrable.

6X(t)

with

The process

a

Z

There is an extensive classical theory

of the characteristic functions of these processes which (4.3.4) hints at. Z

By breaking

or internally with

Z

up as follows (either measurably with

X)

Zb ( r ) = sum of the jumps of

Z bigger than

and

zb(r)

= z(r)

-

zb(r)

b

Section

7.7:

Semimartinaale Integrals

429

Zb

we can show that the characteristic function of so

Zb

that

is integrable.

is a hypermartingale.

Zb

We can also show that

If we let

bounded variation.

is smooth

c = EIZb(l)].

(We may

even

then

[Zb(r)-cr]

Zb

say

has

has bounded

, .

Ixlu(dx) < m for the classical lxl
variaiton i f and only if u

Z decomposes, Z = N+W b = Z (r)+cr; the discussion

would show why

W(r)

and

with is

N(r)

only

= Zb(r)-cr

intended as

background to justify the definition. A

more

modern

justification

to

single

out

the

semi-

martingales is the form of the Doob-Meyer decomposition theorem

Z(r) = Z(0)

that says a decent path submartingale decomposes,

+ W(r),

+ N(r)

N

where

is a local martingale and

W

is

increasing and previsible. We need not use the two decomposition results we just have stated in our development.

If

clearly don’t use.

M = N+W

write

martingale and stating

this

where

W

Now we will state another which we

M

N

is a local hypermartingale, we may is a locally square integrable hyper-

has locally bounded variation.

is

result

to

indicate

why

our

The point of

seemingly

more

general local martingale integrals are no better than a square integrable theory once we combine each with Stieltjes integrals. Let define

X path

be a

Gt-semimartingale lifting of

liftings

of

integrands

analogous to the one in (7.1.4).

relative

Z. to

We shall a

measure

430

Chapter

7: Stochastic InteEration

(7.7.6)NOTATION FOR SEHIHARTINGALE PATH HEASURES: Suppose that

X(t)

6t-semimartingale and as above.

= X(6t) {T~}

+ M(t)

is the

+ U(t)

is a decomposed

M

6t-reducing sequence for

The joint variation process of the decomposition pair

(M.U).

U VM(t,w) = [bM,6M](t,w)

+ 6Var U(t.w)

is finite a.s.. but is not necessarily

,

for

E

U

T b : VM(t.o)

do tend to infinity, [6M.6M](rm)).

lim

n or t

>

t.

bt-stopping times,

n],

for

n

E

*IN,

a.s. (as we see by examining n = U VM(un-6t) n. The unbounded joint

<

and make

variation path measure

>

Y6,

S-integrable for any

However. the internal increasing family of

u n (w) = min[t

t E

qw

of the pair

(M,U)

for

t E Tb

for

t E

is given by the

weight functions

T\T6

for

t € llb

for

t E

T\Hb

and

The internal measures q,[P]

infinite

with

qw

may very well be unbounded, or make

noninfinitesimal

probability.

This

Section

7.7:

431

Semimartinaale Integrals

complicates our use of lifting theorems, but may be technically

*series

remedied by a un.

of truncations with the stopping times

The bounded joint vartation path measure of the

decomposition pair

We know

<

p,(S)

(M.U)

=

p,(U)

is given by the internal series

1 [-2"1 ;{ 1 VM(on(w)} u

:

n E *N].

This measure is analogous to the path measure (7.1).

but the procedure for bounding

of the pair

(M,U)

Yo

measure

u

defined on

as

is the section of an internal is carried on

T

x R.

T6

x R

Y E U

E Ts : (t.,)

in

section

E Y})].

(7.1) ultimately

0

now translate into a.s. properties o f The next proposition shows how to

p,

The

that is,

u-liftings

71,

x R.

although we may consider i t

something about almost surely-lc -almost everywhere,

taking

is more

pa

by

u(Y) = E[p,({t

Just

into

We also define a bounded total variation measure

complicated.

where

q,

of section

p,

works.

7)"

told

us

u-liftings

-

the truncation procedure

ChaDter

432

(7.7.7)

PROPOSITION: Let

then

X

= M+U,

u[Y] = 0

The proof

e t c . b e as a b o u e .

the converse part

Y E Loeb(UxR).

of

w

this proposition

development of

local martingale

integrals in section

u-lifting of a standard process

and then apply the condition

(7.7.8) DEFINITION: Let

= 0 + M(t)

X(t)

+ U(t)

be a

6t-semimarttngale

( d e c o m p o s e d as a b o u e m i t h . 6 t - r e d u c i n g s e q u e n c e

M).

An internal process

G

is

{T~} for

2

(6M , 6 U ) - s u m m a b L e i f

(a)

G

(b)

except f o r a single nulL set o f

a’s,

whenever

is f i n i t e ( t h e u a r i a b l e s

and

t

t

is n o n a n t i c i p a t i n g a f t e r

ouer

is

The importance of the result is seen from the

(7.4). We may take a bounded

H

If

i f a n d o n l y i f f o r a l m o s t all

of

fairly technical. partial

7: Stochastic Integration

U6).

6t.

s

range

Section

433

7.7: Semimartineale Inteizrals

and there is an increasing sequence o f

{a,}

6t-stopping

times

such that

st E [ J Z ( IC(s,o)1216M(s,o)12

<

: 0

s

<

a,(o))

stlG(s,o)l 2dXw(s,o)]

<

I

m.

= E"J[.
Because of Proposition (7.7.7). the proof of (7.4.2) carries over almost intact to show the next result.

(7.7.9) PROPOSITION:

X

Let

X(6t)

2

0

internal

d-dimenstonal

decomposed as aboue.

for a

single

If

F

and

G

are

null

set

of

0's.

for

all

t,

ltF( t)6X( A

6t-sem mart tngale w i t h

2 (61 .6U)-surmable processes a n d

then except infinite

be a

convergence

in

t)

2

ltG( t)6X( t).

u-measure

results

like

(7.4.6) can now be proved for semimartingale sums.

(7.3.7)

and

Chapter

434

M

Since we have chosen

7:

Stochastic Integration

to be a stable

6t-martingale the

infinitesimal Stieltjes sums

have

6t-decent paths. The

peculiar

"integrability"

question

described

after

Example (7.7.1) is taken care of by the next result.

(7.7.10) THEOREM:

X1

Suppose that

= 0

+ M 1 + U1 and

X2

are both decomposed semimartingale liftings

X1

be a

martingale. has

H

If

(AM 2.AU)-summable projections of

of

XI.

is a

Let

At-semi-

is an almost-preuisible process which

(6M 2 ,6U)-summable

a

X2

6t-semimartingale while

= 0 + M2 + U2

u

G2,

u2-Lifting

Yl(s) = 2'

1-lifting

G16X

G1

and

a

then the decent path

and

Y2(t)

=

Zt G2AX

are

indistinguishable.

This

result

is

proved

using

(6.6.12)

-

(6.6.14)

as

described in the special case following Example (7.7.1). The whole standard theory of semimartingale integrals is summarized by

(7.7.11) THE STOCHASTICALLY INTEGRABLE LIFTING THEOREM: Let

X(t)

be a

d-dimensional

which is a lifting of the almost preuisibte process

6t-semimartingale

%-semimartingale

H

: C0.m)

x

R

+

Z(r). IR kxd

An

has a

7.7: Semimartingale Integrals

Section

435

n

0-predictable decomposition pathwise terms

X(t) = 0

Stiletjes

of

+ W(r).

(bML,dM)-summable

the

N =

+ M(t) + U(t)

integrals

projected and

with respect

<

for all

and there is a n increasing sequence o f {p,}

such that

p,

00

An almost-preuisible process

provided

decomposition

these

the

to

the

Z(r) = 0 + N(r)

satisfy:

~iIH(s.w)lIdW(s.o)I

-

below

for

if and only i f the

decomposition

W =

G

Lifting

a.s.

H

r

a.s.

9-stopping

times

while

is

called

conditions

a hold

Z-integrable

for

some

Z(r) - Z(0) = N(r) + W(r).

In this case the stochastic integral

is well-defined (up to indistinguishability) b y the decent path projection o f

436

AFTERWORD

This book might very well have been written in two 'volumes. Had

we

chosen

that

format, volume

two would

various fundamental applications of

have

contained

infinitesimal analysis to

processes arising in the study of stochastic integrals. than doing

this, we have only given the

Rather

'measure-theoretic'

foundations of this branch of stochastic analysis.

It is fair

to criticize our 'definition' of foundations taken by itself, this afterword points

out

pursue in the literature.

some

topics

so

that the reader might

We especially recommend the book by

Albevario. Fenstad. Hoegh-Krohn, and Lindstrom [1985] and the Memoir by Keisler [1984].

(A.l)

Stochastic Calculus

The

rules

for

manipulating

stochastic

integrals

are

different from the rules for manipulating classical integrals. Many of these rules can be proved by the three step procedure:

For

1)

Lift the terms to be manipulated.

2)

Manipulate the

3)

Project the rearranged terms.

example,

M

if

* finite

liftings.

N

and

are

d-dimensional

local

Bt-martingales we may apply transfer to ordinary summation to rearrange the inner product +

u
)

u=v

.

(M(t),N(t))

This gives:

into sums of the form

437

Afterword

,t-6t

,~-6t d ,

+z

6Mi(s)6Ni(s)

s=6t

Decent path projection yields the formula for (A.l.1)

INTEGRATION BY PARTS:

( % ( r ) , k ? ( r ) )=

E(s-)di(s)

We know that every

(I,%) has a local seen that

M(t)

2d-dimensional

+ [%,%](I-).

local hypermartingale

6t-martingale lifting (M.N). We have also 2 6N -summable lifting of the is a locally

left-limit process %(r-)

= S-lim M(t). tfr

This means that the decent path projections of both sides of the internal calculation yield the standard formula shown above.

A similar argument can be applied to prove that i f

fi

are one-dimensional local hypermartingales with

= 0

and if

is a locally

M(0)

and = N(0)

6M2-summable process, then the

quadratic variation "commutes" with integration.

Afterword

438

QUADRATIC VARIATION OF INTEGRALS:

(A.1.2)

The

first

martingale. integral. Stieltjes

integral The

stochastic

is a

second

integral

is

integral yielding a

a

pathwise

Stieltjes

The internal rearrangement of sums is clear and the summability condition needed

H

for

is proved

in

Hoover 8 Perkins [1983], Theorem 7.173. The most famous fundamental theorem of Stochastic Calculus is known as "Ito's Lemma." the

local result

It has various generalizations up to

for discontinuous

semimartingales.

Let

us

consider a special case of the classical formula.

and

Let

B(t.o)

let

f

denote Andersons's infinitesimal random walk

: IR + IR

function.

Whenever

be a twice continuously differentialbe

*IR

X E

is

finite

and

6x

is

infinitesimal, the uniform Taylor "small oh" formula says

f(x+dx)

-

f(x)

= f'(x)6x

where

L(x,~x)Z 0 .

makes

B(t)

1 f"(X)(SX)2 + 5

+

L(X,6X)(6X)

2,

This means that i f the random sample

S-continuous, s o

B(t)

is finite for finite

o t.

then

f(B(t+6t))

- f(B(t))

for an infinitesimal

= f'(B(t))aB(t)

r(B(t)).

the increments of the process

since X(t.w)

1 + Zf"(B(t))6t

[6B(t)lZ = f(B(t,w))

+ L(B(t))6t

= 6t.

We sum

to obtain the

439

Afterword

approximation:

=

1

Z

f(B(0))

t

X(t)

for all finite

6X

t.

+

Itf'(B)6B +

The term

$ If"(B)Bt

1 2 Jf'*(g(t))dt

pathwise integral

t

1 f"(B)6t

f

is a lifting of the

B

whenever

is

S-continuous.

We wish to show *

(A.1.3)

ITO'S LEMMA:

= f(g(0))

f(g(r))

+

or

df(g)

= f'(g)dE

+ $f"(g)dt.

In order to prove this we need to know that the term

lifts

Since

f'

and

the is

corresponding

standard

stochastic

integral.

B

is finite

S-continuous at finite points and

f'(B)

S-continuous a.s., we only need to show that

locally

2

6B -summable.

T

Itf'(B)6B

is

The reducing sequence

m (a) = m A min[t

:

IB(t)l

>

RI]

does the trick. In general i f

M

is an

S-continuous

Bt-martingale, the

A

same sort of argument produces the Ito formula for reader may

find additional details of

Anderson [1976].

M.

The

the classical case in

Afterword

440

(A.2)

Stochastic Integral Equations

A classical stochastic differential equation is something of the form dX(t)

X.

in the unknown process Brownian + IRd

@

f

motion, Rd

vector

dB).

: C0.m)

g(t,b)

(so

+ g(t,X(t))dB

= f(t,X(t))dt

Since

no

B

where x

is a

IRd + IR

is a

and

d x d

classical

g

d-dimensional d : c0.m)

R

x

matrix acting on the

meaning

is given

to

the

unintegrated differentials, an integral equation with an initial condition

Xo(w)

is actually intended.

Ito showed that these

equations have solutions under Lipschitz assumptions on

f

and

A

g.

"Generalizations"

of

existence

"weak

of

Ito's

theorem

solutions," meaning

sometimes

that

there

assert

is a

new

probability space and a new Brownian motion so that a process exists on the new space. very

general

"strong"

X

Keisler [1984] proves the following

existence

theorem.

The

fact

that

X

exists as a process on the original hyperfinite evolution scheme is a nice feature of

this scheme (also see Hoover & Perkins

[1983] and Hoover & Keisler [1982]).

(A.2.1)

THEOREM: Suppose f

:

[O,l] x IRd + IRd ,

g

:

[O.l] x IRd

+

IRd

@

IRd

are bounded Lebesgue measurabLe functions such that

441

Afterword

[det g]-2

is b o u n d e d .

N

B(r,o)

Let

%.

to

be a

d-dimensional

Then for each

Xo(o).

Brownian motion adapted

I(0)-measurable

initial condition

the equation

X(r,o) = Xo(o)

+

+

f(s,X(s.o))ds

g(s,X(s.w))dg(s,o)

h a s a s o l u t i o n o n our h y p e r f i n i t e s c h e m e .

PROOF : See Keisler [1984]. Theorem 5.5. Keisler [1984], Theorem 5.2, also shows very simply that i f f

and

are bounded measurable functions, continuous in the

g

X-variable equation scheme.

(but not needing above

The

has

X

solution

C0.m)-version

[1982] as Theorem 9. past of

a

[det g]-2 on

bounded),

the

original

then

the

hyperfinite

of this result is given in Perkins

Cutland [1980] allows

g

to

depend on the

and further explores applications to control theory

in Cutland C1982, 1983a1. &

Hoover

Perkins

[1983],

Theorem

10.3. show

that

the

stochastic equation

X(r,w)

has

a

strong

Z(0)

= 0.

H

= H(r.w)

+

J:

solution where is an

f(s.o.X(*.o))dZ(s.o)

Z

is

a

semimartingale with

%-adapted decent path process and

f

is a

Afterword

442

previsibly measurable function of the past of the paths in the third variable. The starting point for all these existence proofs

is to

l i f t the functions and known processes, replace the integrals by

* finite This

sums against

"lifted"

equation

solution, e.g.,

X(t)

once X(t),

infinitesimal differences

X(t+Gt)

is known.

f(t.X(t)).

has c

an

inductively

such as

defined

6B.

internal

+ g(t.X(t))aB(t).

X(t) + f(t,X(t))6t

The difficulties involve showing that

g(t,X(t))

are

each

liftings

of

some

classical process . . .

(A.3)

Harkov Processes One omission from this book closely related to the last

section is the study of probabilistic properties of solutions of integral equations.

f

when

and

associated Markov

g

Keisler [1984].

Theorem 6.11. shows that

of (A.2.1) are bounded and continuous, then the

integral equation has a path

solution.

The

internal

continuous

solution

of

the

strongly"lifted"

infinitesimal difference equation must satisfy a property that is not the transform of the strong-Markov property, but rather a property

that

makes

Keisler gives the

its

standard

part

have

that

property.

S-notions of the Markov property, the strong-

Markov property and the Feller property.

He also shows when

various integral equations have solutions with these properties. Markovian properties of stochastic processes are certainly fundamental. but we refer our reader to Keisler's excellent memoir for the corresponding internal notions.

443

Afterword

(A.4) Reshuffling Keisler [1984] proves his existence theorems for Brownian motions on a hyperfinite evolution scheme by first proving them for Anderson's infinitesimal random walk and then appealing to the following "homogeniety property'' of this scheme.

Y

are

continuous

Markov

processes

with

the

If same

X

and

finite

dimensional distributions, then there is an internal bijection which reshuffles the sample space, respects the progressively measurable filtration and almost surely maps

X

onto

saw a faint shadow i f this result in Chapter 4.

Y.

(We

The result is

easily extended to decent path Markov processes, but Hoover & Keisler [1983] have gone far beyond this.)

(A.5)

Universality Keisler r1984-j also proves a "universal" property of the

hyperfinite evolution scheme.

If a stochastic integral equation

has a solution on any space then we may replace the Brownian motion with one on our hyperfinite space and obtain a solution with

respect

to

that

Brownian

motion

on

our

same

space.

moreover, with the same finite dimensional distributions as the original solution.

He proves that for any continuous process on

any space there is a continuous progressively measurable process

on the hyperf ini te

scheme with

the

same

finite

dimensional

distributions as the original. Hoover

8

Keisler

[1982]

generalization of this result.

contains

an

important

They give a logic of "adapted

distribution" which has common fini te dimensional distributions as its zeroth step.

Two Markov processes have the same adapted

Afterword

444

distribution dimensional

if

and

only

distributions, but

have

adapted

same

distributions

The hyperfinite evolution scheme is universal (and full

theory

of

for

adapted

more

measure

processes.

the

filtration

finite

of

for

progressive

the

properties

"saturated")

the

they

if

general

distribution.

Sections 6 and 7 of their paper give a precise meaning to our earlier claim that there is no loss of generality in studying semimartigale integration on our hyperfinite scheme. Works

of

Keisler, Hoover.

Rodenhausen

and

others

study

"probability logic" in greater detail.

(A.6)

Local Time Perkins [1981b] gives impressive results on Brownian local

time.

All of

This is also partly explained in Perkins [1983].

Perkins' articles in the references are worthy of study. of them involve key uses of infinitesimal analysis.

Most

They are

beyond the scope of this book.

(A.7)

Applications The works

[1982,3]

of

Helms

[1982],

are nice uses of

infinitesimal

aimed at scientific applications mathematics to mathematics).

Arkeryd

(rather

[19Sl-] analysis

and

Cutland

in problems

than applications of

Lawler [1980]

has an interesting

treatment of loop-erased random walks that may turn out to have applications.

Kosciuk

[1982]

applicable use of infinitesimals.

gives

another

potentially

The most ambitious discussion

of applied infinitesimal stochastic analysis is contained in the forthcoming

book

of

Albeverio.

Fenstad.

Hoegh-Krohn

and

Afterword

445

Lindstrom

[1985].

physically analysis.

oriented It

remarkably

Their

is

book

treats

applications

of

essentially

little with

a

broad

selection

infinitesimal

self-contained,

this book.

Since

it

of

stochastic

but

overlaps

summarizes

the

research of its authors, we have not mentioned that separately.

(A.8)

Other Infinitesimal Methods The recommended reading above does not adhere closely to

the hyperfinite framework we have chosen.

However, most of i t

is fairly close to our treatment in spirit at least and should Lawler [1980] is written in the

be easy to move into from here. framework of Ne son’s [1977] read

a

little

more

Internal Set Theory, s o i t must be

carefully

[19Sl]

because

Henson and

Wattenberg

problem

infinitesimal measure

of

[l966] book, but

their methods

framework we consider. one.

solve an

of

that

restriction.

interesting

theory raised essentially

technical

in Robinson’s

fall outside

the

Our framework is not the only possible

We do think that i t is a coherent highly promising one for

applciation to

mathematics and science.

Naturally, we also

think i t is interesting in its own right. So now. dear reader, we wish you farewell and good hunting.

We hope these methods help you find solutions to problems that interest you.

References

446

Sergio Albeverio. Jens Erik Fenstad, Raphael Hoegh-Krohn and Tom Linds trom

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,.

[1982] Star-Finite Representation of Measure Amer. Math. SOC., 271. pp. 667-687. Robert M. Anderson & Salim Rashid [1978] A Nonstandard Characterization of Proc. Amer. Math. SOC.69, pp. 327-332.

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Weak

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L. Arkeryd

[1981a] A Nonstandard Approach to the Boltzmann Equation, Arch. Rational Mech. Anal., 77. pp. 1-10. [1981b] A Time-wise Approximated Boltzman Equation, I M A Journal of Applied Math., 27, pp. 373-383. [1981c] Intermolecular Forces of Infinite Range and the Boltzman Equation, Arch. Rational Mech. Anal., 77. pp. 11-21. [1982] Asymptotic Behavior of the Boltzman Equation with Infinite Range Forces, Comm. Math. Phys.. 86, pp. 475-484. [1984] Loeb Sobolev Spaces with Applications Integrals and Differential Equations, preprint.

to Variational

Bernd J. Arnold [ 19821 "Die Verwendung uon Infinitesimalien in der Theorie Der Brownschen Bewegung," Diplomarbeit, Technische Hochschule Darmstadt.

Jon Barwise (editor) [1977] "Handbook o f MathematicaL Logic." North-Holland, Studies in Logic, vol. 90. Amsterdam. Patrick Billingsley [ 19681 "convergence Sons, New York.

of

Probability

Measures, "

John

Wi ley

&

Leo Brieman [l968] "Probability," Addison-Wesley, Reading. Rafael V. Chacon. Yves Le Jan and S. James Taylor [1982] Generalized Arc Length for Brownian Motion and Processes, preprint with appendix by Edwin Perkins.

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C. C. Chang & H. Jerome Keisler [1977] "Model Theory," North-Holland Studies in Logic & Found. of Math., 73, Amsterdam. Nigel J. Cutland [1980] On the Existence of Strong Solutions to Stochastic Differential Equations on Loeb Spaces, 2 . Wahrsch. Verw. Gebiete. 60. pp. 335-357. [19Sl] Internal Controls and Relaxed Math. SOC.. 27. pp. 130-140.

Controls,

J.

London

[1982] Optimal Controls for Partially Observed Stochastic An Infinitesimal Approach, Stochastics, 8. Sys tems : pp.239-257. [1983] Nonstandard Measure Theory and Its Applications, Bull. o f London Math. SOC.. 15, pp.529-589. Martin Davis [1977] "Applied Nonstandard Anaylsis," Wiley-Interscience, Pure and Appl. Math. Series, New York. Claude Dellacherie and Paul-Andr6 Meyer [ 19781 "Probabilities and Po tent ial , " Studies, Amsterdam.

Nor th-Hol land,

Math.

J. L. Doob [1953] "Stochastic Processes," John Wiley York.

&

Sons, Inc., New

Nelson Dunford and Jacob T. Schwartz [1958]

"Linear Operators Part I." Interscience. New York.

Stewart N. Ethier and Thomas G. Kurtz [1980] "larkou Process: Existence and Approximation." Preliminary version of a book. William Feller [l968] "An Introduction to Probability and Its Applications, Vol. I." Third Ed., John Wiley & Sons, Series in Prob. & Math. Stat., New York.

I. I. Gihman and A. V. Skorohod [1974] "The Theory o f Stochastic Processes," Springer-Verlag. Grundlehren band 210. New York.

B. V. Gnedenko and A. N. Kolmogorov [196S] "Ltmit Distributions f o r Sums o f Independent Variables,'' revised ed.. Addison-Wesley, Reading.

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L. L. Helms and P. A. Loeb [I9791 Applications of Nonstandard Analysis to Spin Models. 3 . Math. AnaL. AppL. 69. pp. 341-352. [1982] A Nonstandard Proof of the Martingale Convergence Theorem, Rocky Mountatn 3. Math. 12. pp. 165-170. [1982b] Bounds on the Oscillation of Spin Systems, J. Math. Anal. Appl.. 86, pp. 493-502. C. Ward. Henson [1979a] Unbounded Loeb Measures, Proc. Amer. Math. SOC. 74, pp. 143-150. [1979b] Analytic Sets, Baire Sets and the Standard Part Map, Canad. 3. Math. XXXI, pp. 663-672. C. Ward Henson & L. C . Moore, Jr. [1974] Nonstandard Hulls of the Classical Banach Spaces, Duke Math. 3. 4 1 , pp. 277-284.

C. Ward Henson 8 Frank Wattenberg [198l] Egoroff's Theorem and the Distribution of Standard Points in a Nonstandard Model, Proc. Amer. Math; SOC. 81. pp. 455-461. Edwin Hewitt & Karl Stromberg [1965] "Real York.

and

Abstract

AnaLysts."

Springer-Verlag.

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A Normal Form Theorem

preprint.

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Lw,p, with

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Adapted Probability Distributions, preprint.

Douglas N. Hoover & Edwin Perkins

[1983] Nonstandard Construction of the Stochastic Integral and Applications to Stochastic Differential Equations I and 11. Trans. Amer. Math. SOC.. 275, pp. 1-36 & 37-58. A. E. Hurd [198l] Nonstandard Analysis and Lattice Statistical Mechanics: A Variational Principle, Trans. Amer. Math. S O C . , 263, pp. 89-110.

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[1983] "Nonstandard Analysis Recent Deuelopments," Springer Verlag Lecture Notes in Math., 983, New York. Nobuyuki Ikeda & Shinzo Watanabe [I9811 "Stochastic Differential Equations and Diffusion Processes." North-Holland Math. Library, vol. 24, Amsterdam. H. Jerome Keisler

[1976]

"Foundations

of

Infinitesimal Calculus."

Prindle. Weber

& Schmidt, Boston.

[1977] Hyperfinite model theory, in the volume "Logic Colloquium 7 6 . pp. 5-110. North-Holland, Amsterdam. C19841 An Infinitesimal Approach to Stochastic Analysis, Hem. Amer. Math. SOC.. vol. 48. nr. 297.

S. A. Kosciuk [1982] Nonstandard Methods in Diffusion Theory, Ph. D. Thesis. University of Wisc.-Madison. [1983] Stochastic Solutions to Partial Differential Equations. in the vol.. Nonstandard Analysis - Recent Developments, Hurd [1983] above.

K. Kunen [1979]

Personal communication.

K. Kuratowski [l966] "Topology I & 11." Academic Press. New York.

A. U. Kussmaul [ 19771 "Stochas t ic Integra t ton and General tzed Mart tngales , " Pitman Publishing, Research Notes in Math., London. Gregory F. Lawler

[1980] A Self-Avoiding Random Walk, .Duke Math. J . 655-693.

47(3).

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[1980a] Hyperfinite Stochastic Integration I. 11. 1 1 1 , and Addendum, Math. Scand. 46, pp. 265-292, pp. 293-314. pp. 315-331, pp. 332-333.

[1982] A Loeb-Measure Approach to Theorems by Prohorov, Sazonov and Gross, Trans. Amer. Math. S O C . , 269, pp. 521-534. [1983] Stochastic Integration in Hyperfinite Dimensional Linear Spaces. in the vol., Nonstandard AnaLysis - Recent Deuelopments, Hurd [1983] above. [1980d] The Structure of Hyperfinite preprint series, Univ. of Oslo.

Stochastic

Integrals,

Peter A. Loeb

[1972]

A Non-Standard Representation of Measurable Spaces. L,,

*

and L,.

in the vol. "Contributions to Non-Standard

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[1975] Conversion from Nonstandard to Standard Measure Spaces and Applications in Probability Theory, Trans. Amer. Math. S O C . 211, pp. 113-122. [1979a] Weak Limits of Measures and the Standard Part Map. Proc. Amer. Math. SOC. 77, pp. 128-135. [1979b] An Introduction to Nonstandard Analysis and Hyperfinite Probability Theory. in the vol. "Probabilistic. Analysis and Related Topics 2 , " Bharucha-Reid (editor). Academic Press. New York. M. Loeve

[1977]

"Probability Theory I & 11." Springer-Verlag. New York.

W. A. J. Luxemburg [1973] What is Nonstandard Analysis?, in a supplement to Amer. Math. Monthly 80, pp. 38-67. Michel Metivier & J. Pellaumail "Stochastic Integration." Academic Press series [1980] Probability and Mathematical Statistics, New York.

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[1982a] On the Construction and Distribution of a Local Martingale of Given Absolute Value, Trans. Amer. Math. SOC., 271, pp.261-281.

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APPENDIX

A PRIMER OF INFINITESIMAL ANALYSIS

This Appendix is a collection of results that are useful in various parts of the book. They are given here in detail to serve to

as a further help at learning to apply the logical principles Analysis.

(APP.l.l) PROPOSITION: A

*natural

n €ON

,

n € *N

number

,

o r unlimited

n > m

is

either

for every

m E

standard,

ON

.

PROOF:

We

offer a

Principle; below, Take any limited

proof as an

direct

(APP.1.2)(4),

n

*N

,

Leibniz'

we suggest a different

i.e., a *natural number n

for some (standard) m E N'

n < m

application of

.

proof.

such that

Apply Leibniz' Principle to

the bounded sentence N, ( x < m

vx and 1,

conclude

...

,

m

imply

that

n

x

or

= 0

...

or

x = m )

equals one of the standard

numbers

0,

.

(APP.1.2) REMARK:

*

N

"looks

times copies of

like" N

followed by a dense

Z

,

n

belong to

and

0

ordinal

0

is uncountable.

Specifica11y : (1) If

m = n

m

,

-m

and

- n(

Im

< 1

,

(apply Leibniz' Principle to the 'same' statement in (2) If n € *N

n

*N

are also in

is unlimited and

*

N

m € N'

and unlimited.

,

then

n + m

Therefore, around

then N ). and each

Appendix

454

N

n E

unlimited

(m+n)/2

.

N

m,n E *N

(3) If

Z embedded into the

there is a whole copy of

unlimited part of

are unlimited, then the *integer part of

is also unlimited. Hence, between two disjoint copies of

(4) Once we have shown (l), (APP.l.l)

you can give another proof

using the standard part map: r = st n ;

limited, call

In

- [r]l

and

therefore, n

=

-

In

5

,

nr

,

[nr]

1

,

-

and

n

is

because it is the sum of an smaller

than

1

r , s E ‘(0,l) are standard

,

nr

at an infinite distance appart

ns = nr(ns-r

-

1)

.

ns

of

are each

Hence, the hypernatural

Z

[ nS] lie on two different copies of

.

This

proves that there are uncountably such copies.

(APP.1.3) PROPOSITION: (a) The

set

of

standard *natural

numbers

‘N

is

\ UN

is

external.

(b)

*N

The set of unlimited *natural numbers

external. (c) and

‘X

‘JX c *X (d)

For any set =

*X

,

or

,

X E

X

‘X

is external

either

X

is a finite set

and

the

inclusion

,

of

unlimited

is strict. The

;

. then the hyperreal numbers

(why?), and

other, because

N

of

+ Ir - [ r l l ,

is unlimited and

r < s

real numbers,

rl

standard real number

[ r ] E ‘N

n E *N

(5) If

unlimited

a

n E

if

*

then

which is strictly smaller than infinitesimal

.

Z

Z there is also another (disjoint) copy of

numbers

1

sets of limited scalars

scalars, of infinitesimals

0

,

0

the map

st

,

and

the

1

Appendix

455

,

=

relation

are all external entities.

PROOF : claim that for a subset of an internal set

First, we the

property

of

P(V) :

assume

e

for

*X P sentence

T,V

being internal is equivalent to

,

T

v 6 X Vt6Xp, P' to obtain

hence

*

xP' T

if

internal

apply

teP(v)

xp,

V t e

are

p ;

some

t/

tlvE

V

*

t6

Leibniz'

iff

P(V)

Principle

to then

to

the

( V x ~ t , x e v ) ,

( V x e t, x e v ) ,

iff

T E

is internal, then

belonging T C V ;

and

,

V

*

P(V)

(the

converse

is

obvious). Proofs it

is

of 'externality' are best handled by

contradiction:

usually convenient to show that a set does not have

some

property which is known to hold (by an appropiate application

of

Leibniz' Principle) for internal sets. N :

(a) Consider the following true statement in

E P(N),

V T

*

Its is

*

*

( T is bounded in N)

transform says that if

T

implies

(T has a maximum)

is an internal subset of

*

bounded, i.e., bounded by some member of

maximum,

i.e.,

a

N

*

.

N and

, then it has a

maximum (writte down the whole sentences

in

detail if you do not feel sure about the last assertions). Now regard then

it

would

bounded.

But

member of N '

U

N

have a maximum then

,

N ;

as a subset of

m+l

m

,

if it were

because

it

is

would also be limited, and

internal, certainly hence

a

by (APP.l.l).

(b) The reader can work out a proof similar t o the last one,

by considering the statement W

T & P(N), T # 0

implies

T has a mimimum.

Alternately, sentences, always

one

finite

t r a n s f o r m of a p p r o p i a t e obvious

then

to

X

e

is a f i n i t e set.

X

X = {xl,..,xn)

Now,

E

X

be

X

saturation

infinite,

A c

i n c l u d e d as

property

[A : A

c U x&

i n t e r s e c t i o n , then

x

that

xn ) * ‘X is

is also internal, s o a

of

part

Leibniz’

the s e t

i s a member

x

if

must belong t o ‘X \ {XI

is external,

‘X

*

\ A is finite]

‘X

and t h i s i s absurd:

n o t empty;

‘X

x =

or

and assume

P r i n c i p l e ( s e e ( 0 . 2 . 3 ) ( b ) and ( 0 . 4 . 2 ) ) ,

n

of

given

(APP.l.l)

...

( x = *xl or

then every f i n i t e subset

Hence

external,

f o r some p , t h e n p 2 1 and E X P ’ 9 and by L e i b n i z ’ P r i n c i p l e ,

iff

X E * X

9

let

internal; the

*x _c *xp-l

so

I

be

sets

The f o l l o w i n g proof

i s a g e n e r a l i z a t i o n o f t h e proof of

‘X

x 5 xp-l * vx xp-l

is

has

*N\‘N

internal

would be i n t e r n a l .

‘N

above. I f

by

*

by

boolean o p e r a t i o n s w i t h

internal s e t s ;

( c ) Assume =

can show,

that

give

otherwise

*X

1

Appendix

456

of

this

.

and t h e r e f o r e t h e i n c l u s i o n

U

X c *X

(See a

of an e x t e r n a l s e t i n t o an i n t e r n a l s e t h a s t o be s t r i c t .

h i n t f o r a more d i r e c t proof o f t h i s f a c t i n E x e r c i s e (0.4.5).) (d)

If

internal; -1

x-x (0 \

0

were i n t e r n a l ,

then

u

a similar argument works f o r

N =

*

N

R\ 0

n 0 would a l s o be

.

i s i n t e r n a l ( i n d e e d , even s t a n d a r d ) , i f

{0}

would be i n t e r n a l and) t h e n

*

R \ 0

Since t h e o

map

were i n t e r n a l

would be i n t e r n a l as

well.

i s n o t d i f f i c u l t t o s e e t h a t f o r a map t o be i n t e r n a l i t

It is

necessary

(apply

0 --->

the R

that

both i t s domain and i t s

Internal Definition Principle),

cannot be i n t e r n a l .

range

s o the

be

internal

map

Again by t h e same p r i n c i p l e ,

st if

were i n t e r n a l , t h e s e t o f i n f i n i t e s i m a l s would be i n t e r n a l t o o :

: FJ

1

Appendix

In

451

the

elementary the

rest

of this Appendix,

*

properties of

R

we are to

introduce

that appear very often

some

throughout

book and have in common that they are translations to

this

setting of analogous standard topological properties, but keeping the standard tolerances, whence the

prefix.

IS-!

(AF'P.1.4) DEFINITION:

A hyperreal function

-a

*R

in

if

x,a f dom(f)

-a c

We

is said to be S-continuous implies

f

x E D

if

f

(D C - dom(f)

x = a

is defined for such x

a

relative to

D

f(a)

at

(and

is S-continuous

and

is S-continuous relative

continuous at

=

f(x)

in particular). We say

(and f(x)

say

a

s

5 D c- *R

relative f(a)

x

f

imply

f(x)

in particular).

9 g if f

is

S-

each

in

D

for

a

in particular).

for hyperreal

Similar definitions apply defined

on

subsets of arbitrary metric

case,

x

a

a

means

that

the

spaces

*distance

functions (in this

d(x,a)

is

infinitesimal).

(APP.1.5) PROPOSITION:

Let

f

an internal set. if and

D -C dorn(f)

be an internal function and Then

f

is S-continuous relative to

D

aR+

,

only i f for every standard positive

there exists a standard positive for all x

,

y

in D

Ix-yl < e

be

0

in

OR+

in

E

,

such

, implies

If(x)-f(y)l

<

E

.

that

1

Appendix

458

PROOF: If

f

is S-continuous and

~ ( € 1= { e

'R+

in

6

E*R+ : x,y E D & ~x-yt <

e

is fixed, the set

is internal by the Internal Definition Principle.

.

T(E) 2 o "*R+ = o + noninfinitesimal standard

in

8

.

6

every standard positive standard positive

E

T(E)

This proves the

x,y € D

if

is in

8

.

,

Hence,

By hypothesis,

,

so there is a

condition.

E-8

x = y

and

I

E

~ ( € 1contains a

is external,

8 < 6

Any

.

T(6)

Conversely,

o+

Since

<

imply If(x)-f(y)I

Ix-yl < 8

then

<

If(x)-f(y)l

E

for

for every

.

, and f(x) = f(y)

(APP.1.6) COROLLARY: Let internal each

x

f

be an internal function and

set of limited points. in

D

and if

f

If

f(x)

D c 0

is limited

an for

is S-continuous relative to

A

then

be

,

D

, . A

f :D/=----.. 'O/=

st f(x)

for

is uniformly continuous, where

x € st-'(;)

is the infinitesimal

hull

f(x) = of

f.

PROOF : A

The

map

,

st-'(;)

by

f

is

well-defined

S-continuity.

disturbed by changing

<

to

The 5

E

on

-

0

equivalence

classes,

condition is

at most

by taking standard parts.

A

proves that

f

is

(APP.1.7) REMARK: The

function

external: consider

E-0

continuous uniformly on

D

That

.

. f

may not be uniformly continuous if

f(x) = l/x

on

0\ o

D

is

.

(APP.1.8) DEFINITION: (a) Let

f be a hyperreal function and let

D

5 R

1

Appendix

be U

459

s u b s e t of i t s domain t h a t c o n t a i n s a

a

(r,s)

,

.

r , s € ‘R

with

the right S - l i m i t

Then we s a y t h a t

of 2

interval

b € ‘R

is

g ,

within

,

f(x) = b

S-lim

real

xs r XED if

for

every s t a n d a r d p o s i t i v e

e ,

positive x 6 D

standard

a

is

such t h a t

and

Similarly,

there

E

r << x

r+e

<

implies

left

we d e f i n e t h e

-

If(x)

S-limit

b( <

E

.

f ( x ) and

S-lim

the

x+s x€D (bilateral)

S-limit

f(x)

S-lim

(for

.

t € ‘(r,s))

x+ t x6D Finally,

we w i l l drop any r e f e r e n c e t o

t h e whole domain o f

r e a l numbers and xn

is

.

(xn:n € ‘N)

(b) I f

of

f

whenever i t

D

b € ‘R

i s an e x t e r n a l sequence of hyper-

,

we s a s y t h a t

is t h e S - l i m i t

b

Y

b = S-lim x n+m n ’ if

f o r every s t a n d a r d p o s i t i v e

t h e r e is a standard

E

nE

such t h a t n € ‘N

and

n > n

we

say

that

And E

€ ‘R

t h e r e i s an

m,n € ‘N

and

nE E

imply (x,) U

m,n > n

N

is

Ixn

-

bl

S-Cauchy

c

E

.

for

if

every

such t h a t imply

Ixm

-

xnl <

D

an

E

.

(APP.1.9) PROPOSITION:

Let subset

f

be a n i n t e r n a l f u n c t i o n and

of i t s domain of d e f i n i t i o n such t h a t

D

internal U

3

(r,s)

.

Appendix

460

Then

in

order that the standard real number

righ't S-limit of

f

at

r

and sufficient the following:

, such that for all x

r

x = r

it is

the

necessary

x1 € D ,

there is an

x1

=

,

€ D

x > x1

and

D ,

within

be

b

1

.

f(x) = b

imply

PROOF: Assume

positive

that

x € D D

Call

=

E

,

=

is another

DE :

x € DE

F

can

take

-

If(x)

D,

bl <

.

E

imply

<

If(x)-bl

€1,

is external, that is why in order

all points y

the

Saturation

where the inequality

holds, regardless of whether they are finitely apart from

o r not). Then the family

has

the

finite

=

[F, :

intersection

smaller than that of

X

,

E

€ OR+]

property

therefore

and

cardinal

n [F,:E€'R+]

strictly

is not empty.

is easy to see that any member of this intersection is one of

those

x1

that we were looking for.

Conversely, suppose there is an x1 in the statement. Then, given any

{e is

standard

that we

of internal sets to apply

F

It

each

and

x > y

and <<

Principle, we include in

r

for

,

'R'

OE

r < < x < r+OE imply

and

build a family

r < y

Then

such that

(notice that the relation to

.

S-lim f(x) x+ r x€D

{y € D : r < y < r + 8 }

= {y €

F

there

E

smaller than

b

€ *R+: x € D

internal

and,

and by

E

satisfying the condition

E 'R'

,

the set

x1 < x < r+O imply hypothesis, contains

finitesimals, which is external; take as

8,

If(x)-bl the

set

<

€1 of

in-

the standard part

of one of the noninfinitesimal members of this set.

2

Appendix

461

(APP.l.lO) DEFINITION: A set

-A

D

c_

st(D) 2 A

if

there

exists

a :

x = a

that

D

,

an

.

,

*A

with

A 5 Rd

, is called S-dense j . ~

i.e., if f o r every standard

x

in

When

D

which is infinitely

is the whole space

A

a

in

close

,

A to

we

say

subsets

of

Rd

is S-dense.

The

same

definitions

make

sense

for

arbitrary metric spaces.

(APP.l.ll) REMARK: The The

Lemma (2.1.1) might have gone here in our

standard

development.

part of an internal set is always closed, so

dense" means roughly that the closure is everything. so

important in section (2.1),

'IS-

Its role is

that we didn't want experts

who

are not reading this Appendix to miss it.

(APP.1.12) EXAMPLE: A n example

of an internal S-dense subset of

*H2

is

2 T2 = {(x,y) € R2 : 3 h,kE*Z, -n Sh,kSn2 & x=h/n,y=k/n]

where

is an infinite *natural

n

*finite, it has

S

If

(2n2+1)2

Principle

sequence

s(.)

for

k>l

and

(verify this). If

I

*

Since

=

D

fact,

T2

is

0

,

R ,

we may

use

S = { s k :lSkSn} for some

sl<sk<sk+l<sn

a(sl)

*

finite subset of

to write

with

In

elements.

is an S-dense,

Leibniz'

number.

.

then

is S-dense,

is a standard finite interval, length(1) = ~ [ ~ ( :s s )

a(sk) = ~

If we let is an

s1

internal

~

-

function

is negative infinite.

(a,b]

es n

internal

,

*I]

for example, then

.

s

~

-

Appendix

462

(APP.1.13)

EXERCISE:

Use length

1

t h e t e l e s c o p i n g sum p r o p e r t y t o v e r i f y t h e formula above.

We can u s e

approximate

t o b u i l d Lebesgue measure,

CY

s e e Chapter 2. Verify

t h e comments above about an S-dense

finite

set

S

and prove t h e c o n v e r s e remark as f o l l o w s . I f S = {sk : k E

is a

*

{sk}

N,

l<=k<=n)

f i n i t e s e t r e p r e s e n t e d by t h e i n t e r n a l i n c r e a s i n g

sequence

is negative i n f i n i t e ,

positive

and

if

s1

i n f i n i t e s i m a l when then

*

S

sk

i s S-dense i n

i s f i n i t e and R

sn

sk-skdl i s is positive

infinite,

.

s i m i l a r n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r a

Give

S = {sk} t o be S-dense i n

Our

*[0,1]

set

.

f i n a l b a s i c example t o i l l u s t r a t e t h e i n t e r p l a y between

i n t e r n a l and e x t e r n a l c o n c e p t s can be used t o c o n s t r u c t measure on

(APP.1.14) If

w i t h a * f i n i t e measure ( s e e Chapter 2 ) .

Rd

PROPOSITION: T

i s an S-dense * f i n i t e s u b s e t of *Rd

t h e r e i s an i n t e r n a l f u n c t i o n I c Rd

Lebesgue

is

{xERd : a j 5

a bounded XJ

[O,ll

a:T---->

d-rectangle

(for

,

then

such t h a t i f

example,

I

=

5 b J , lSjSd}), then

d-voi(I) =

l[a(t)

: t

E

T

n *I]

.

PROOF:

For f i n i t e *Rd

rn

in

U

N

g i v e n by t h e r e c t a n g l e s

c o n s i d e r t h e p a r t i a l paving

P(m)

of

1

Appendix

463

for

k

E

S-dense,

set

for

E

s(k,m)

such t h a t

*Zd

each

n *I(k)

T

-in2

.

Let

of t h ese s e l e c t i o n s .

d

d k +1

.

Since

w e may s e l e c t a

single

12jsd

= {s(k,m) :

s(m) When

k

,

2 kJ < m

I ( k ) € P(m)

[F,->

...

[ k: , ky l + l )X

I(k) =

is finite,

m

point

lkJl 6 m 21

s o is

be t h e

S(m)

t h e r e f o r e i t i s i n t e r n a l ( i t i s d e s c r i b e d by t h e s t a t e m e n t o r x=s

. .. o r

or

and

"x=s 1

x=snll).

The p r o c e s s o f forming t h e paving

P(m)

is i n t e r n a l ,

since

i t s d e s c r i p t i o n can be i n t e r n a l l y f o r m a l i z e d . The s t a t e m e n t

"there

an

is

internal subset

S c T

e x a c t l y one p o i n t o f each r e c t a n g l e o f formalizable:

s E

is external,

unique

s

The Is(x)

t h e r e i s an i n f i n i t e

in

F i x such an

=

S

such t h a t

P(m)"

contains

S

is a l s o i n t e r n a l l y

31s

e sn

n

E *Zd

k

.

I) U

N

.

Since

and an i n t e r n a l

,

kJ 2 n 2

with

,

k j / n 5 sJ < ( k J + l ) / n

0

N

selection there is a

for

lSj4d

.

f o r t h e r e s t o f t h e argument.

S

indicator 1

so that

i s i n t e r n a l and c o n t a i n s

such t h a t f o r e v e r y

S C_ T

E p(m))(

*P(T))(v I

q(m) 1

N :

{m

,

q(m)

q(m) =

(3

Thus

is

T

if

function

x E S

,

of

S

,

Is(x) = 0

is i n t e r n a l provided we

x 6 S

if

restrict

, its

domain t o an i n t e r n a l s e t :

Is = { ( x , y ) E T x { O , l } : y = l i f xES We t a k e o u r w e i g h t i n g f u n c t i o n

Now l e t for

16jsd

I

.

c1

= Is/"

& d

.

be a bounded d - r e c t a n g l e i n

Let

n .(x) = x j J

.

y=O i f xefSl

Rd

,

denote p r o j e c t i o n

say

lxJl 2 B

on

the

jth

2

2Bn

by

c o o r d i n a t e and l e t

the

*

E *Z : k / n € I T ~ ( * I ) },] p . = '[{k J c a r d i n a l i t y o f t h a t s e t . W e know t h a t pj

1

Appendix

464

transfer, so For

pj/n

lSjSd

is finite.

we know : [ k/n,(k+l)/n)

€ *Z

p.-1 6 '[{k J

nj(*I)l]

and

E

'[{k

*

hence the

*Z

: [ k/n,(k+l)/n)

number of little boxes inside d

j=1

*

I

,

s

,

I

,

is bounded

J

is bounded above by

.

II (pj+l)

j=1

between

r

-number of little boxes that touch d

,

,

6 pj+l

r

II (p.-1) while the

below by

n nj(*I) # 011 *

these

is

a

finite sum of products

The of

difference

less

than

d

Each

pj/n

is

d

factors pj finite

(since the

and

(s-r)/nd = By

o

and

is infinitesimal,

therefore

the

difference

.

the *finite additivity of volume and monotony,

*(d-vol)(*I)

*

l/n

II p.-term cancels out). j=1 J

5 s/nd

*

.

Since we also know

(d-vol)( I) = d-vol(1) d-voi(I)

,

r/nd

r 5 # [ S n *I] 5 s

we have our result,

ZIa(t) : t E T n *I]

.

2

,

& ULTRAPOWERS THAT ARE ENLARGEMENTS

APPENDIX

The

goal of this Appendix is to construct a special kind of

nonstandard models, in

the so-called enlargements,

that are needed

Section (0.4) as a first step in the process of defining

superstructure extensions (0.4.2),

(0.4.3)

that, we define

for which

the

the

important principles

and (0.4.4) are shown to hold.

In order to do

the ultrapower of a superstructure with respect

an ultrafilter, and then we show that for suitable ultra-

to

filters, this construction produces enlargements.

(APP.2.1) DEFINITION: J

Let U

5

be a nonempty set. A family of subsets of

U,V E U

(2) if

,

is called an ultrafilter when

P(J)

(1) if

J

€ U

U

,

then and

U n V EU V

,

satisfies U

V

C_

J

,

then

u ,

V € (3) for

every

subset

V ' SJ ,

complement J \ V € U ,

either

V € U

or

its

and not both.

(APP.2.2) REMARK: Nonempty (1)

and

prove

(2)

families of nonempty subsets of above are called filters.

J

It is not

that

satisfy

difficult

to

that 'ultrafilter' is the same as 'maximal filter' for the

relation

of inclusion, C

.

A standard application of

Zorn's

Lemma shows that each filter is contained in some ultrafilter.

Appendix

466

2

(APP.2.3) DEFINITION: Let

X

be a superstructure (see

infinite set, and

€u

g

J

x ,

into

f,g E

xJ ,

{j € J : f(j) = g(j)} € U

iff

J

an

.

If

call

{j € J : f(j) € g(j)} E U

iff

f =U g

J

an ultrafilter of subsets of

U

f,g are functions from f

(O.l.l)),

.

(APP.2.4) EXERCISE: are some simples exercises for the reader to get

Here

to these definitions. Show that for any (1)

=u

(2)

f

(3) f (4)

f, g

iff

{j E: J : f(j) f g(j)} E U ;

gU

iff

{j E: J : f(j)

g

f cUg

k

a l l €u-members of

(i.e.,

{j € J : f(j) 5 g(j))

,

f(j) = g(j)

and

then

f =

U

g

€ U ;

X

(3.8.3)],

x'

(see

, with x'(j)

that

in

transformed sentences,

into

=

want

to replace

.

super-

Stroyan-Luxemburg,

or Barwise [1978, Ch.

=

x

X

into

A.3, XJ

Theorem

given by

,

for all j E J

analog of the Transfer Principle (0.2.3)(a)

*

j's,

.

we know that for the embedding of --->

many

will be a fixed

Xo consequence of gas' Theorem

a

x an

are €u-members of

.

the rest of this Appendix,

[1976, Theorem 3.1]),

f

except for finitely

structure over a fixed ground set As

g(j)} 6 U ;

if the ultrafilter is free, that is, if n [ U : U E U ] =

0

For

,

f,g E X J

is an equivalence relation;

g) iff (5)

used

holds, provided

is changed into

eu

and

=

When we deal with nonstandard extension, though, we

EU,

=u

by actual 'belongs to'

and

'equals

2

Appendix

467

to'. That is why we make the next construction.

(APP.2.5) DEFINITION: Let

be any ultrafilter over

U

collapsing function

M

Xo

and call

Yo

class; let

.

The

Mostowski

is defined as follows.

(i) Take an element in

J

M(f)

f

of

whose images are

XJ

the corresponding

be the set of those

all

=U-equivalence

equivalence

classes

interpreted as new individuals. (ii) Assume

M(f)

for each

the elements in

xP

not in

p 2 0

that for some f & XpJ ; X

P+l

\

X P

we

have

defined

we are going to define it :

' M(f) = IM(g) : g

if

f

is in

J Xp

, g EU

X

for

and is

P+l

f}

(APP.2.6) DEFINITION: With the same notations as above, call structure on the ground set

where j E J

,

and define

* :x----

the constant function x ' ( j ) = x The ultrapower X J / U is the image of X

XI

.

Yo

the super-

Y

is

> Y

for

all

under

*.

(APP.2.7) REMARK: The

map

*

is a superstructure extension.

This means (see

its definition in the comments before (0.4.6)): (i)

*

is injective; the reader is invited to

supply

a

proof of this. (ii)

*

satisfies part (a) of Leibniz' Principle; this

is

Appendix

468

not

difficult

to

of

prove, but it is outside the scope

2

this

Appendix, since a much more careful treatment of bounded formulas is needed.

For a full proof of this and the Internal

Principle, we

refer

to any of the

following

Definition

books:

Stroyan-

Luxemburg [1976, Chapter 3, Sections 4 & 81, Davis [ 1977, Chapter Sections 7 & 81,

1,

Barwise [1978, Chapters A.6 & A.31; and for

an elementary version, to Keisler [1976, Section 1D*l

.

(APP.2.8) REMARK:

Moreover, whenever the ultrafilter chain

contains a descending

U

* is not

of sets with empty intersection, the map

(prove

so it becomes a meaningful extension.

it),

Examples

of

such ultrafilters are not hard to find; maybe the simplest one is a

maximal

filter

of parts of

N

that contains

the

so-called

Frbchet filter {A 5 N : N \ A

(apply

Zornls

is finite]

Lemma to prove the existence of

such

a

maximal

filter). However, we are interested in superstructure extensions that enjoy the stronger property of being enlargements, of

(0.4.6).

through

The

the

consistent

existence of enlargements can

Compactness

if

Theorem

("a

set

each of its finite subsets is

of

in the

be

sense

established

sentences

is

consistent"),

see

Robinson [1966, Chapter 21 for that approach; instead, we prefer to

show

stronger not

only

that

for

'adequate' ultrafilters

than that of the descending chain), a

true extension, but also

original superstructure.

an

(with a

property

the ultrapower enlargement

of

is the

2

Appendix

469

DEFINITION:

(APP.2.9)

We

say

the

ultrapower

e v e r y nonempty f a m i l y intersection

i s adequate

XJ/U

of s u b s e t s of

R

property ( i . e . ,

X

of

s € XJ

so t h a t

R

V B E R

3 U E U ,

for

with th e f i n i t e

such that every f i n i t e

h a s nonempty i n t e r s e c t i o n ) ,

family

when

sub-

there is a

map

.

B 2 s ( U )

(APP.2.10) EXAMPLES:

Let

(1)

X

,

be t h e s e t of f i n i t e p a r t s of t h e power s e t

J

J = Pf(P(X))

,

and

an u l t r a f i l t e r t h a t c o n t a i n s

U

of the

sets { j € J : A € j i s We c l a i m t h e u l t r a p o w e r

J X /U

A E X .

i s adequate.

t h e f i n i t e i n t e r s e c t i o n p r o p e r t y , then f o r each A . = n[B € R : B

s:J--->X

B € R

, i t is obvious t h a t

C a l l a binary relation

s u b s e t o f i t s domain,

finite

b € X

element

.

the

relations finite

x

concurrent i f f o r

,

S C_ dom(r)

every

t h e r e is always an

,

(a,b) € r

.

be t h e C a r t e s i a n p r o d u c t o f t h e f a m i l y

J

{Pf(dom(r)) : r € i.e.,

r €

such t h a t V a € S

Let

, the set

.

j E J

B z s ( { j € J : B € J}) (2)

J

be a c h o i c e f u n c t i o n

s ( j ) € Aj

Then f o r e v e r y

j

has

jl

J

i s n o t empty. Let

REP(X)

If

x

s e t o f a l l maps of

subset

i s a concurrent r e l a t i o n } j

d e f i n e d on t h e s e t o f

X

and such t h a t t h e image

of

i t s domain.

j(r)

( I t i s obvious

of

that

concurrent

r

is

a

concurrent

2

Appendix

470

r e l a t i o n s do e x i s t ,

so

i s n o t empty.) Next,

J

t a k e as

U

an

u l t r a f i l t e r that contains the s e t s i s a concurrent r e l a t i o n , then j o ( r ) c j ( r ) I . Then t h e u l t r a p o w e r X J / U i s adequate: l e t R E P ( X ) be a

U ( j ) = { j € J : i f r€X 0

family

the f i n i t e intersection property,

with

and

define

the

following binary r e l a t i o n : ( B , x ) E ro

ro i s a member of set

n j(ro)

and is c o n c u r r e n t :

X

n o t empty.

is

x E B € 8 ;

iff

Let

f o r each

--- >

s : J

j

be

X

€ J

, the

a

choice

function

s(j)e nj(ro) , For

every

B

if

j

Then,

therefore,

in

,

R

take a

,

cU(jB)

B 2 s(U(jB))

,

s

.

J

jBE J

the set

by d e f i n i t i o n of

E

j

so that

j B ( r o )= {B]

.

j(ro)

;

i s a member

B

s(j) € B

.

of

We have shown t h a t

.

(APP.2.11) PROPOSITION:

Every adequate u l t r a p o w e r i s an enlargement. PROOF: Let

A

be any member of

a *finite entity

in

F

X

.

We need t o show t h a t t h e r e i s

such t h a t

XJ/U

'A

C_ F

.

The f a m i l y IF € Pf(A) has

the

,

a E A

f i n i t e i n t e r s e c t i o n property.

a d e q u a t e , t h e r e i s a map

s : J --->X

s(U) 5 f o r some

: a € FI

U

in

U

.

{F

Pf(A)

Since th e ultrapower s o t h a t f o r each

is

a € A

: a € F]

Then { j € J : a E s ( j ) }€

By t h e d e f i n i t i o n of

,

M

,

u

it follows that

.*

a € M(s) €

*

Pf(A)

.

Appendix

Thus,

M(s)

471

is *finite and contains all

More simple ultrafilters, (APP.2.8),

‘A

= (*a

: a € A}

like the one mentioned in

. Remark

enjoy weaker forms of the Saturation and Comprehension

Principles (0.4.2) and (0.4.3). Stroyan-Luxemburg

[19761

properties of this type.

for

We refer the interested reader to

a

more

detailed

discussion

of

absolutely continuous measure: 103 adapted - (D-): 278 - (F-): 278, 342 adequate: 469 a.e. = almost everywhere algebra of events: 3 almost - commutative diagram: 122 - everywhere: 75 - - convergence lemma: 315 - previsible - - measurability theorem: 282 - process: 333 (basic): 333, 337 - set: 330 (basic): 330, 337 - previsibly measurable: 282 - progressive measurability theorem: /279 - progressively measurable: 282 S-continuous - - function: 121 - lifting: 121 surely: 75 analytic set: 127 Anderson's - extension: 152 infinitesimal random walk: 20, 210 Lusin theorem: 122 approximate volumes: 115 archimedean property: 24 a.8. = almost surely atom: 8

-

-----

-

-

-

basic - almost-previsible - - process: 333 - set: 330 - previsible set: 329 bounded hyperfinite measure = limited /hyperfinite measure - internal formalization: 22 Brownian motion: 210, 20, 364 Burkholder-Davis-Gundy inequalities: /372

cadlag: 216 card+(X)-saturated: 15 cardinality function: 2, 20 carrier (near-standard): 148 (p-finite or nu-finite): 77 Cauchy's inequality: 369 change of variables theorem: 143 characteristic function: 169 (joint): 180, 181 Chebyshev's inequality: 157 coarser sampling lemma: 345 compact convergence metric: 257 compactness theorem: 468 complete measure space: 64 - product: 153 comprehension principle: 37, 41 concurrent: 469 conditional expectation - - (external): 106 - - (internal): 107 - probability - (hyperfinite extension): 289 - - (internal): 289 continuous at zero: 332 martingales: 409 - (P-): 330 - path projection: 206, 259 convergence - in distribution: 146 in probability: 201 - on compact subsets: 257 countably additive: 101 cumulative distribution function: 169 cylinder set: 214

-

-

-

-

-

-

-

-

-

D-adapted: 278 De Moivre's limit theorem: 33 decent path: 216 - lifting - (At-):239, 263 theorem: 239, 263 - - - (nonanticipating): 293, 340 - - - (S-integrable): 244, 264

-

--

Index -

473

I

-

projection: 229, 263

- - theorem: 232, 263 - - - (nonanticipating): 291 -

sample: 228 (At-): 229, 263 derived: 291 - set lemma: 291 determined - at time t: 267 - before the instant r:268 - during the instant r: 268 direct limit of successive enlarge/ments: 39 distribution - (Cauchy): 196 - function = cumulative distribution /function - - (cumulative): 169 - - (joint): 180, 181 - (standard): 171 - measure - - (internal): 180 - - (joint): 181 - (Poisson): 192 Doob's inequality: 320 D-projection: 239 D-sample: 228, 263 Dunford-Pettis criterion: 111

-

-

Eberlein-Smulian theorem: 112 enlargement: 39 entity: 8 essential bound: 81 event determined at time t - - (internal): 267 (measurable): 267 - before the instant r: 268 during the instant r:268 expected value: 106 extended finite lifting theorem: 79 external - cardinality: 38, 41 conditional expectation: 106 - entity: 17

-

--

-

-

F-adapted: 278, 342 filter: 468 filtration - (previsible): 268, 341 - (progressive): 268, 341 finite: 14 cardinality: 13 - function lifting lemma: 77

-

2 (i 1-): 39 intersection property: 35 - measure: 101 - number: 23 - VS. limited: 23 - VS. near-standard: 257 finitely additive: 101 - bounded function: less = << redictable set: 321 FLY: 84, 93 F-martingale: 342 forward l/h-average: 230 Fourier transform: 169 F-predictable set: 321 F-stopping time: 298, 341 Fubini theorem (Keisler's): 162, 164 function - projection lemma: 71 - (u-finite): 87 fundamental theorem of integral cal/culus: 30

-

-

-

half standard part map: 247 Henson's - lemma: 42 - set: 127 Hoover-Perkins theorem: 419 Hoover s - half: 158 - strict inclusion: 157 hull (infinitesimal) - of a function: 241, 458 - of a space: 93 hyperfinite - conditional probability: 289 - evolution: 199, 258, 265, 348 - - scheme: 199, 258 - extension of an internal measure: 44, 57 filtrations: 278 - measure (limited or bounded): 44 - - (unlimited or unbounded): 58 - probability - - measure: 105 - - space: 105 hypermartingale: 312 - lifting theorem: 313 - (local): 342 hyperreal number: 16

--

Index -

474

ILL:93 independent: 186, 187 - increments: 210 indicator function: 101, 107 indistinguishable (P-): 213 individual: 8 infinite: - number: 23 - vs. unlimited: 23, 24 infinitely close - and near-standard laws: 184 - laws: 184 infinitesimal: 23 - hull - - of a function: 241, 458 - - of a space: 93 - random walk: 20, 210 integrable - process ( 2 - ) : 435 lifting lemma: 92 - set: 57 - (uniformly): 243 integration by parts: 437 internal: 17, 40 bounded formalization: (22) cardinality: 38 conditional - expectation: 107 probability: 289 definition priciple: 22 distribution measure: 180 ev nt determined at time t: 267 h (i 1-): 39 measure = (internal) *finite measure pathwise measure: 355 process: 199 product measure: 152 random variable: 169 uniform probability: 52 vs. standard: 17 1-norm: 85 ( a - ) : 40 inversion formula: 178 iterated integration lemma: 356 ItB's lemma: 439

-

-

joint

- characteristic function: 180 - distribution: 180, 181 - - measure: 181

- quadratic variation process: 368 -

variation

- path measure - - (bounded): 431 - - (unbounded) : 430 - process: 430

Keisler's Fubini theorem: 162 kernel: 125 Kolmogorof metric: 217, 260 law: 182

- (infinitely close): 184 - - (and near-standard): 184 Lebesgue decomposition theorem: 105 lifting: 249 Leibniz' principle: 5, 15 lifting: 71, 122, 204 (almost S-continuous): 121 (D-): 239 (Lebesgue): 249 (martingale): 313 - (local): 344 (metric): 82 (S-continuous path) : 213, 260 (S-integrable): 92 theorem (decent path): 239, 263 - (nonanticipating): 293, 340 - (S-integrable): 244, 264 (extended finite): 79 (finite function): 77 (hypermartingale): 313 (integrable): 92 (local martingale): 344 (martingale): 313 (metric): 204 (nonanticipating): 288 - decent path: 293, 340 - dominated Lebesgue: 295 - Stieltjes: 364 (predictable): 328 (S-continuous path) : 214, 260 (semimartingale): 427 (set): 68, 270 (S-integrable) - decent path: 244 - Lebesgue: 252 (Stieltjes) - (differential): 357 - (nonanticipating): 364 - (summable): 361 (stochastically integrable): 434 (stopping time): 298, 341 (uniform): 72 (6U-oath): - 259 (un$form): 72 (6M -path): 386 (6t-bounded variation): 355 (at-decent path): 239, 263 (6U-path): 358 (6U-summable path) : 361

-

(ll-):

76

Index

475

limited hyperfinite measure: 44 number: 23 set: 57 'finite measure: 43 Littlewood's principles: 43 local - hypermartingale: 342 - martingale - - (At-): 342 lifting: 344 - - - theorem: 344 - - projection theorem: 344 - time: 44 locally 6M -summable: 401 Loeb: 44, 57 Eos's theorem: 466 Lusin's separation theorem: 128

- (finite): 101 - (finitely additive): 101 - (hyperfinite): 102 - (hyperfinite probability): 106 - (inner): 44, 50, 56 - (internal) = (*finite) - (internal distribution): 180 - (internal pathwise): 355 - (internal product): 152 - (joint distribution): 181 - (joint variation path) - - (bounded): 431 - - (unbounded): 430 - (limited hyperfinite): 44 - (limited *finite): 43 - (non-atomic): 51 - (outer): 44, 50, 56 - (positive real): - preserving: 118 - (quadratic path variation): 385,

Markov - inequality: 157 - process: 442 martingale - (F-): 342 (hyper-): 312 - - (local): 342 - lifting: 313 - - (local): 344 - - theorem: 313 - (semi-): 420 - (At-): 311 - (At-local): 342 - - (stable): 414

- (saturated): 64 - (sigma finite): 64 - (singular): 102 space - (complete): 64 - - (semifinite): 64 - - (sigma finite): 64 - (S-tite): 148 - (total quadratic variation): 386 - (total variation): 431 - (uniform 'finite): 52 - (unlimited hyperfinite): 58 - (unlimited *finite): 56 - (Wiener): 212 - (*finite): 43 metric: 199 - (compact convergence): 257 - (Kolmogorof): 217 - - vs. uniform: 217 - lifting: 82 - - theorem: 204 of uniform convergence on compact /sets: 257 - projection: 203 - - theorem: 203 - space - - (complete): 200 - - (separable): 200 - (uniform): 200 - - vs. (Kolmogorof): 217 monad: 24 monotone class lemma: 166 Mostowski collapsing function: 467 mutually singular measures: 102

-

--

h

-

-

(*):

305

(*sub-): 305 (*super-): 305 maximal function: 318, 368 measurable - event determined - - at time t: 267 - - before the instant r: 268 during the instant r: 268 - function: 71 - (previsibly): 278 - - (r-almost): 282 - (progressively): 278, 342 - - (r-almost): 282 random variable: 160 - set: 44, 50, 56 - - (universally): 137 measure - (absolutely continuous): 103 - (bounded hyperfinite) = (limited /hyperfinite) - (complete): 64 - (complete product): 152

--

-

/399

-

-

-

Index -

416

nearly surely: 75 near-standard: 201, 203, 222, 258, /261

-

carrier: 148 law: 184 vs. limited or finite: 202, 257 Nelson's internal set theory: 35 nonanticipating: 288 - decent path - - lifting theorem: 293, 340 - - projection theorem: 291 dominated Lebesgue lifting theorem:

-

/295

-

lifting theorem: 288 Stieltjes lifting theorem: 364 nonarchimedean field: 24 n. S. = nearly surely number - (finite): 23 - (hyperreal): 16 - (infinite): 23 (infinitesimal): 23 - (limited): 23 - (unlimited): 23 - (*natural): 18

-

path: 206 (decent): 216 liftigg - - (6M -): 386, 400 - - (At-decent): 239 - - (SIJ-): 358 - - (6U-summable): 361 - projection - (continuous): 206 - (decent): 229, 263 - sample - (decent): 228 - - - (At-): 229, 263 stopping lemma: 301 pathwise measure: 355 - projection: 373 paving: 127 (semicompact): 127 P-continuous: 330 polarization identity: 370 Polish space: 82 polyenlargement: 35 polyenlarging extension: 15, 35 predictable: 321 - (F-1: 321 (finitely): 321 - (i-): 321 lifting theorem: 328 previsible

-

-

-

-

-

- (basic): 329 - filtration: 268, 341 - measurability theorem: 279 - - (almost): 282 previsibly measurable: 278 - (n-almost): 282 probability: 106 - conditional: 289 - logic: 444 - measure: 106 - of an event: 3 - (uniform): 52 process - (almost-previsible): 333 - (basic almost-previsible): 333, 337 - (Bernouilli): 27, 233 (Cauchy): 196,235 - (internal): 199 - (Markov): 442 - (Poisson): 26, 224 (predictable): 321 - (stochastic): 213 - (variation): 368 - - (joint): 430 - - (joint quadratic): 368 - (2-hntegrable): 435 - (6M -summable): 388 - - (lyally): 401 - ((6M ,GU)-summable): 432 progressive - filtration: 268, 341 measurability theorem: 279 - (almost): 282 progressively measurable: 278, 342 - (n-almost): 282 projection: 71 - (continuous path): 206, 259 (D-): 239 - (decent path) : 229, 263 - (metric): 203 - (pathwise): 373 theorem - - (decent path): 232, 263 - - - (nonaticipating): 291 - - (function): 71' - (local martingale): 344 - - (metric): 203 - (S-continuous): 207 - - (semimartingale): 427 - (At-martingale): 312

-

-

-

-

-

-

-

quadratic variation: 368 (joint): 368 lemma: 376 measure: 386, 399 of integrals: 438

-

Index -

Radon-Nikodym derivative: 105 random variable internal: 169 - measurable: 169 reducing sequence: 343 representation theorem for random /variables: 174 reshuffling: 177, 443 Robinson's sequential lemma: 26

-

S-absolute continuity = S-AC S-AC: 84 sample (D-): 228, 263 (decent path): 228 - (At-): 228 (At-decent path): 229, 263 - (Vt-): 345 saturated sigma-algebra: 64 saturation principle: 36, 40 S-bounded t-variation: 354 S-Cauchy sequence: 89, 459 S-completeness: 124 S-continuous - function: 147, 258, 457 (almost): 121 - lifting (almost): 121 - path lifting: 213, 260 - - theorem: 214, 260 - projection teorem: 207 S-dense: 115, 461 - in *A: 461 section of a function: 154 of a set: 154 semicompact paving: 127 semifinite measure space: 64 semimartingale: 420, 426 - integrals: 425 - lifting: 427 theorem: 427 - projection theorem: 427 - (6t-): 427 semimetric: 200 - of convergence in probability: 201 separation theorem: 128, 254 set lifting lemma: 68, 270 sigma finite measure: 64 S-integrable: 85 decent path lifting theorem: 244 Lebesgue lifting theorem: 252 !.I-lifting:92 S-isometry: 94 Skorohod lemma: 234 - topology: 217, 260

-

--

-

--

-

411

S-law: 183 S-limit: 459 lemma: 242 SL : 85 S-MC: 84 S-monotone convergence = S-MC Souslin - operation: 125 - scheme: 125 - - (decreasing): 126 S-separated jumps: 220 stable At-local martingale: 414 standard: 17, 40 - bounded formalization: 13 - part: 24, 25, 201 - - (discrete): 17 - - of a finite number: 25 - - of a set: 117 - - of a vector: 25 - - of an infinite number: 25 - - (weak-star): 149 - (C.-): 40 stationary increments: 210 Stieltjes lifting theorem - (differential): 357 (nonanticipating): 364 - (summable): 361 Stirling's formula: 29 S-tite measure: 148 stochastic - calculus: 436 - differential equation: 440 integral equation: 440 - process: 213 stochastically integrable lifting /theorem: 434 stopping time - (F-): 298, 341 lifting lemma: 298, 341 - (At-): 297, 341 superstructure: 8 extension: 39

-

-

-

-

time deformation: 216, 261

- (measure of): 216, 261 total - quadratic variation measure: -

variation measure: 431 transfer principle: 5, 15

ultrafilter: 465 (free): 466 ultrapower: 467 (adequate): 469

-

386

Index -

uniform lifting theorem: 72 norm: 9, 146 uniformly absolutely continuous: 243, 244 bounded: 243 - integrable: 111, 243 unique extension set: 57 universality: 174, 443 universally measurable: 127 unlimited hyperfinite measure: 58 number: 23 *finite measure: 56 upcrossing lemma: 307 urelement: 8

-

At-maximal function: 318 At-reducing sequence: 343 A t-sample: 228 6bsemimartingale: 427 At-stopping time: 297, 341 &&path lifting: 358 - lemma: 259 6U-summable path lifting: 361

-

*: 5, 35, 467

variance: 157 variation - (classical): 351 (6t-bounded): 354 lifting: 355 - (6t-locally bounded): 426 (joint): 430 (joint quadratic): 368

--

weight function: 43 (signed): 146 Wiener measure: 212

-

*archimedean axiom: 24 *bounded: 455 *binomial theorem: 28 *cardinality: 20 *continuity: 7 *finite: 15, 40 cardinality: 38 - measure: 43 (limited): 43 probability: 108 - set: 15, 40 sum: (20), 43 *martingale after At: 305 maximal inequality: 307 'maximum: 455 'natural number: 18 *number of elements: 19, 20 *submartingale after t: 305 *supermartingale after t: 305 'transform: 6 of a statement: 14

--

-

2-integrable: 435 Vt-sample: 345

6M2-

- path lifting: 386, 400 - summable process: 388 -(6M-p(locally): 401 ,6U)-summable: 432 &-bounded variation: 354 - lifting: 355 At-decent path sample: 263 At-local martingale: 342 6t-locally bounded variation: 426 At-martingale: 311 projection theorem: 312

-

: 5, 35, 467

# : 2, 20 u : 17

= : 23

-<<< ::2323, 103 [.] : 27 tm : 25 T, D, 6t: see 'hyperfinite evolution' TA : 318, 341