Multidimensional Diffusion Processes

1.4. Martingales and Compactness

41

Proof. Given t^ and ^2, let rf and rf be, respectively, the smallest and largest multiple of h such that tf >ti and f^ ) - x(rf, co)\ < p. On the other hand, neither |x(tj, co) - x(t^, co)| nor |x(rf, co) - x(t2, co)| can exceed 9i(oj\ and so the proof is complete. G By analogy with hypotheses 1.4.2 and 1.4.3, we now state appropriate hypotheses for a P on (Q, . # ) which is concentrated on Q.^. 1.4.8 Hypothesis. For all non-negative f e CQ{R'^) there is a constant Af >0 such that (f(x(jh)) + Af(jh), J^jh, P) is a non-negative submartingale. 1.4.9 Hypothesis. Given a non-negative f e Co(R^), the choice of Aj in 1.4.8 can be made so that it works for all translates off The analogue of Lemma 1.4.4 in this context is the following lemma, whose proof is identical to the proof of Lemma 1.4.4 when one takes remark 1.2.11 into account. The details are left to the reader. 1.4.10 Lemma. For any S which is a multiple ofh and any n >0, Pi^+i -'^n^

^l^t'J

< SAp/4

a.s.

We are now ready to prove the analogue of Theorem 1.4.6. 1.4.11 Theorem. Let {h„: n>0} be a non-increasing sequence of positive numbers such that h„-^0 as n^ oo. Let {P„: n > 0} be a sequence of probability measures on (Q, Jt) such that P„ is concentrated on Qf,^. Assume that each P„ satisfies hypotheses L4.8 and L4.9 (with h = h„) and that the choice of the constants Af can be made independent of n. If for each T > 0 and e> 0

(4.5)

lim n-»QO

X ^»(l*(0' + i)K) - AiK)I > e) = 0, 0<jh„
and (4.6)

limsupP„(|x(0)| > / ) = 0, I >"oo

n>0

then {P„: n > 0} is precompact. Proof The proof here goes by analogy with the proof of Theorem 1.4.6. We must show that (4.7)

limlirnTPJ d^O n-*ao

for all r > 0 and p > 0.

sup

I 0 <s
|x(f) - x(5)| > 2p | = 0 J I

42

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

By Lemma 1.4.7, pj

sup

I \

Ix(0 - x{s)\>2p\<

P„id*(p) p/2).

0<s
The term P„(ST(p) < S) is estimated in exactly the same way as P(S.(p) < ^) was in Theorem 1.4.6, only one now uses Lemma 1.4.10 instead of Lemma 1.4.4. In this way, one proves that lim sup P„{S*(p) <3) = 0. 5^-0

n>0

On the other hand, by hypothesis,

P„{ei>P/2)<

I

Pn(\x{j+l)K)-x{jh„)\>p/2)

0<jh
^0 as n - • 00

D

L5. Exercises 1.5.1. Let £ be a non-empty set and ^ a collection of subsets of E such that if A and B are in ^ then A n Bisin^. Show that the smallest class j ^ of subsets of £ such that (0 ^ c j ^

(a) Ee^ (Hi) A, B e ^, A cz B implies B\A e ^ \iv) A, Be^, AnB=^0 implies A KJ B e ^ (v) {A„}f e^.An^ A„^ 1 for « > 1 implies {Jf

A„e^

coincides with a(^), the smallest or-algebra containing ^. The proof of this fact is very much like the monotone class theorem (for details see Dynkin [I960]). As an application, show that if j f is a class of bounded functions on E into R such that (i) Xc e -^ for all C e ^ (a) 1 e Jt {in) Jf is a vector space (iv) If {/„}f ^ Jf is a nondecreasing sequence of non-negative functions with sup„,, /„(^) < 00, then/= lim„^^ /„ is again in Jf then J^ contains all the bounded measurable functions on (£, o'(^)). 1.5.2. Let (X, p) be a metric space and denote by Up(X) the class of all bounded, uniformly continuous functions on {X, p) into R. Show that the smallest class of functions from X into R which contains Up(X) and is closed under bounded pointwise convergence is B(X), the set of bounded Borel measurable functions on X.

1.5. Exercises

43

1.5.3. Let (£, ^) be a measurable space. Let ^ e £ be a point. The atom A{q) containing q is defined by A(q) = n [B: B e ^, q e B]. Give an example to show that A(q) need not be in ^ . However if ^ is countably generated and j^/ is a countable subalgebra generating J^ show that A(q) = n [B: B e ^, q e B], and is therefore always an element of ^. 1.5.4. Let (E, J^, P) be a probability space and l e t / ( - ) and g{') be two nonnegative random variables on it. Assume that g is integrable and that P[f(')>X]<jE[g('):f{')>Xl

A>0.

Show that if r > 1, then g e E(P) implies/e E(P) and that

The proof of this inequality turns on a clever application of the formula

£[/'] = j imYPidq) = r IJA-•?[/(•) > X\ dX which is valid for any non-negative/and any r > 1. See Theorem 3.4 in Chapter 7 of Doob [1952] for details. 1.5.5. Let (£i, J^i), (£2, ^2) and (£3, ^^3) be three measurable spaces. Let F: £1 X £3 -• R be a measurable map. Let ^l{q2, dq^) be a signed measure on (£3, ^2) for each ^2 ^ ^2 and a measurable function of ^2 for each set in ^ 3 . Show that the set AT c: £1 x £2 defined by N= [(quqi)' \ 1/^(^1,^3)1 1/^1(^2, ^^3) < 00 is a measurable subset of (£1 x £2, J^i x ^2) and the function J £(^1, ^3)/i(^2» ^^3) is a measurable function of (^1, ^2) on the set N. Deduce from this that if 6(t, q) is a progressively measurable function and rj(ty q) is a progressively measurable continuous function which is of bounded variation in t over any finite interval [0, T], then Z(r, q) is again a progressively measurable function where Z(r, ^) = f'^(5, q)tj(ds, q) if •'0

= 0 otherwise.

f' I ^(5, ^) I I ;y I (ds, q) < co •'0

44

1. Preliminary Material: Extension Theorems, Martingales, and Compactness

1.5.6. Let £ be a non-empty set and rj a map of [0, oo) x £ into a Polish space (X, d). Define on E the a-field J^ = (T[t]{s): s > 0]. Show that if/is an ^ measurable map of E into a Polish space M, then there exists a ^^z+ -measurable map F of X"^^ into (M,^) and a sequence {t„}f £ [0,oo) such that /(^) = F(^(fi,^), ...,^/(r„,^)•••)»

qeE.

Next, assume that ^/(^ (?) is right continuous for each qeE and define ^t = o-[^(5): 0 < 5 < r]. Given a measurable function 6: [0, oo) x E^M such that for each t, 6(u •) is J^,-measurable, show that there exists a ^[o,x) X ^j^2+-measurable map F of [0,oo) x X^^ and a sequence {t„}5° ^ [0,oo) such that for all t > s and q £ E 0(s At,q) = F(s A r, rj(t^ A r, ^),..., ^(r„ A r, ^),...). In particular conclude that for each fixed r, 6(s A t, ^) is measurable in (5, q) with respect to ^[o, n x -^r» ^i^d hence ^( •, •) is progressively measurable. 1.5.7. Let (£, ^, P) be a probability space and I c J^^ be a sub tr-field. Let (7, ^ ) be a measurable space and F: E x Y-^ R a measurable function (relative to J^ X ^ ) such that sup^^^y £^[|F(*, y)\] < 00. Show that a version G(q, y) of £^[F(', y)|I] can be chosen so that G(*, •) is L x ^ measurable. Suppose now that we have a map /: £ -• 7 which is I-measurable. Assuming that £''[|^(%/(;))|]< 00, show that £ V ( % / ( - ) ) | 2 ] = G(.,/(.))

a.e.

1.5.8. Suppose (9(t), i^,, P) is a martingale on (£, i^, P). Let J^,+o = C]s>t ^s where .^^ is the completion of #"5 in (£, .^, P). (That is J e .^^ if and only if there is an .4 in ^, with A AA a B where Be^ and P(B) = 0.) Show that {6(t), ^t+o, P) is a. martingale. 1.5.9. Use Theorem 1.2.8 to show that if (9(t), #",, P) is a continuous real valued martingale which is almost surely of bounded variation, then for almost all q, 6(t) is a constant in t. Note that this conclusion is definitely false if one drops the assumption of continuity. 1.5.10. Suppose (9{t), ^,, P) is a martingale on (£, J^, P) such that sup,^o £(^(0)^ ^ ^' Show that E{9{t))^ is an increasing function of ? with a finite limit as r -• 00. Use this to show that d{n) tends in mean square to a limit ^(00) as n-* CO. Next, use Doob's inequality to prove

sup 1^(0-^(5)1 >e

<-,Em^)-e'(s)i

In particular 6{t) -• 0(oo) a.e. as t ^ 00. Finally, show that if T is an extended stopping time relative to {#"{}, then £^ [O(co)\^t] = 9{x) a.e. P. This is an especially easy case of Doob's Martingale Convergence Theorem.

1.5. Exercises

45

1.5.11. Let (£, ^) be a measurable space and J^,, f > 0 a non-decreasing family of (T-fields such that ^ = o-(lJ, J^,). Given /I £ [0, oo) x £, we say that A is progressively measurable if XA{'^ •) is a progressively measurable map from [0, oo) x £ into R. Show that the class of progressively measurable sets constitute a (T-field and that a function/: [0, oo) x £ -• (X, ^) is progressively measurable if and only if it is a measurable map relative to the (T-field of progressive measurable sets. 1.5.12. With the same notation as above, let i : £ -> [0,oo] be an extended non-negative real valued function such that for each r > 0, {^: T(^) 0, the map O, restricted to [0, r] x £ is a measurable map of ([0, t] x £, ^[o. t] ^ ^t) into itself. Since the second component of O^ is the identity map it is enough to check that T(^) As: [0,0 x £ ^ [0,r] is ^[o,f] x # i measurable. Since T(^) A S cannot exceed r, we need only check that for each a < r the set {(s, q): T{q) A s < a} is in ^[0, f] X <^r • Clearly {T(^) A S

< a} = ([0, t] X {q: x(q) < a}) u [0, a] x £.)

1.5.13. Let A(t\ r > 0 be a non-increasing family of subsets of £ such that A(t) G ^t for each t > 0. Consider the set A=

[j(t,A(t))

=

{(t,q):qEA(t)}.

f>0

Define B(t) = A(t - 0) = P|,<, A(s) for r > 0 and B(0) = A(0). Show that if A is progressively measurable then B = [j(t,B(t))

=

{(t^q):qeB(t)}

t>0

and B\A are progressively measurable too. (Hint: Consider the function/(r, q) = ;f^(r, q). From the fact that/(r, q) is progressively measurable show that/defined by /(t,q)=/(t-0,9)=lini/(i(l-^|,<j| is again progressively measurable. Identify/as XB)

Chapter 2

Markov Processes, Regularity of Their Sample Paths, and the Wiener Measure

2.1. Regularity of Paths Suppose that for each n > 1 and 0 < tj < • • • < r„ we are given a probability distribution F„,...,,„ on the Borel subsets of (R^f. Assume that the family {Ptu...,tJ is consistent in the sense that if {sj,..., 5„_ j} is obtained from {t^, ..., t„} by deleting the kth element t^, then P^^^ ..,s„-i coincides with the marginal distribution ofPf^^ j^ obtained by removing the kth coordinate. Then it is obvious that the Kolmogorov extension theorem (cf. Theorem 1.1.10) applies and proves the existence of a unique probability measure P on (R'^f^- "^^ such that the distribution of (il/(ti), ..., lA(^n)) under P is Pti,...,t„- (Here, and throughout this chapter, i/^ stands for an element of (R'^f^' "^^ and V(0 is the random variable on (R'^f^^ "^^ giving the position of ij/ at time t.) As easy and elegant as the preceding construction is, it does not accomplish very much. Although it establishes an isomorphism between consistent families of finite dimensional distributions and measures on a function space, the function space is the wrong one because it is too large and the class of measurable subsets is too small. To be precise, no subset of (R^f^' "^^ whose description involves an uncountable number of fs (e.g. {ij/: supo
2.1. Regularity of Paths

47

The question now is: how does one go about determining when the P associated with {Pti,...,t) gives Q outer measure one? The answer is contained in the next lemma. 2.1.2 Lemma. Let P be a probability measure on ((R'^f^' *\ ^(/jd)[. oo). Then Q has P-outer measure one if and only iffor every bounded, countable set 5 ^ [0, oo): (1.1)

P({^'- ^ \s is uniformly continuous}) = 1.

{Here ij/ \s stands for the restriction ofij/ to 5.) Proof First suppose that Q has P-outer measure one. Given S, note that As = {il/: il/\s is uniformly continuous} is a Borel set and that Q^ As. Hence P(As) = 1. Next assume that (1.1) holds for all 5 and let /I be a Borel set containing Q. By exercise 1.5.6, there exists a measurable function F on ((R'^Y^) and a countable set T = {t„: n > 1} ^ [0, oo) such that

Z^(^) = F(^(rO,...,^(0,...),

i^e(RT"^'-

Thus if 5^ = r n [0, iV], f)N=i {^'- ^ |s^ is uniformly continuous} £ A. But, by (1.3), P({^'. ij/ \s^ is uniformly continuous}) = 1 for all N; and so, by the monotone convergence theorem, P(A) = 1. This shows that Q has P-outer measure one. D We are now going to develop a criterion on {P,^ ^J for testing whether the associated P satisfies (1.1). The basic method which we will use goes back to Kolmogorov, although the elegant approach that we employ here is due to Garsia, Rademich, and Rumsey [1970]. It must be admitted that as a tool for studying Markov processes, Kolmogorov's criterion is rather crude by comparison to the machinery which we developed in Section 1.4. On the other hand, it has the important feature that it depends only on the two dimensional marginals P^ ^, and not on any higher order structural properties of the process. This fact makes it more ubiquitous than more refined results. 2.1.3 Theorem. Let p and ^ be continuous, strictly increasing functions on [0, oo) such that p(0) = ^(0) = 0 ami lim, , ^ ^(r) = oo. Given T>0 and (pe C([0, 7], R'l if

then for 0 <s
(1.3)

|#)-^(5)|<8jJ V ' ( ^ j p W .

48

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

Proof. Define d.^ = T and

'^yn^h Since /Q I{t)dt < B, there is a to € (0,^_i) such that I (to) < B/T. We are now going to choose a non-increasing sequence {t„:« > 1} ^ [0,to] as follows. Given f„_ 1, define d„^ i by p(d„_ i) = ip(r„_ J. Choose t„ e (0, ^„_ i) so that (1.4)

I(t„)<2B/d„.,

and

This can be done because the set of f G (0, ^„_ j) on which either one of these inequalities can fail must have measure less than d„-1 /2, and so there is a point in (0, ii„-1) at which they both hold. Clearly

Thus d„^ 1 < ^„+1
p(t„) = 2p(d„) = 4(p(^„) -

Wn))

<4(p(d„)-p(d„^,)). Combining (1.4), (1.5), and (1.6) with the fact that d„ < ^„_i, we see that mn) - Cl)(tn+l)\ < '¥-H2I{tn)/dn)p(tn - tn-fl)
p(d„^,))

<4^-'(4B/d'„)(p(d„)-p(d„^,)) dn

<4

'V-'{4Blu^)p{du). ^A

Summing over n > 0, we now get (1.7)

|^(to)-(0)| < 4 f

^-\ABIu^Wu).

Replacing (r) by (j)(T — t) in the preceding argument, we conclude that (1.8)

\(t>(T)-(l>(to)\ <4

\\-'{4B/u')p(du).

2.1. Regularity of Paths

49

Adding these, we arrive at \ct>(T) - 0(0)1 < 8

(1.9)

j\-'(4B/u')p(du).

Note that (1.9) now holds for any T, p, and 0 such that (1.2) is satisfied. In particular, if 0 < s < r < 7 are given and we define 4>(u)==(pis-\-^-^-u\

M6[0, T],

and PW = P ( ^ - « 1

we [0,7],

then

Thus, replacing 0 by 0, p by p, and B by B in (1.9), we have

|^(r)-^(0)|

<8\\-'(4B/u')p{du), •'o

which becomes (1.3) after the obvious change of variables. D

2.1.4 Corollary. Let (£, ^, P) be a probability space and 0: [0, co) x E-^ R^ a ^[0,00) ^ ^-measurable function such that d(*, q) is continuous for all q e E. If for each T >0 there exist number a = aj > 0, r = r^ > 0, and C = C7- < oo such that

(1.10)

E[\eit) - 9{sW] < c\t- s\^+\

0 < s,r < r,

then for any 7 = 7^ e (2, 2 + a^) and X > 0: (1.11)

P(

sup

IMziM,J)_(4Ar)^C.A

where p = Pj, = (y^- 2)/r^ andA = Aj = ^l^l

\t - sP""""^ ds dt.

50

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

Proof. From (1.10) we know that (t) - 0(s) I \^

1

It(1fc1!^["|


Thus,

cn^f^f"-)-^'^ T

.T

and by Theorem 2.1.3,

1^(0-^(5)1 < 8 [ (^J du^i^ ^^ (4Xy''\t-sf, y-2

0<s
if

cn^r^)-—• 2.1.5 Corollary. Let Pbea probability measure on ((R'^f^' '^\ ^(Rd^io,«)) and suppose that for each T > 0 there exist numbers a = ar > 0,r = r j > 1 + a j , and C — CT < oo such that (1.12)

E[ Iil/(t) - il/(s) n < CI r - 5 P•'^

0 < 5 < r < r.

Then Q has P-outer measure one. Proof. According to Lemma 2.1.2, we must show that if T > 0 and S is a countable subset of [0, r ] , then P({il/: yff \s is uniformly continuous}) = 1. Given 5, choose for each N >\ points 0 = to, ^ < ' *' < ^N, N = 7" so that if S^ = {r.^: 0 < i < N}, then S^ c 5^+1 and S c Q * 5^. Then' (1.13)

P({^'' ^ \s is uniformly continuous}) ^lin,.imp(Lsupl^f:if)i^-%(4AH), A/QoN^Qo\! s,teSN r~'^l y —-^ !/

where 7 and )9 are chosen as in Corollary 2.1.4. But if ^^^\') is defined by Nu\ (^f+l,Ar - t)\p{ti,f^) + (t - ti,N)w{U+UN) \p [t) — , ti+l,N — ti,N

^ ^ , ^ , ti^N < t < ti+iN ,

2.2. Markov Processes and Transition Probabilities

51

and \l/^^\t) = il/(T) for t > T, then ij/^^^') is continuous and an easy calculation shows that (1.12) implies E[ I iA<^>(0 - il/^''\s) n < 3TI r - s P ^^

0 < 5 < f < 7.

Thus, Corollary 2.1.4 applied to ij/^^^ yields PIL.

sup

lMz:^^A(4Ap|)
and so we see that the right hand side of (1.13) is one.

D

We now have all the ingredients necessary for the following theorem of A. N. Kolmogorov. 2.1.6 Theorem. Let {Pti,...,t} ^^ ^ consistent family of finite dimensional distributions. If for each T > 0 there exist numbers a = a r > 0, r = r^ > 1 + OCT, and CT < oo such that (1.14)

j ^ \y - xfP,^,(dx

X dy) < C^^l? - s\^^\

0<s
then there is a unique probability measure P on (Q, Ji) such that

(1.15) for alio
p(x{t,) G r „ ..., x(0 E r„) = p,,.... j r , x ••• x r„) ••• < t„ and Tj, ..., r„ e ^R,.

Proof Since J^ = a[x(t): t > 0], the uniqueness is obvious. To prove existence, we proceed as follows. First construct P on ((R*'}^' °°\ ^^Rd)io, «>), corresponding to {^ti, ...,r„}» ^s in the first paragraph of this section. Then (1.14) implies (1.12); and therefore, by Corollary 2.1.5 and Lemma 2.1.1, P determines a restriction F to (Q, ^(/?^)[o, oo)[Q]) given by F(A n Q) = P(A), A e ^(^^[o. «>. Since ^ = J^(K
2.2. Markov Processes and Transition Probabilities In the theory of Markov processes, the consistent family {Pn, ...,tj arises in a special way. To be precise, we define a transition probability function as a function P(s; x; r, F), 0 < s < r, X e R**, and F e ^^d, satisfying: (i) P(s, X, t, •) is a probability measure on (R^, ^^d) for all 0 < 5 < t and

52

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

(a) P{s, •, r, r ) is ^;jd-measurable for all 0 < s < r and V e ^^d, (Hi) ifO<s
P(5, x;u,r)

= j P(u y\ w, T)P(s, x; r, dy).

Equation (2.1) is known as the Chapman-Kolmogorov equation. A transition probability function should be thought of as giving the conditional distribution of a process at time t given that at time s the process was at x. It turns out that, even if one specifies the initial distribution of the process, a stochastic process is not uniquely determined by insisting that it have the preceding property relative to a given P{s, x; t, •). However, if one goes one step further and demands that P{s, x; r, •) be the conditional distribution of the process at time t given the process before time 5 and that at time s the process was at x, then the process is uniquely determined as soon as its initial distribution is given. We now have the following formal definition. 2.2.1 Definition. Let P(s, x; r, •) be a transition probability function and /i a probability measure on (R^ ^^j^). A probability measure P on ((R**)^®' *^ ^(Rd)io.oo)) is called the Markov process with transition function P(s, x; r, •) and initial distribution fx if (2.2)

F(iA(0) G r ) =/z(r),

Te^Rd-.

and for all 0 < s < r and r e ^^^i (2.3)

P(iA(r) e r | G[^(U): 0 <

M

< 5]) = P(s, 1/^(5); t, Y) (a.s., F).

If P(s, X; r, •) = P(t — 5, X, •), the corresponding Markov process is said to be time-homogeneous. We now turn to the question of the existence of a Markov process with given transition function and initial distribution. 2.2.2 Theorem. Let P(s, x;t,')be a transition probability function and /j, a probability measure on (R*^, ^^d). Define (2.4)

Fo(r) = //(r),

Te^Rd,

and for 0 < f ^ < • • < r„+1: (2.5)

P^„...,w.(A) = J.^- JP(tn,yn;tn+udyn+i)Ptu...,tn(dyi X ... X dy^),

where A e ^^Rd)N+\. Then^ {F,j ^J is consistent; and a probability measure P on ((/?'*y°' ®\ ^(j?d)io.oo)) is the Markov process with transition function P(s, x; r, •) and

2.2. Markov Processes and Transition Probabilities

53

initial distribution fi if and only if P is the probability measure on ((R^f^' ®\ ^(Rd)io,oo)) having {Pti,...,t„} as finite dimensional distributions. In particular there exists one and only one Markov process with given transition function and initial distribution. Proof The consistency of {P,j J is immediate from the Chapman-Kobnogorov equation (2.1). Now suppose P on ((R^f^''^\ ^^Rd)io,oo)) has finite dimensional distributions given by (2.4) and (2.5). Given t > 0 and T e ^^i, it is clear that P(iA(0) G A, ^(t) e r ) = [ P(0, y; r, r)Fo(^>') •'A

= £^[F(0,tp(0);^,r),tp(0)GA], and therefore: P(iA(r) e r I (7[iA(0)]) = P(0, ,A(0); ^ r )

(a.s., P).

Next let 0 < 5 < r and r e ^^a be given and suppose that 0 < MJ < • • • < M„ = 5 and T i , . . . , r„ e ^^d are chosen. Then

p(^A(t/i)eri,...,^A(«„)er„,^(r)6r) = ^«,....,„„,r(ri x " x r „ x r ) \\

Pis;y„;t,r)P

rix...xr„ 'PI = £''[P(s, ^s); t, n

il,(u,) € r „ ..., ^(u„) 6 r j .

Thus an easy application Exercise 1.5.1 implies that (2.3) holds. Finally, assume that P on ((R^f^' *\ ^^Rd)io,oo)) satisfies (2.2) and (2.3). We want to check that its finite dimensional distributions {Q,j^ ^,J are {P^^^ J. Clearly (2.2) implies Qo = PQ- TO complete the identification we use induction on n. If n= 1 and t^ = 0, we have aheady checked Q^^ = P^ j. If n = 1 and ^i > 0, then for T i e ^Rd:

Q,,(r,) = P(iA(ri) e r j = £^P(0, iA(0); r^, T,)] = J P(0, >;; r„ r,)Qo(^y) = | m

yi h. r,)Po(dy)

= ^r.(r,). Now assume that Q,, / : (Ry -• K, we have

t = Pti

t - Then for any bounded measurable

54

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

In particular, if r^, ..., r„+ ^ e ^^d and

/(yi. •••. yn) = xu(yi) ••• XrSyn)P(tn^ 3^n; t„+u r„+i), then, by (2.3),

\\

P(tn^ yn; ^«+i, ^yn+i)^ri

J^yi x ••• x ^y„)

rix...xr„+i

= Pu,...,t„.A^i x . . . x r „ ^ i ) . Thus the proof can be completed by another application of Exercise 1.5.1.

D

Of course, there is no reason why a Markov process should always have to be realized on ({R^f^- °°^ ^(R4)io.ao)). In fact, we want the following definition. 2.2.3 Definition. Let (£, J^, P) be a probability space and {J^^: r > 0 } a nondecreasing family of sub R*^, the triple (^(t), J^,, P) is called a Markov process on (E, ^) with transition probability function P(s, x; t, •) and initial distribution ^ if i(t) is ^,-measurable for all r > 0 and (2.5)

P ( ^ ( 0 ) G r ) = /i(r),

Te^Rd.

and (2.6)

F((^(r) G r IJ^,) = F(s, ^(5); r, F)

(a.s., P)

for all 0 < 5 < r and T e ^^d. Notice that if £ = (R'^y^- °">, J^^ = (^R^T' '^\ and J^, = (T[II/(U): 0 < M < r], then the preceding definition is consistent with the one given in 2.2. L The case in which we will be most interested is when E = Q^^ = Ji,^^ = ^^, and ^(t) = x(t). In fact, if (x(r). Jit, P) is a Markov process on (Q, Jt\ we will call it a continuous Markov process. 2.2.4 Theorem. Let P(s,x; t, •) be a transition probability function such that for each T > 0 there exist a = a j > 0,r = r^ > 1 + a r , an^ C = CT for which (2.7)

sup [ \y - yil'Pit,,

yu t2. dy)
0
2.2. Markov Processes and Transition Probabilities

55

Given a probability measure ^ on (R^, ^Rd\ there is a unique probability measure P — P^ on (Q, J^) such that (x(t), M^, P) is a continuous Markov process with transition function P(s, x;t,') and initial distribution fi. In particular, for each s > 0 and X E R^, there is a unique probability measure P^ ^ on (Q, Jt) such that (2.8)

P,, ^(x{t) = X for all

0
and (2.9)

P , , M h ) e r I ^ , J = P(t, x(t,); t2, r )

(a.s., P,, J

for all s
Proof Let Q on ((R'^f^' *^ ^(Rd)[o, «>) be the Markov process with transition function P(s,x;t,) and initial distribution fi (cf. Theorem 2.2.2). By Theorem 2.1.6, Q admits a restriction P^ to (Q, J/) having the same finite dimensional distributions. Since J^^ = ((T[\I/(U): 0 < M < t])[Q], it is obvious that P^ is the desired measure. (The uniqueness of P^ is clear, since its finite dimensional distribution must be those of Q.) Finally, suppose 5 > 0 and x e P** are given. Define

I

Sy^

if0
Pis,yi;t2.-)

ifti

P(ti.yi;t2,

•)

<s
if5
Then F(ti, ^i; ^2, •) satisfies (2.7). It is easy to check that the desired P^^c is the unique probabihty measure on (Q, J^) such that (x(r), J^^, P^^^) is a continuous Markov process with transition function F(sy x; t, •) and initial distribution S^. • A particularly important apphcation of Theorem 2.2.4 is to the situation in which (2.10)

P(s, x; r, r) = f g,(t - s, y - x) dy, •'r

where (2.11)

J.

^.(^'^) = 7^^"'^''^'^-

It is an elementary exercise to show that (2.7) is satisfied in this case with a = 1, r = 4, and C = d. Thus we have the following famous theorem of N. Wiener.

56

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

2.2.5 Theorem. For each s > 0 and x e R*^ there is a unique probability measure iirf\ on (Q, M) such that irjf,(x(r) = x,0
G r I ^ , J = f g,(t2 -h,y-

x(t,)) dy

(a.s., iTf)

for all s Qfor (s, x) G [0, 00) x R^, by x(r, Ts, ^o)) = x((t - s) V 0, co) -f X.

Then TTg = iT^^^ o T'^K Also, if {ex, ...,ed} is an orthonormal basis in R^ and %i : R^ -^ R is defined by ntx = (x,e,), 1 0] are mutually independent under P and P 071^^ = i^^^^ for 1 < i < d. Finally, the following are equivalent statements about a probability measure P on (Q, J^) : (i) p = ^c^), (a) £^e'<«'^<'»] = e-^'^l'/^^ for all t>0 and Oe R^ and M, is independent of G\x\f) — x{s)\ t > s] under P,for all 0 < s < 00, (Hi) for alln>UO
2.3. Wiener Measure This section is devoted to the development of some of the important properties of Wiener measure. For reasons which will become clear in Chapter 4, we want to couch our discussion in slightly more general terms. Thus we introduce now the next definition. Definition. Let (£, ^, P) be a probability space and {J*^,: t > 0} a nondecreasing family of sub tr-algebras of ^. Given 5 > 0 and a function j8: [0, 00) x E -• R^, we will say that (P{t), ^^, P) is a d-dimensional s-Brownian motion (alternatively, when there is no need to emphasize d or {J^,: r > 0}, ^(•) is an s-Brownian motion under P) if

2.3. Wiener Measure

(/) (a) (in) (iv)

57

P is right-continuous and progressively measurable after time s, p is P-almost surely continuous, P(P(t) = 0 for 0 < t < s) = 1, for all 5 < ^1 < ^2 and F e ^^d P(p(t2) 6 r I J ^ , J = f g,(t2 - t , , y - Pit,)) dy (a.s., P), •'r where g^ is given in equation (2.11).

If s = 0, we will call {P{t\ ^ , , P) a Brownian motion, Clearly (x(r), J^t, i^%) is an s-Brownian motion. In fact, (x(t\ M^, 1^1%) is the canonical s-Brownian motion in that if (P(t\ ^t, P) is any s-Brownian motion and P o j?~ Ms the distribution o(p(') under P on Q (note that by (n), q^ p(', q)isa. map of a set having full P-measure into Q, and therefore P o p~^ {$ well-defined on (Q, e/#)), then P ^ jS" ^ = T^^f\). The next lemma gives a partial answer to the question of why one likes to consider other versions of Brownian motion besides the canonical one. 2.3.1 Lemma. Let (£, ^, P) be a probability space and (p(t), ^^, P) an s-Brownian motion. Denote by #'(J^,) the completion of^(^t) under P and use P to denote its own extention to ^ . For t > 0, set J^,+o = n<5>o ^t+d- Then (P(t), ^ , + 0 , P) is again an s-Brownian motion. Proof. Obviously, all we have to check is that P{P(t2) e F | ^^j+o) = jr 9d(h — h^ y - Pih)) <^y (a.s., P) for all s < tj < ^2 and F e ^j^^. To do this, it is certainly sufficient to show that for all 0 e CQ(R^) and A e ^tx + o^ (3.1)

E\(P(t2)\A] = E' \gd(t2-t,.y-P{t,))(i>(y)dy,A

Note that (3.1) is obvious if ^ € ^ij. If .4 € #ii+o, then, since A e J^,+e for all £>0,

E'[(t>{P{t2 + a)), A]- =T7PE \ 9d(t2 -tuy-

P(ti -f e))(P(y) dy, A

Since j5(-) is right-continuous, we can now let e \ 0 and thereby get (3.1).

D

We now want to prove one of the basic properties of Brownian motion, namely: "it starts afresh after a stopping time." The first step is the following lemma.

58

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

2.3.2 Lemma. Let (P(t), ^^, P) he an s-Brownian motion. Given to > s, define PtoiU Q) = P(t + ^0» ^) ~ P(h, q\t >0 and q e E. If is a bounded Jt-measurable function on Q, then (3.2)

£"[4) o ^,„ I ^ , J = E^"\]

(a.s., P).

Proof. We need only prove (3.2) for
CO G Q ,

where 0 < fj < ••• < t„ and 01, .., 0„ e Q(i^'^). That is, we must show that if tQ>s, Ae J^jo' n>l, and (^i, ...,„ e Q(R''), then for 0 < fi < • • • < r„: (3.3)

£l(/>i(Ao(^i)) • • • UPto(tn)\ A] = E^'\ct>Mh)) • • • „(x(t„))]F(^).

To this end, let w > s and <j) e Q(/?'') be given; Then for D > M and A € ^ „ : E-'WM'

A] = f Elmv + u)- fS(u,

q))\^MP(dq)

= f j f 4>(y - P{u, q))gAv, y - P(u, q)) dyy(dq) = ( j {y)gAv, y) dyj^i/i). Thus (3.4)

E'{cj>{pM \^u] = \ {y)g,(v, y) dy

(a.s., P).

In particular, (3.4) proves (3.3) when n= \. Next suppose that (3.3) holds for n. Let 0 < fi < ••• < r„+i and (/>i, ..., (/>„+i e Cfc(R'') be given. Applying (3.4) to M = r„ + ro, u = r„+1 - r„, and 0(>') = 0„+ i(y + z), we have: £:''[^„+i(A„+J^n+i-0 + ^)l^r„+J

= I 0n+ i{y)9d(tn^ ^-t^,y-z)dy

(a.s., P).

59

2.3. Wiener Measure

Thus, since (3.3) holds for n: £l0i(/J,„(ti))-<^„.i(A„(t„.i)M] ''A

^UKKq))-n(P.otn,q))P(dq)

''A

X j (l>n+i(y)9d(tn+i -t„.y-

x ( 0 ) \p(A)

= £ni(^(^i)) •• 0„(x(r„))(/>„,iWr„,O)]P(A) Thus the induction is complete.

D

2.3.3 Theorem. / / {P(t\ i^,, P) is an s-Brownian motion and T is a stopping time satisfying T > s define p^(') by: \P(t^
if T(g) < O)

10

if

PXUq)T(q)=oo.

Then for A e ^^ and O a bounded J^-measurable function on Q into K, we have (3.5)

E^[
n (r < oo}).

In particular, (3.6)

E^[a)o^Jj^^] = £^ <"»[(!)] on

{i < oo}

(a.s., P).

Proo/. It is certainly enough to check (3.5) when 0 is a bounded continuous function on Q. But in that case, £''[0) o j5,, A n {T < oo}] = lim f^O) o j5,„, /I n {T„ < oo}],

60

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

where s -\- ^^ T„=

if

X < CO

if

X = CO.

n ' 00

But A e ^^ implies A n {T„ = s -{• (k/n)} e ^s+(k/n) for all /c > 1, and so E'[op^^,An{x„
Y.E k=

fc=l

\

\

"1/

= £^^10]F(.4 n {T„ < oo}). Since {T„ < oo} = {T < oo}, this completes the proof. Q

2.4. Exercises 2.4.1. Extend Theorem 2.1.3 in the following way. Let p and ^ be strictly increasing continuous functions on [0, oo) such that p(0) = ^(0) = 0 and lim^i^o ^ ( 0 = ^ Let L be a normed linear space and/: R^ -^ La. function which is strongly contin uous on B(a, 2r) for some a e R^ and r > 0. Show that

Wr)Va,r)

\

p{\x

-

y\)

J

implies that 2\x-y\

\f(x) -f{y)\\ < 8 (^

Ud+lj^\

^\-^y(du\

X, y e B(a, r),

where 7 = 7^=

inf

inf

B(x, p) n B(0, 1) .

xeB(0, 1) 0
P

The proof mimics that of Theorem 2.1.3. Set

Given distinct points x, y e B(a, r) set p = \x — y\ and proceed as follows: (a) Choose c e B{(x + y)/2, p/2) n B(a, r) so that

and set

XQ =

)^o = ^»

2.4. Exercises

61

(b) Given x„_i and y„-i, define d„-i and e„_i by: p(4-i) = i p ( 2 | x „ _ i - x | )

and p(^„-i) = ip(2|y„_i - yl),

respectively, and choose x„ e B(x, X _ i) n B(a, r),

>;„ e B(};, ie„_ J n B(a, r)

so that /(x„) < 2"-'B/yd^,

and 4 ' ( M ? ^ ^ ^ ^ ^ j < 2"- '/(x„. 0 / K - .

/(y„)<2''-'B/y^..

and ^(''p}!;!,!^/;";;!*)^'' ) ^ 2"" '/(j'„- .V?^-.-

(c) Conclude that

ll/(y)-/(c)|| < 41^ ^ P - ' I ^ J P W 2p

/4''+iB\

||/(x)-/(c)||<4j^ ^-|_^jp(d„). 2.4.2. Using Corollary (2.14), show that if ^ is a set of probability measures on (Q, J^) such that lim supF(|x(0)| >L) = 0 and for each T > 0 there exist a^ > 0, r^ > 1 + ar, and Cr < oo with sup E^[ Ix(t) - x(s) l"^] < Cr{t - s)^ •'^

0 < s < r < T,

then ^ is precompact. This observation suggests the following derivation of the existence of W^'^K Let {X„: n> 1} be independent Z?*^-valued normal random variables with mean 0 and covariance / on some probability space (£, J^, P). Given w > 1, define 0„: £ -)• Q by:

x(r, 0„(^)) = ^ ( | x , 4- «(r - ^ ) ^ M . i ) , (where ^ i ^fc = 0), and set P„ = Po^;K

Check that

sup E^'i I x(t) - x(s) n < C(t - s)\

0<s
62

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

for some C < oo and that the finite dimensional distribution of P„ coincide with that of i^^'^^ Sit times k^/n, ..., ki/n, where / > 1 and 0 < /c^ < ••• < /c,. Hence conclude that P„ -• i^^'^^ as n -> oo. 2.4.3. Prove Theorem 2.2.6. 2.4.4 Prove that if P(') is any Brownian motion, then

E[\m-p(s)n==c,\t-s\\

o<s
for any r > 1. Deduce from this that for any 0 < a < 1/2, p(') is almost surely Holder continuous with exponent a on any finite time interval. 2.4.5. Given A > 0, define 5^: Q -^ Q by: x(r, 5^co) = A" ^'^x(h, (o). Show that ir^"^^ is invariant under S^ (i.e., iT^"^^ = iT^'^^ « S^^). Using this fact, prove that a Brownian motion is almost surely not Holder continuous with exponent 1/2 even at one time point. With a little more effort one can show that Brownian motion has an exact modulus of continuity: (231 log ^ | Y'^ (cf. McKean [1969]). 2.4.6. Let a > 0 be given. Using Theorem 2.3.3, derive the following equality. ir^^\x(t) >a) = ii^^^^\x(t) >a,T
= iiT^^^z < t)

where T = inf{t > 0: x(t) > a}. Conclude that ds. ir^'Hr
(inyi'l

Next, use this formula to derive the joint distribution under i^^^^ of x(t) and T. Finally, show that ir^'H sup |x(s)| >a\<2dl~\ \o<s
I

—e'^^/^rd

\7C/

a

2.4.7. Observe that by the strong law of large numbers, lim

p(n) = 0

(a.s., P)

if (P(t\ ^t, P) is a ^-dimensional Brownian motion. Combining this with 2.4.6, show that lim tp(l/t) = 0

(a.s., P).

63

2.4. Exercises

Use this, along with Theorem 2.2.6, to check that if

at) = tp(i/t\

t > 0,

then the distribution of (^(•) under P is again iT^'^K That is, ((^(r), J^;, P) is a Brownian motion, where ^[ =
fiS sup S-^0 0<s
P ^ ^ ^ l

= 1.

/')c i^_ A I

|r-.|^. ^2(51og-j

The difficult part of the result is to prove that r^

(4.1)

ir^'^ lim — \-*0

\x(t)-x{s)\

sup -"*-

= 1. 4-^—-f^ / . w / 2 <^ 1

0 < s < t < 1 I -)c i _ „

A I

|r-s|^<5 ^2^l0g-j

The other part is a simple estimation of the probability of f]i=i A,^(p), where A^(p) are the independent events A,(p) =

4)-^-^)

< (1-^X2-"^^ log2")i/2

and 0 < p < 1 is arbitrary, (cf. McKean (1969)). It has been pointed out to the authors by R. Wolpert that a slightly weaker version of (4.2), namely (4.2)

^(1)

iim imi

sun sup

^iO 0 < s < r < l

M L : --YJ2 i ^ <^ ^8 = 1, i^c

i^„

A I

\t-s\<S (2^l0g^j

can be deduced from Theorem 2.1.3. To this end, take il/(u) = e" - 1 and for fixed a > ^ take p{u) = (aM/(log l/u)y'^. Check that E^'

I0:*(^af)"'i

<

00.

Next, apply Theorem 2.1.3 to show that lim ir^^^

1 + log :(t) - x(s)\ < 4a^'- j ' " ^ -^ log(l + ^ ) ^ ^ dw

fltoo

0 < s < r < 1 = 1.

64

2. Markov Processes, Regularity of their Sample Paths, and the Wiener Measure

That is to say, there is a finite random variable B such that 1

lo -

'" ^ 'K) for all 0 < s < r < 1 with r - s < ^. By L'Hospital's rule 1 + log

"•"

/

l\l/2

(.logl)" and therefore — hm

sup

\x{t)-x{s)\ 16a"2 L1I—1^<

almost surely. Since a > ^ is arbitrary, this implies (4.2). If one is willing to replace 8 in (4.2) by 16, then one can take €>(«) = e"^'^ and p(u) = otu^'^ for some a > 1. The details are somewhat simpler in this case.

Chapter 3

Parabolic Partial Differential Equations

3.1 The Maximum Principle In Chapter 2, Section 2.2, we showed how one can start with a transition probability function P{s, x; r, •) and end up with a Markov process. The problem is: where does P(s, x; r, •) come from? The example we gave there, namely: (1.1)

P(s, x; t,r)=

\ QII

-s,y-x)dy

•'r

is a natural one from the probabilistic point of view because of its connection with independent increments and Gaussian processes. It turns out to be natural from another point of view as well: the theory of second order parabolic partial differential equations. The connection between the P(5, x; r, •) in (1.1) and partial / G CY,{R^) and differential equations is well-known and easy to derive. Namely, if <> /(s, x) = j glT -s,y-

x)(p{y) dy, s < T,

then (1.2)

\ lim /(5, ') = (p

whe re A is Laplace's operator

i'dxf That is, / solves the backward heat equation with terminal data (p at time T. Besides the kinetic theory of gases and Einstein's famous articles on Brownian motion, there is a purely analytic reason why it is not surprising that the heat equation should be the source of a transition probability function. This reason is the " weak maximum principle " for parabolic equations.

66

3. Parabolic Partial Differential Equations

3.1.1 Theorem. Let a and b be bounded functions on [0, oo) x R'^ with values in S^j and R'^, respectively. Define

IffeC^'

^([0,

T) X R^) is bounded below and satisfies:

(1.4)

|^ + L J < 0 , 0 < 5 < r lim /(s, •) > 0,

then f is non-negative. In particular, iffeC^' ^([0, c,ge Q([0, T)) satisfy: (1.5)

f^ + L J + c(s)f> -g(sl

T) x R'^) n C„([0, T] x R**)

0 < s < T,

then for 0 <s
/(5, x) < ||/(r, .)||exp|jjc(u) du^ + fj^(r)exp||^(u) t/t/j^L

Proof. First assume that / satisfies (1.7)

f^ + L J < 0 ,

0<s
OS

If (so, XQ) G [0, T) X R** is a point with the property that (1.8)

/(5o, xo)
(5, x) e [so, T) x R'',

then we would have: df ^ (so, Xo) > 0,

V^/(5o, Xo) = 0, and

((aS;^^»'^»0),„.<_/'^' where V^^ denotes gradient in the x-directions. Since ^so/(5o. ^o) = X Trace(a(so, Xo)H^(so,

XQ))

+
V^/(SO, XO)>,

and

3.1. The Maximum Principle

67

where

«*""-((sS-'"')),.,„., is the Hessian matrix of/, we conclude that: ^ ( s o , ^ o ) + ^5o/(5o,^o)>0,

since Trace(^B)>0

if

A,BeS^.

But this contradicts (1.7), and therefore there is no (s©, XQ) e [0, T) x R*^ for which (1.8) holds. Next suppose that / satisfies (1.4). Given S >0 and g > 0, set fs,e(s.x)=f(s,x)-hS(T-s)-h8e-^\x

12

Then ^fs,s

Sf

^5

^5

. __-., .2 S — se ^\x

and J-sfs, e = -LJ+ se ^(Trace a(s, x) 4- 2(b(s, x), x » . Hence, ^

+ LJ,,, < - ^ + £e-^(Trace(a(s, x)) + 2 < 6 ( 5 , x ) , x > - |x|2).

Therefore, for each ^ > 0, we can choose e^ > 0 so that/j ^ satisfies (1.7) for e < £^. Now suppose that /^ ^(s, x) < 0 for some (s, x) € [0, T) x R'^. Then, since lim.^r f^ As, ')>0 and /^^^(s, x)-^ +oo as |x| -• oo, there must be a point (so, Xo) e [0, T) X R*^ at which (1.8) obtains, and this is impossible if (1.7) is to hold. We have therefore proved that for all S > 0 and all 0 < e < e^,/^ ^ > 0 in [0, T) X R'^. From this it is clear t h a t / > 0 in [0, T) x R'*. Finally, suppose / G C^'^{[0, T) x R'^) n ^([0, T] x R^) and c,g € ^([0, T]) satisfy (1.5). Then \\f{T,)\\-f{s,x)Qxp{-J satisfies (1.4), and so (1.6) follows.

c(u)du)-\- f g{t)Qxpf- f D

c{u)du\dt

68

3. Parabolic Partial Differential Equations

We will make repeated use of Theorem 3.1.1 throughout the rest of this chapter. However, the reader should notice that Theorem 3.1.1 is a uniqueness theorem, and therefore it is not very powerful except in conjunction with an existence theorem. Combined with an existence result one can then prove the following: 3.1.2 Corollary. Let L^ be defined as in Theorem 3.1.1. Assume that for each t>0 and (p e CS(R'') there is an fe C^'2([0, t) x R*^) n Cj,([0, t] x R'^) such that f(t, ')^(p and (df/ds) + LJ = 0, 0 < s < r. Then for each t>Oand(pe C^(R'^) there is exactly one suchf. Moreover, if for 0 <s < t and x e R'^we define A,, ^(x) on CQ^(R'^) into R^ by As,t(x)(p = /(s,x), then As,t{^) determines a unique non-negative linear functional on C(R') =Le

C(R'): lim (p(x) = 0 1^1-

such that \As,Mcp\ Finally, if we define T,t,0<s
(peC(R').

<\\(PI

C(R'^) by

then 7^, ^ is a non-negative contraction on C(R'^) into itself lim^^^ 7^ ,(/)(x) = (p(x)for allxeR'^^and Ttuh = Tti,t2 o Tt2,ti ,

0 < ti < ^2 < ^3 .

Proof. The uniqueness o f / i s immediate from Theorem 3.1.1. In fact, from that theorem we see that min<^(x) < / ( s , •) < max(/)(x),

0 <s


This proves that A5^(jc) is a non-negative linear functional on Co(R'^) and that

\K.{xyp\<MSince Co(R'^) is dense in C(R% this completes the proof of the assertions about

KMIn order to prove that 7^, < is a non-negative contraction on C(K'') into itself, all we have to show is that T.^.cp e C(R'') for cp e C^(R''). Since, if cp e CS(R% / ( s , x) = Tst(p(x) is the unique C^' ^([0, t) x K'^)-function tending to (p as s/'t, it only remains to check that such a function must tend to 0 as |x | -• oo. Thus, the proof reduces to showing that if M > 0 and cp^ e CoiR*^) satisfies: 0 <(pM
3.1. The Maximum Principle

69

/M e Ct 2([0, t) X R') n Q([0, r] x R') satisfying (a/M/^s) + L, /M = 0, 0 < s < r, with/jvf(r, •) = (PM» has the property that/M(s, •) e C(R'') for all 0 < s < r. To this end, note that we can find positive numbers A and B such that for all XQ G R*^ the function : il/Js, x) = Ae^^'-'\t - 5) + e'^'-^^ \x - Xo|^

(s. x) e [0, t) x R^

satisfies (dij/^^/ds) + L^I/^^Q < 0, 0 < 5 < r. Thus if p = ^ |xo | > M + 1, then, since P^
0<s
In particular, A

lA

0 ( t - s) = ^ «»<'-»(' - s) P 1^0 I for I Xo I > 2M -h 2, and this completes the proof that T^ , cpj^ e C(R''). Finally, we must show that

To this end, let (p e C*(R'*) be given and set/(5, •) = Tg, t 1} ^ C?'(R'') such that (p„-^f(t2, ') uniformly, and set/„(s, •) = r,,,2
and lim (/(s, x) -/„(5, x)) =f(t2.

x) - (p„{xl

X 6 R'.

s/t2

Thus, by Theorem 3.1.1, sup

\\f{s,.)-f„(s,-)\\<\\f(t„-)-,p„(-)\\-^0

0<S
as n-^ 00. That is: 7;,. ra = f(tu •) = lim Utu •) = lim 7;,,,, (^„ = 7;,,,, o 7;^ ^^ <^, n-»oo

which is what we needed to show.

n-*oo

D

70

3. Parabolic Partial Differential Equations

A two-parameter family {T^ ,: 0 < s < t} of operators on C(R'^) having the properties given in Corollary 3.1.2 is called a time-inhomogenous semi-group of non-negative contractions on C(R^). It is clear that if the linear functional A^ ,(x) given by A^ t(x)(p = 7] , (p(x) admits the representation (1.9)

A,^ t(x)(p = f P(s, x; t, dy)(p(y%

0 <s
and

x e R',

where P(s, x; r, •) is a probability measure, then P(s, x; r, •) is a transition probability function. Our next result shows that the {T^ ,: 0 < s < r} constructed in Corollary 3.1.2 admits a representation of the sort in (1.9). 3.1.3 Corollary. Under the same assumptions on L, as those in Corollary 3.1.2, there exists a unique transition probability function P(s, x; t, •) such that (1.9) holds. Moreover, there exist numbers A and B, depending only on the bounds on the coefficients of L^, such that: (1.10)

\ \y-^

r^(5, ^; t. dy) < Ae^^'-'\t

- s)\

0<s
and

x e R'.

Proof. As we said in the preceding paragraph, the existence of a transition probability function P{s, x; t, *) will be established once we show that for each 0 < s
A., tM(p = \ (p(yHdyl

q> e C(R').

We must show that /z(j?^) = 1. To this end, let cp^ e CS'(R'^) be chosen so that 0 < (PM < 1, <)Pjvf = 1 on B(xo, M), and T,^t(PM(xo) = / M ( 5 , XQ) >^-Tr2^^''''~'\^-'\ M and so ^(R^) = 1. It remains to prove (1.10). For this purpose, let r > 0 and XQ e R^ be given and define y{s, x) = Ae^^'-%t - sf -h A(t - s)e^^'-'^ |x - XQ |^

3.2. Existence Theorems

71

There is a choice of A and B, depending only on the bounds on the coefficients of L such that dy ^ 4- L,y < 0, 0 < 5 < r. Thus, if {(p„}f ^ Co{R^) is a sequence of non-negative functions such that (p„(x)/'\x — XQ |^ X G R**, then, by Theorem 3.1.1, Ts,t(pn
0<s
In particular, j (p„(y)P(s. xo; U dy) = r,,,(^„(xo) < Ae^^'-'\t - sf. The estimate (1.10) now follows from the monotone convergence theorem.

G

3.2. Existence Theorems Theorem 3.1.1, and its corollaries, appears to be a good mill with which to turn out transition probabihty functions. However, like any mill, it requires grist before it can produce; and the grist in this case comes from the theory of partial differential equations. The following theorem is of just the sort that we need; in fact, it gives us more information than is required. We state it here without proof, because the proof is quite intricate (the parametric method is the one usually employed) and we will not be relying on it for anything outside the present section. A good derivation can be found in the book of A. Friedman [1964]. 3.2.1 Theorem. Let a: [0, oo) x R** -* S^ and h\ [0, oo) x i?** -• R^ he hounded functions for which there exist numhers OL>0\0 <x\e\^for all (s, x) e [0, oo) x R'^ and 9 e K^ (ii) ||a(s, x) - ait, y)\\+ \h{s, x) - h(t, y)\
^^-lif(-^)8^^^i''(-^)iThen there exists a unique positive function p(s, x; t, y), 0 < s < t and x, y e R**, which is continuous jointly with respect to all its variables and has the property that if (p e C^iR'^) and g e Cf ([0, 00) x R% then for each t > 0 the function (2.1)

f(s, x) = I p(s, x; r, y)(p(y) dy-\- j du ^ p{s, x; u, y)g(u, y) dy

72

3. Parabolic Partial Differential Equations

is in C5' ^([0, r] x R^) and satisfies

withf(t, ')=^ (p. In particular, L, satisfies the conditions of Corollary 3.1.2, An easy consequence of Theorem 3.2.1 is the following. 3.2.2 Corollary. Let L, be given as in Theorem 3.2.1 and let P(s, x; r, ') be the associated transition probabilityfiinctionguaranteed by Corollary 3.1.3. Then for any 0 <s
F

Rd,

where p(s, x; t, y) is the function described in Theorem 3.2.1. Also, if t >0 and feCt ^([0, t) X R') n Q([0, t] x R% then (2.2)

jf(t,y)P(s,x;t,dy)-f(s,x)

Proof The identification of p(s, x; t, •) as the density of F(s, x; r, •) is immediate from the fact that if ^ = 0 in (2.1), then the corresponding / satisfies (df/ds) + L,f=0, 0<s
\l/(s,x) =

jf(t,y)P(s,x;t,dy) d

f>„i(^^ + L„ |/(M, y)P(s, x; M, dy) is in C^' ^([0, t] X R"^) and satisfies aiA df ^ + L,iA = ^ + L,/, with i/^(r, • ) = / ( ? , •). Thus, by Theorem

0<s
0 < s < f, 3.1.1, iA(s, •) = / ( s , •) for all

D

The rest of this section is devoted to the derivation of a result due to Oleinik. The point of Oleinik's theorem is to get existence theorems when the coefficient a is degenerate (i.e., (i) in Theorem 3.2.1 fails) but there are more stringent smoothness requirements on a and b. We will need the following lemma.

3.2. Existence Theorems

73

3.2.3 Lemma. Let a: R^ ^ S^ be a function having two continuous derivatives. Assume that Xo = sup{|D^a''^(x)| : 1 < ij
and

xe R^}

and x e R^: \iDa''^)(x)\ < (2Ao)i/2(a"(x) + a^^M)^/^

Moreover, if u is any symmetric d x d-matrix, then: (2.4)

(Trace((Da)(x)w))2 < 4d^XQ Trace(Ma(x)w),

xeR"^.

Proof To begin with, let (p e C^(R^) be a non-negative function such that a = supj^g^i |(p"(x)| < 00. Then for any x E R^ and all y e R \ we have, by Taylor's Theorem 0<(p(x-^y)<

(p(x) + (p'(x)y + ^ y\

In other words, the quadratic a/2y^ + (p'(x)y + (p(x) is negative for no real y. Hence, from the elementary theory of discriminants, ((p'(x)y-2oi(p(x)<0, and so (2.5)

\cp'{x)\<{2ay'Hcp{x)r'\

We now apply (2.5) to the functions (p±(x) = a"(x) ± 2a'\x) + aJ\x). Since (p + (x) = <e, ± e^, a(x)(e, ± Cj)}, where {e^, ..., Cj} is the standard basis in R^ \(p'±(x)\ <4/lo. Hence

\cp\(x)\<(Uoy'HcpAx)y". But a'^(x) = i((p+(x) —
<4'%,W +
74

3. Parabolic Partial Differential Equations

To prove (2.4), we can assume that a(')is diagonal at the point x in question. We then have, by Schwarz's inequahty: (Trace((Da)Wt/))2 = [ ;^((Da)M)'^M'^

< d'lko i

(a%x)

+

a^\xW^f

d

= Ad^Xo Trace(wa(x)M). D Before presenting Oleinik's resuh, we introduce some standard notation from the theory of partial differential equations. Given cp on [0, oo) x R'^ into K\ let 'P.=j^

If a — (ai, ..., a^) is a multi-index of non-negative integers, we define | a | = Xi ^i and

Sometimes (p^""^ is used in place of D'^cp. Finally, for n > 0, define

i(s,-)\\. |a|<«

3.2.4Theorem. L^r a: [0, oo) x R** -• S^,fe:[0, (x^)R^ -• R\ andc: [0, oo) x i?** -• R^ be bounded continuous functions, and set

Assume that a e C5''"([0, oo) x R*^) for some m > 2 and that b, c e for some n > 1. Given T > 0, cp e Q(R'^), and C^'"([0, (X))x R'^)

3.2. Existence Theorems

75

g E C?'"([0, T) X R'^l suppose that f e Cl ^([0, T] x R^) satisfies f^ + LJ+c(s,

')f=-g.

0<5
mthf(T, ') = (p. If, for some 0 < / < m A«, fe C?''([0, T] X R') n C^''-^2([0, T) x R% thenf G C^' '([0, T) x R**) an^ r/iere exist numbers Ai and Bi such that (2.6)

11/(5, Oll^^) < MWcpf^ + sup \\git,-)f^)e^'^^-'\

0 < s < r,

s
Moreover, the constants Ai and Bi in (2.6) can be chosen to depend only on /, d, and the bounds on the spatial derivatives of a up to order I v 2 and those ofb and c up to order I. Proof Given a with |a| < /, we use Leibnitz's rule to derive:

(2.7)

/?• + L, /<" + ^ X' «»<4 /:i? + E c, , /«« = - g^^\ ^

k

p
where ^ means summation over those I
Then, from (2.7), we obtain:

|a| = l /?
|a| = /

We must estimate

To this end, note that, by Lemma 3.2.3, 2

{<{ f'ijf

= | T r a c e | | ^ //<'*> j j < Ad^'X, Trace(//<'">«//<'">)

76

3. Parabolic Partial Differential Equations

where Z/^^*^ is the Hessian matrix

of /^^*> and X, = sup{ | {9, d^a/dxUs, x)ey \/\e\^: 0 e R'\{0}}. Thus

(s, x) e [0, oo) x R' and

where AQ = niaxj
\|a| = l

/

fc

where

Hence

since: t>0

We have now arrived at: (2.9)

w, + L,w + Ci w + 2 X Z Ca, /, /^^y^'^ + 2 ^ f V |a|=/ ^
|a| = /

Next, observe that

+ C3ii/(s,-jir"

> 0.

3.2. Existence Theorems

77

where C2 and C3 depend only on the bounds on the coefficients c^^ p. Also

|a|=/

+ W||g(5,-)||^'^

Using these relations in (2.9), we arrive at: (2.10)

w, + Lsw + (Ci + C2 + C3||/(s, Oll^'-^^ + \\g{s, ')f^)w

We can now use induction on / and Theorem 3.1.1 to get (2.6).

D

Theorem 3.2.4 is the basic result of Oleinik on which our existence theorem for degenerate parabolic equations turns. However, before we can apply it, we need the following addendum to Theorem 3.2.1 (cf. Friedman (1969)). 3.2.5 Theorem. Let a: [0, 00) x R"^ -^ S^ and b: [0, co) x R'^ -^ R'^ be bounded continuous functions having bounded continuous derivatives of all orders. Set

If for some a > 0, (6, a(s, x)e} > a I ^ 1^

(s, x) e [0, 00) x R''

then for each t>0 and cp e C^lR'^) there R'') n C°"([0, t) X R') such thatf(t, •) =

exists

and

0 e K^

an / e C ^ ^ ( [ 0 , r ] x

0<s
We are now ready to prove the main result of this section. 3.2.6 Theorem. Let a: [0, 00) x R'^ -^ S^ and b: [0, 00) x R'^ -^ R"^ be bounded continuous functions having two bounded continuous spatial derivatives. Set

Then there exists a unique transition probability function P{s, x; r, •) such that P(s, x; ',r) is ^(s ^fmeasurable, for all (5, x) e [0, 00) x R"^ and F e ^^d, and (2.11)

j / ( f , y)P(s, x; r, dy) -f(s,

x) = j ' ^u 1 1 £ + L„J/(w, y)P(s, x; w, dy)

78

3. Parabolic Partial Differential Equations

for all 0 < s < t, X G R*^, and f e C^' ^([0, oo) x R"^). Moreover, for each (s, x) e [0, oo) X R**, there is a unique probability measure Ps^j, on (Q, . # ) such that: Ps, xMt) = X, 0 < t < s) = 1 and

Ps,A4t2) e r | ^ , J = P(ri, x{t,); t2. n

(a.s., P,,J

for s 1} and {b„: n > 1} so that (i) a„ and b^ have bounded continuous derivatives of all orders, (a) for each n> 1 there is an a„ > 0 such that

(e,a„(s,x)ey>oi„\e\' for all (s, x) e [0, oo) X R'^ and d e R^ (Hi) for all r > 0 ||a„(s, x) - a(s, x)|| -}- |fe„(s, x) - b(s, x) | -• 0 uniformly on [0, r] x R'' as n -• oo, (iv) there are bounds on the first two spatial derivatives of a„ and b„ which are independent of n. Define {L": n > 1} accordingly, and let {rj,,: 0 < 5 < f} be the associated timeinhomogeneous semi-group on C{R'^) given in Corollary 3.1.2. Given t > 0 and (p 6 C^(R'), set /„(s, •) = r;,,
uniformly on [0, t) x R''. This proves that for (p e CQ(R% and therefore also for (p e C(R'^), T"t(p(') converges uniformly on [0, t] x R** to a limit, which we denote by r,,,
on CiR'^y We now prove that at most one P(s,x;t,) satisfying (2.11) exists. Indeed, given t>0 and (p e C^iR^^l define /„(s,-) = T^^^^P as before. Then (2.11) holds with/„ replacing/. Since (df„/ds) -\- Lsf„-*0 uniformly on [0, t) x R'', we have

\(p(y)P(s,x;t,dy)-f„(s,x)-^0

3.3. Exercises

79

as n -^ 00, and therefore: (2.12)

j(p(y)P(s,x;t,dy)==T,,Mxy

That is, if P(s,x;t,') exists, (2.12) must hold for all cp € C(R'^), and therefore there is at most one such P{s,x;t, •). Conversely, to show that P(s, x; t, •) exists, let P„(s, x; r, •) be the transition probabihty function associated with E. By Corollary 3.1.3, there exist A and B, independent of n, such that (2.13)

j b - ^ \^Pn{s, x; t, dy) < Ae^^'-'\t - s)\

0<s
and

x e R'.

Thus {P„(s, x; r, •): w > 1} is (weakly) compact on (R^, ^^d) for each (s, x) e [0, oo) X R"^ and t > s. Since T:, ,(^(x) = I (^(^)P„(s, x; r, dy%

cp e C(R'l

and Tj" ^ T^^^ strongly, it follows that P„(s,x;r, ) converges weakly to a limit P(s,x;r, •); and for this Hmit, Equation (2.12) obtains. In particular P(s,x;r, •) is a transition probabihty function (since {T^^: 0 < s < r} is a semigroup), P(s, x; f, r ) is jointly measurable in s, x, and f (5 < r) for each F G ^^d, and [ |y - X |^P(s, x; r, dy) < Ae^^'-'\t - s)^,

0<s
and

x e R^

where ^ and B are the same as in (2.13). In view of these remarks, it remains only to prove that P(s,x;r, ) satisfies (2.11). But, by Theorem 3.2.1, (2.11) is satisfied when P(s, x; r, •) is replaced by P„(s, x; r, •) and L. is replaced by L". Since L" f-^ L. /uniformly on [0, t) x R"^, we conclude that (2.11) must be true for the P(s, x; t, •) just constructed. The existence and uniqueness of {Ps,x'' (s, x) e [0, 00) x jR**} is now an easy consequence of Theorem 2.2.4. D

3.3. Exercises 3.3.1. Let L be a linear operator from C*(P'') into CiR*^) having the following properties: (i) L is local (i.e., if (p(x) = 0 in some neighborhood of a point x^ in R'' then (a) L has the maximum principle, (i.e., ifcpe C'^(R'^) and (p has a local maximum at the point x^ then (Lcp)(x^) < 0).

80

3. Parabolic Partial Differential Equations

Under these conditions show that L must be of the form (L(p)(x]

2^

" w+z^x:^" dx

^ ^ dx; dx

where a^\x) and b^(x) are continuous functions on R'^ and {a'^(x)} is a non-negative definite matrix for each x. Proceed as follows for the proof: (1) First show that for any constant function c, Lc = 0. (2) If (pEC^(R^) has the property that for some x^ e R'^, \(p(x)(p(x^) I = 0( IX - x^ 1^) as IX - x^ I -• 0, then (by considering (p(x) + e | x - x^ |^ for suitably small e) conclude that (L(p)(x^) = 0. (3) Define ^ x ) = (L(pj)(x) and a'^(x) = (L(p,(pj)(x) - b^(x)(Pi(x) - b'(x)(pj(x\ where (p,(x) = Xj, i.e. the rth coordinate of x. Then verify by trying functions of the form Yfi ^ii^i — ^?)Y that {a'^(x^)} is positive semidefinite for each x^. (4) With the above definition of a'^(x) and b^(x) (using (2) and Taylor's formula) verify the form of L. 3.3.2. Prove the following extended maximum principle: Let a: [0, oo) x R'^ ^ S'^ and b: [0, oo) X R'' -* R'^ be bounded and continuous. Let 7 > 0 be given. Let/: [0, T] X R'^ -^ R he a. bounded continuous function such that there exists a sequence of functions {/„} in C^^([0, T] x R'^) such that lim sup I f„(t, x) -f{u x) I = 0 n-*QO

0
xeR''

and lim sup 0

0
dt

+ LJ„(ux)

<e

where L, is the operator associated with (a, b). Prove that sup I /(r, x) I < £(T - r) + sup | /(T, x) xeR''

xeRd

(Hint: Apply Theorem 3.1.1 to the functions {±f„{U x) -¥ 8'(t - T)} with

£' > £.)

3.3.3. Let a: [0, oo) X R*^ -)^ Sd and b:[0,cc)xR'^be bounded and continuous. Let a have two spatial derivatives which are uniformly bounded. Let b have one spatial derivative which is uniformly bounded. Construct a sequence b„ such that b„-*b uniformly on [0, oo) x R**, the first spatial derivatives ofb„ have a uniform bound independent of «, and each b„ has two uniformly bounded spatial

3.3. Exercises

81

derivatives. Denote by Lj"^ the operator corresponding to [a, b„] and by P„(s, X, r, dy) the transition probabihty corresponding to L\"^ constructed in Theorem 3.2.6. Show that for each 0 < 5 < r < o o a n d ( p G C(R'^) \im(r::\(p)(x) = hm [ (p(y)P„(s, x, t, dy) n-*oo

n-»oo

= (Tst(p)(x) exists uniformly in x; and show that 7^ , is an inhomogeneous semigroup on C{R^). Prove also that (7^ ,(^)(x) = J (p(y)P(s, X, t, dy) for some transition probabilities P(s, x, r, •), and the Markov process corresponding to P(s, x, r, •) can again be realized in the space C([0, oo): R'') of continuous functions on R*^. (Hint: Define for fixed (p e Co(R'') and 0 < r < 00 u"(s, x) = (P:i\(p)(x) for

0<s
Note that from the construction of P") in Theorem 2.3.6, the first order spatial derivatives of u^"^ are continuous and are bounded, independent of n. Observe that the maximum principle of 3.3.2 applies to each u^"\ The rest of the proof is similar to the proof of Theorem 3.2.6.)

Chapter 4

The Stochastic Calculus of Diffusion Theory

4.1. Brownian Motion Let (£, ^, P) be a probability space and (p(t\ J^,, P) a ^-dimensional Brownian motion. The purpose of this section is to point out some of the properties that (P(t), ^t, P) possesses in common with a much larger class of stochastic processes which we will be calling Ito processes. Since we are going to be giving a completely rigorous derivation of these properties in the more general context of Ito processes, our treatment here will be somewhat informal and proofs will not be complete. The first topic that we want to take up is that of equivalent characterizations of Brownian motion. Given 0 e R^, define ^^(x) = ^'^^' ^>, x e R*^. Then an easy calculation shows that (1.1) E'[ee(P(t2))\^n] =

eMt,))oxp

^-^-J^\0\2 0 < ti < t2 and ^ 6 R'^.

In fact, (1.1) is equivalent to (iv) in Definition 2.3.1. Another, more concise, statement of (1.1) is that (1.2)

(XiQ(t), J^j, P) is a martingale for all 6 e R'

where X,e(t) = ^xp[K9.my

+ -2\0\'tl

Thus (1.2) is equivalent to property (iv) in Definition 2.3.1. A second way of stating (iv) in 2.3.1 is the following. Starting with (1.1), we have j^E'[eMt))\^,;\ for t>ti. Thus for ta > ' i :

= -^e,(^(t,))exp

t - t

-\0

84

4. The Stochastic Calculus of Diffusion Theory

4.1.1 Theorem. Let (£, ^, P) be a probability space, {^,: t >0} a non-decreasing family of sub o-algebras of ^, and j5: [0, oo) x £ -> K** a right-continuous, P-almost surely continuous, progressively measurable function such that P(P(0) = 0) = 1. Then the following are equivalent: (i) (p(t), ^t, P) is a Brownian motion, (a) (Yf(t), ^^, P) is a martingale for allfe Ci(R% (Hi) (XiQ(t), ^t, P) is a martingale for all 6 e R^. Theorem 4.2.1'below shows that Theorem 4.1.1 is a special case of a general result. Next, assume that d — 1. We want to discuss the possibility of defining (1.5)

" (Ois) dp{s):

The problem is that j?(% q) is a.s. not a function of bounded variation. This fact can be easily seen from Exercise 1.5.6, since (P(t), ^t, P) is a continuous martingale. One can also see it from the following simple computation. Let t^^ „ = k/2", 0
k=0

Then E[W„] = 1, and a simple argument, using the Borel-Cantelli Lemma, shows that W„ -• E[W„] a.s. Hence the quadratic variation on [0, 1] of )?(•, q) over the dyadics is equal to one a.s., and therefore P(% q) is a.s. not of bounded variation. This means that we cannot take an entirely naive approach to the definition of (1.5). To see how one can get around this sticky point, suppose that 0 is a smooth function on [0, oo) and think of (1.5) as being defined through an integration by parts. Then i'o 0(s) dp(s) = p(t)e(t) - p{0)e{0) - ^'o P(s)0'(s)ds. An elementary calculation shows that E\j'o e(s) dp(s)] = 0 and E[(l'o e(s) dp(s)y] = ^'o e^(s) ds. Hence 6 -^ jo 6(s) dp(s) establishes an isometry from l3([0, t]. A) into I}(E, P): where X is Lebesgue measure. One can therefore extend the definition of jo 0(s) dp{s) to cover all 6 e L^([0, t], A). This procedure was first carried out, and used with great success, by N. Wiener. The situation is somewhat more comphcated when 9 depends on q as well as s. To illustrate the sort of phenomenon encountered here, consider 6(s, q) = ^(5, q). Suppose first that we attempt to define

(1.6)

f' p(s) dp(s) = lim ' x 'Pih, nMh^ 1. „) - Pih, „))• •'0

n^oo k = 0

85

4.2. Equivalence of Certain Martingales

By elementary manipulation, we have that

Xm,„Mr,,,,„)-^(r,,„)) k=0

fc = 0

i^ni)-i^^(0)

^

^

i^^(l)-iAO)-i

a.s.

as n -• 00. Thus the lack of bounded variation manifests itself here in the appearance of the term — ^. We will see later on in this chapter that — ^ results from the fact that dp(s) = ''(ds)^'' and therefore dp^s) = 2p(s) dp(s)-h (dp(s)f = 2p(s) dp{s) H- ds (cf. Ito's formula Theorem 4.4.1). It is important to notice that putting the increments of Brownian motion "in the future" as in (1.4) makes a difference (this is another manifestation of the absence of bounded variation). For instance, one can easily check that

(1.7)

lim I ^(h.umh^un) - P(Kn)) = i^'(i) - m o ) + i

n-^oo k = 0

Although the difference here seems to be small, it turns out that (1.6) is a far preferable way to define jo P{s) dp(s) than the one given by (1.7). Suffice it to say that the advantage with (1.6) is that the following relations hold:

j P{s)dp(s)=

0

and

|jJ/J(s)
4.2. Equivalence of Certain Martingales Let (£, ^, P) be a probabihty space and {<^,: r > 0} a nondecreasing family of sub (T-fields of ^. Let 5 > 0 be arbitrary and a: [s, oo) x E-*S^ and b: [s, oo) X E -^ R^ be bounded progressively measurable functions. For each (t, q) €[s, co) X E the components of a(r, q) will be denoted by {a'^(r, q)} and those of b(t, q) by {^(u q)].

86

4. The Stochastic Calculus of Diflfusion Theory

For any function/(x) in the space C^(R'^) of twice continuously differentiable functions we define L, / for r > s by

(LM)f)(x)-\ i ^'(t^^)A^(^) + ZHM) ^^ dXi dxj dxi 2.. We note that if (^(•, •) is any progressively measurable function from [s, 00) X E~^R^ then (Lj(q)f)(i(u q)) defines another progressively measurable function of t and ^ for r > s and q e E. Usually we will suppress the variable q and write (^(r), a(t), b{t\ L, for our objects. We shall suppose that we are given a function i(t, q) mapping [s, oo) x £ into R** which is progressively measurable, right continuous in r, and almost surely continuous in t. The main result of this section is the following theorem which proves that various types of relations between L, and ^(t) are equivalent. 4.2.1 Theorem. For any J( •), a( •),/?(•) satisfying the conditions described above, the following are equivalent: (i) f(i(t)) - J^ (L„/)({(w)) du is a martingale relative to {E, J^,, P)for t > sjor allfeC^(R'). (ii) f(t, i(t)) - jl ((d/du) + L„)/(w, i(u)) du is a martingale relative to (£, J*",, P) for t > sJor allfe C^ ^([0, oo) x R'^). (Hi) f(t, i(t))exp[~][((d/du)-\-L^)f/f)(u,i(u)du] is a martingale relative to (£, J^f, P) for t>s and for all f e C^ ^([0, oo) x i^**) which are uniformly positive. (iv) IfOeR'^ andge Cl ^([0, oo) x R'') then Xe,,{t) = exp U , ^W -^(s)-l j\e

+ Vg,a(u)(e +

-f^\~^ +

(v)

Hu)du) + g(t,Ut)) Wg)y(u,a^))du

L^yu,au))du

is a martingale relative to (£, ^ , , P)for times t > s. IfOeRUhen Xe(t) = exp <^, f(t) - ^(s) - j b(u) du}--\

(vi)

<0, a(u)ey du

is a martingale relative to (E, ^ , , P)for t > s. IfOeR^^then Xie(t) = exp K^, m

- ^(s) - \'b(u) duy + i j\o,

is a martingale relative to (£, J*^,, P)for t > s.

a(u)ey du

4.2.

Equivalence of Certain Martingales

87

Moreover if any of the above equivalent relations holds, then for each t > s (2.1)

P(^sup | { ( „ ) - ^ ( . ) - / " M . ) . . | ^ A )

^(2.)exp(-^^)

where A = sup,>s^^g£ sup|^l = i (6, a(t, q)6y. In particular, for any r > 0, exp rsup

(2.2)

\i(u)-i(s)


and the constant C depends only on t — s, r, A and B where B=sup \b(t,q)\. t>s qe E

Proof. Assume (i). Let/e C^{[0, oo) x R''). Then for s < ti < tj and /I e ^„ E[f{h,at,))-f{t„at,));A] = EUih, i{t,)) -fit,, ^(f,)) +/(t., ^(t^)) -fit,, ('{l^)iu,at^))du;A

I

iit,)); A]

(L,f)(t,,i(v))dv;A

= E [(l)iu,Ciu))du;A] +E

[C [(!)(«'«'^))-(IK ^^")) du: A

+ E \'\Kf)iv,iiv))dv;A •"(1

f\L^f)it„av))-iKf)iv,m)dv;A = £|n£+L„/j(u,^(u))
88

4. The Stochastic Calculus of Diffusion Theory

region, namely: t^ < w < D < ^2. It is obvious that the vahdity of the last equality for all fe CQ([0, CO) X R'^) implies the validity of the same equality for all feCt ^([0, 00) X R"^). Hence (i) implies (ii). Next assume (ii) and let/e Cl' ^([0, oo) x R**) be uniformly positive. Take

a(f) =/(t, i{t)) - j j l + L„/j(«, iiu)) du and ri(t) = exp

-i;(»ti^)(..*))*

Then a(t)n{t) - j\(s) dr,(s) =/(f, at)) exp f- | J ^ ^ ^ ) ± ^ J ( u , ^u)) dui Applying Theorem 1.2.8, we see that (ii) implies (Hi). Assume (Hi) and assume for the moment that {(s) = 0. Obviously (iv) would follow if we were allowed to take f(t, x) = exp[<0, x> -I- g(t, x)]. We cannot do this because/is unbounded and not uniformly positive. To circumvent this problem, we choose for each M > 1 a uniformly positive function f^ in Cl'^([0, co) x R*^) such that /Af(t, jc) = exp[<^, x> + g(t, x)] for X < M and define XM = (inf [t > s: sup \^(u)\ > M]) A (M vs). s
Then by (Hi) and Corollary 1.2.7, (Xe,g(r A TM), ^r» ^) is a martingale for r > 5. Clearly Xe^ g(t A TM) -• Xe^ g(t) a.e. F as M / " oo for each t > s. Thus we can establish (iv) if we can show that {XQ^ ^(f A TM): M > 1} is uniformly integrable for each t > s. But Xlg(tA.TM) = ^2e.2s(^ATM)exp <

| ' " " < ^ + V^, a(M)(^ + Vg)}(u, ^u)) du

CX2e,2g(t^'^M)

and £[^20,2g(t A T^)] = £[^20,2^(5)] = exp[2^(5, 0)]. This completes the proof when i(s) = 0. To remove this restriction, we define i'(t) = i(t) — ^(s). From (Hi) and Exercise 1.5.7 it follows that for uniformly positive functions / in

Cl\[0,^)xR<^)

/(U'W)exp[-((i^^/^^

))du

4.2. Equivalence of Certain Martingales

89

is an (£, ^^, F) martingale for t > s. From what we proved earlier, it follows that for all e eR" and g e C^ ^([0, oo) x R'^), X;,^(r) is an (£, <^,, P) martingale for t > 5, where XQ^ g(t) is the expression defined in the same way as X^ g(t) with ^'{t) replacing i(t). If one defines g^'(t, x) = g(t, x' + x) for x, x' e R*^ we have for s


m'e,,^t,)\3^t^

= X',,,^{t,)

a.e.

Again using Exercise 1.5.7

£W,...,('2)|^,J = ^^.,Jt.)

a.e.

Clearly X'Q^ g^^^^(t) — XQ^ ^(t) for all t > s, and we have therefore proved {iv). Obviously (v) is just the special case of (iv) when ^ — 0. To see that (v) implies (vi) it is enough to show that (v) implies (2.1) and (2.2). Indeed, if the estimates (2.1) and (2.2) hold, then both sides of the equality E[X,(t2);A] = E[Xe(t,);A] for s 0 be given. Then by Theorem 1.2.3 and (v) we obtain p( sup <^, ^(f) - ^(s) \s
(b(u)du}>A •'s

/

.exp(-pA + ^^4rA)j. Taking p = X/A(t — s) we get

Replacing ^ by — ^ we obtain a similar bound for the infimum. Combining the two, we have P ( sup \i(u) - ^(s) - rb(v)dv\ >yi] <2d,^-X'/2Adit-s) \s
Js

I

/

90

4. The Stochastic Calculus of Diffusion Theory

Since

p( sup b ) - {(s) \s<M
(b(u)ds\>A *'s

I


/

sup \ij{t) - ^s) - (b'{u) du\ > A/V5). \s
*^s

1

/

This proves (2.1), and (2.2) is readily obtained from (2.1) by estimating the integral in terms of the tail probabihties. It remains to show that (vi) implies (i). First we observe that, by elementary Fourier analysis, it suffices to prove (i) when f(x) is of the form /(x) = exp[K^, x}] for some 0 e R'^. Set a(r) = Xie(t) and r]{t) = exp

e.iis) + / b{u)du\-^

f (e,a{u)e)du

By Theorem 1.2.8 (x(t)rj(t) - I oc(u) driiu) is an (£, J*^,, P) martingale for r > s. It is easy to check that the last expression reduces to

mt))-j\L.f){i{f))du with/(x) = exp[i<^, x>]. This proves (i).

n

4.2.2 Corollary, Let ^(t) satisfy any one of the equivalent conditions in Theorem 4.2.1 and assume that £:^[e^l^<'^l] < oo, for all X>0. Suppose that fe C^'^([s, T) x R"^) n C([s, T] X R%for some T > s, and that there are constants Cj and Cj such that f its first time-derivative, and its first t\/\;o spatial derivatives are bounded by

f(tAT, i(tAT)) - j " ^ ( | ^ + L„j/(u, i(u)) du is an (£, J*^,, P) martingale after time s. In particular, for all 0 e R^: (2.4)

e,i{t)-i{s)-(b{u)du

4.2. Equivalence of Certain Martingales

91

and

(2.5)

U i(t) - i(s) -i^h[u)du\ - \\e, a(u)ey du

are (£, J^,, P) martingales after time s. Proof. Extend/to R x R"^ so that/(r, ')=f(s, *) i( t < s and/(r, ')=f(T, •) if / and i/^ e Co'((— 1, l))so t > T. Use/to denote this extension. Choose functions <> that [ and il/ = I on[-j,j].

(/>((t2+ \x\y^^)dtdx=

1

Define
and

Finally, define/„ by fn(t.x) = il/„(t,x)((P„*f)(t,x). Then/„ e CS'iR x R'^). Moreover/„(r, x)-*/(r, x) for all (r, x) e [s, T] x R'', and /„, dfjdt, dfjdxi, and d^fjdx^dx^ are all dominated by Cj^^^'^'. Finally, for s < ^1 < ^2 < T' and A e ^^^\ £ U ( f 2 , c ( r 2 ) ) - / „ ( r i , ^ ( r O M ] = ^ ' j^''(|^ + L„)/„(^^(u))tit.,/l] Using these facts in conjunction with estimate (2.2) and the hypothesis that £P[^A|4(.)ij ^ 00 foj. all A > 0, we see that

/(r A T, ^ t A T ) ) - f " ' | £ + L„j/(u, ^i.))./!. is indeed a martingale. The rest of the proof is easy and is left to the reader. D

92

4. The Stochastic Calculus of Diffusion Theory

4.3. Ito Processes and Stochastic Integration Let (E, J^, P) be a probability space, {^t- ^ > 0} a non-decreasing family of sub
a: [s, oo) x E-^Sj,

b\ [s, oo) x £-•/?''

progressively measurable functions with the properties assumed in Theorem 4.2.1. If {, a, and h are related by one of the equivalent conditions given in Theorem 4.2.1, we will say that ^ is an ltd process (on (£, J^,, P)) with covariance a and drift b after time s. This sentence will be abbreviated by the notation ^ ^ ^d(^, b) when there is no need to emphasize (£, J^,, P). The purpose of the present section is to develop the theory of stochastic integration with respect to an Ito process. Since it is clear that a Brownian motion is an Ito process, the theory which we develop must take into account the pitfalls which we pointed out in Section 4.1. The approach which we will adopt is basically due to K. Ito, and, by modern standards, it is very classical. We begin our discussion with two simple observations. In the first place, the starting time s of an Ito process plays no essential role. Indeed, if ^ ~ ^d{^, b) on (£, J^,, P), then it is clear that | - J^(a, b) on (£, J^,, F), where: |(r, •) = (^(r + 5, •),

a(t, •) = ci(t + s, •),

i(r, •) = fe(t + 5, •), and ^, = ^ t + s

As with the theory of martingales in Section 1.2, this observation enables us to restrict our attention to the case 5 = 0 although we will state our theorems for general s. Our second observation is that if (^ ~ -^d(a, b) and we define ^'(t) = ^{t) — ji b(u) du, then ^' ~ ^^(a, 0). Thus, if we can assign a meaning to di(t) integrals when ^ ^ ^d(^, 0), then we can assign one to d^(t) for ^ ~ ^5(a, fe), namely: ''di(t) = d^'(t) + b(t) dt'' For this reason, until further notice, we will take b = 0. Let (£, ^, P) and {J^,: r > 0} be given. A function 0: [0, oo) x E^R^ is said to be simple if 6 is bounded, progressively measurable, and there is an integer n > 1 such that e(t) = e{[nt]/n) for all t > 0. If (^ - J^(a, 0) on (£, J^,, P) and 6 is a simple function, we define jo <^(w), d^(u)} by

(3.1)

| W . , ( u ) > . I (.(;-), (^f-±l.,)-^(^-M))

where n > 1 is any integer for which 6(t) = 9([nt]/n\ t > 0. Notice that in keeping with the warning in Section 4.1 the increment appears in the future. (It is easy to

93

4.3. Ito Processes and Stochastic Integration

check that the definition does not depend on n so long as ^(0 = 0{[nt]/n), t > 0.) If 0 < ti < t2, we define

Ui

-^0

'0

4.3.1 Lemma. Ifi^ ^J(fl, 0) and 6 is a simple function, then Jo <^(w), d^(u)} is a right-continuous, almost surely continuous progressively measurable function. Furthermore

•'o (j^'<e(u), di(u)}J

- j\e(u),

a{u)e(u)} du,

and 1 r' Xe(-)(t) = exp j <0(M), d^u)} - - J <%), a(u)e(u)} du are all (£, ^t, P) martingales. In particular, if

f(t)=f'<%w^(")> •'o and a(u) = {e(u),a(u)e(u)), then f ~ J^?(a, 0). Finally, if 6^ and 62 are simple functions and Xi and X2 ^^^ ^^<^l numbers, then Aj ^1 -4-^2^2 ^^ ^ simple function and:

\\x,e,(u) 4- A2^2(w), d^(u)y = ^,f<e,(u), d^u)} + A2|'<^2("), d^u)}. Proof That

Ut)=(<e{u),dau)^ is right-continuous, ahnost surely continuous, and progressively measurable follows immediately ft-om the corresponding facts about 1^. We will now show that

94

4. The Stochastic Calculus of Diffusion Theory

f ~ ^?(a, 0). Once this has been done, we can use Corollary 4.2.2 to complete the proof of everything except the final assertion. Since it is clear that if A e R, then

•'o we will know, by (v) of Theorem 4.2.1, that J ~ ^?(5, 0) once we show that Xe^.)(t) = Qxp

m-\fa(u)du

is an (£, J^,, P) martingale. To this end, choose n so that 9(t) = 6([nt]/n\ t > 0, and let k/n 1. Then

^{t2)-^(h)=(e{^),Ht2)-i{h))Since 9(k/n) is a bounded ^^^-measurable function, we can apply estimate (2.2), Exercise 1.5.7, and (v) of Theorem 4.2.1 to conclude that: exp <e(k/nii(t2)-ati)} U'\e(k/nl

a(u)0(k/n)} du

1

a.s.

and therefore that E[XeUh)\^n]

= Xe^.it,)

a.s.

It is now an easy matter to remove the restriction that k/n < t^ 0. Then X,9,(t) + X2e2(t) = X,Oi([nt]/n) + /l2 0 2 ( M H

^ > 0,

when n = Ml ^2. Thus X^ 6^ H- A2 ^2 is simple and it is easy to check that

One just has to write out jQ{ei(u),di{u)),i = 1,2, in terms of the partition determined by n. D

95

4.3. Ito Processes and Stochastic Integration

The next lemma demonstrates that a large class of functions on [0, oo) x £ into R"^ can be approximated by simple ones. To be specific, let a: [s, oo) x £ -• 5^ be a bounded, progressively measurable function after time s and define @ 5(a) to be the set of progressively measurable 9: [s, oo) x E^R^ such that j id{u\ a(u)e(u)y du

< 00

for all T > s. We now prove: 4.3.2 Lemma. Given 9 e @ J (a), there exists a sequence of simple functions {9„:n> 1} such that for every T > 0. (3.2)

i\e{u) - dM a{u)(e(u) - 0„(u))> du=

lim E M->00

/O

0.

J

Moreover, if C=

sup f >0,

\9(t, q)\ < 00,

qeE

then the 9^s can be chosen so that sup

\9,(t,q)\


t>0,qeE

for all n> 1. Proof First assume that

C=

sup 19{t, ^) I < 00 t>0, « e £

and that 9{\ q) is continuous for all qeE. Then we can take 9„(t, q) = 9([nt]/n, q), and clearly {9„: n > 1} has the desired properties, including the fact that suPt>o,«€£ |^n(f> q)\ > 0, (/> = 0 off [0, 1], and J 0(r) dt = 1. Extend ^(•, ^) to ( - oo, oo) by setting 9(t, q) = 0 for t < 0, and define 9„(t,q) = nf

(t>(n(t - u))9(u, du, q)

n > 1 and

t > 0.

Then, 9„ is progressively measurable, 9„(\ q) is continuous, and sup,>o,qeE |^n(^ ^)| ^ <^- Morcovcr, by elementary properties of approximate identities, lim [

\9„(t,q)-9{t,q)\'dt^Q

96

4. The Stochastic Calculus of Diffusion Theory

for all T > 0. Combining this with the preceding, we see that the lemma has been completely proved in the case that supj>o,,e£ \0(t, q)\ < oo. In order to remove this restriction, it is enough to show that if ^ € (§)?(«), then there are bounded e„ 6 @J(a), « > 1, such that (3.2) holds. But this is easy. Simply take 0„(t.q) =

Xi-n,n,mt.q)\)e(t.qy

Then e„ e @ J(a), 9„ is bounded, e„(t, •) -• e(t, •) a.s. for each t > 0, and <^„(r, q\ a(t, q)e„(t, q)} < <^(r, q\ a(t, q)0(t, q)} for all (r, q). Thus, by the Lebesgue Dominated Convergence Theorem, (3.2) holds. D Aside from one technicality, we now have the basic machinery needed to complete the definition of the stochastic integral. Let 6 e @J(a) be given. By the preceding lemma, we can choose simple ^„'s for which (3.2) obtains. Define Ut)=

(<eMdi(u)y,

t>0

and n > 1.

By Lemma 4.3.1, for all m, n > 1:

L{t)-^M and - j <e„(u) - 0 » , a(u)(0„{u) - 0„(«))> du

{L(t)-Ht)r

''o

are martingales. Hence, by Doob's inequality (Theorem 1.2.3):

sup

\L(t)-L(tn

<4E j ie„(u) - OM^ a(u)(9„(u) - OM)} du

0
and so, because of (3.2), lim E m->oo n-+oo

sup luo-uor

= 0

LO
for all T > 0. In particular, there is a sub-sequence {J„, (•)} such that ^„, (% q) converges uniformly on finite intervals for q outside a P-null set N. It is at this point that we encounter the aforementioned technicality. Namely, it would seem reasonable to define

) <%), di(u)y = lim e„, (t) 0

n'-^oo

4.3. Ito Processes and Stochastic Integration

97

off the set N, and let it be defined measurably, but otherwise arbitrarily, on N. We could even guarantee that the resulting function be right-continuous, and it would certainly be almost surely continuous. However, it would not, in general, be progressively measurable relative to the original family {J^,: t > 0} because the set N forces one to " anticipate the future." The usual way in which this difficulty is avoided is to complete the a-algebras J^, as suggested in Exercise 1.5.8. However, this solution is not entirely suitable for us, since the completion is a function of the underlying measure P and in our applications the measure P changes, whereas the (T-algebras do not. Thus we will work a httle harder and obtain a definition of jo <^(M), d^(u)y which, besides being right-continuous and almost surely continuous, is progressively measurable relative to {^^i t > 0}. The essence of the procedure that we have in mind is contained in the following lemma. 4.3.3 Lemma. Let {r]„: n> 1} be a sequence of right-continuous, almost surely continuous, progressively measurable function on (£, #^,, P) into R^. If lim sup PI sup \rj„(t) - ri„(t)\>e = 0 «-*oo n>m

\0
for all s>0 and 7 > 0, then there is a right-continuous, almost surely continuous, progressively measurable rj on (E, ^ , , P) into R^ such that lim Pi sup \rj„(t) - ri(t)\

>eUo

n-»QO

I

\0
for alle>0 and T > 0. Proof Without loss of generality, we can and will assume that for each t > 0, P lim sup sup \rj„(u, q) - //^(w, ^) | = 0 I = 1. \m-»ao n>m 0
I

Since each r}„ is progressively measurable, it is easy to see that A defined by A = {{t, q): lim^_.„ sup„>^ supo<„<, \rj„(u, q) - ri„(u, q)\=0] is progressively measurable (cf. Exercise 1.5.11), Moreover, if A{t) = {q: {t,q) € A}, then PiA(t)) = 1; and if Jq = {t: {t,q) G A}, then Jq is an interval and fl(%q)= lim^„(-,^) M-+00

uniformly on compact subsets of Jq. It is clear that rj is progressively measurable on A. Hence, by Exercise 4.6.8 below, t] admits a right-continuous, almost surely continuous, progressively measurable extension to all of [0,oo) x E. Finally, one easily checks that limPJ, sup \n„(t) - rj(t)\ > 8 l = o

n-^oo

\0
\o
for all e > 0 and T>0.

D

98

4. The Stochastic Calculus of Diffusion Theory

In view of the preceding discussion and Lemma 4.3.3, we have now proved the next result. 43.4 Lemma. If 6e@ J (a), then there exists a sequence {6„:n> 1} of simple functions and a right-continuous, almost surely continuous, progressively measurable function J such that hm E

j ie(u)-0„(u),4umu)-0„iu))ydu

0

and lim£

sup 0
for all T > 0. Obviously, it is our intention to take the function f described in the preceding lemma to be our definition of Jo <^(M), di{u)y. However, before we can do that we must check that this definition does not depend in an important way on the choice of the simple approximants 4.3.5 Lemma. Let 6 e @d(a). There exists a right-continuous, almost surely continuous, progressively measurable function J such that sup

lit) - \\e'{u), dau)>\ < 4£ f <.e(u) - e'(u),«(«)(%) - e'(«))> du

0
for all simple functions 6' and all T > 0. In particular, there is, up to a set of P-measure 0, exactly one such function and it is the one given in Lemma 43.4. Proof Let {0„: n>\) and J be given as in Lemma 4.3.3. If S' is a simple function, then sup l(t) - \ ie'(u), di(u)} 0
<{E

sup Ut)-\<0Au),d4(u)y\ ^ I

0
sup

f'<0'(«) - 0M di(u)y\

o

sup Ut)-(<,0Mdau)>, p
+ 21E f i0'{u) - 0M a(u){e'{u) - 0„{u))} du 2 £ J <0'(«) - 0{ul a(M)(e'(«) - 0(u))y du

•

D

99

4.3. Ito Processes and Stochastic Integration

As we noted at the beginning of our discussion, anything that we can do when 5 = 0 is immediately extendable to general s. Thus, if s > 0, we say that 6 : [s,Go)xE ^ R^ is simple after time s if 0(-fs) is simple relative to {^t+s : ^ > 0} and we define J^(9(u),di{u)) accordingly for ^ e J^J(fl,0). It then follows that if i ^ -^i(«,0) and 9 e ©J(a), then there is a unique, up to a P-null set, rightcontinuous, almost surely continuous, progressively measurable function | after time s such that sup s
f ie(u) - e'(u\ a(u){e(u) - e'(u))y du for all simple functions 0' after time s and all 7 > 0. This function 1 is what we call the stochastic integral of 6 with respect to <^, and we denote it by j ^ <^(M), d^(u)y. The next theorem is now immediate. 4.3.6 Theorem. For 6 e ®d(a) and ^ - J^5(a, 0),

and (j'<0(u), dC{u)}J - ^id{u), a(u) du are (E, J<^,, P) martingales after time s. Moreover, ifO^ and 92 ^^^ in @d(a), then for Ai, ^2^ R

\\x,9,(u) + X292Hdau)y ^,\\9M

d^My + ^2f<02H

da^)}

a.s.,

and so

sup s
j'iOM dau)} - [iOM dau)}\ <4E \ (0Au) - 02H a{u)(0,{u) - e^iu))} du

100

4. The Stochastic Calculus of Diffusion Theory

In the case that 6 e (H)a(a) satisfies (3.3)

sup <0(r, q), a(u q)0{U q)} < oo, t>s,qeE

we can say more about J^ <^(M), di(u)y. 4.3.7 Theorem. Suppose ^ - J\{a, 0) and 6 e ®%a) satisfies (3.3). Let ?(r) = J^ <0(M), d^(u)y and a(t) = <0(r), a(t)e(t)y. Then J - Jf\(a, 0) on (£, J^„ P). Proq/! It is enough to carry out the proof when s = 0. We will do sofirstunder the assumption that 9 is uniformly bounded. In this case, we can choose simple ^„'s having the same bound as 6 such that (3.2) holds. Then Lit)=(<9„{u),da'')> •'o

tends in probability to|'(t) as n-> oo; and, by Lemma 4.3.1, X'm

= exp

xUt)-^\'a„(u)du

is an (£, J^,, P) martingale for all X e R, where a„(t) = <^„(r), a(t)e„(t)y' Clearly X^J^^t) -> X;,(t) in probability, where X,(r) = exp Xm-^(a(u)du Thus, by (t;) of Theorem 4.2.1, the proof will be complete in this case if we can show that X^''\t)\ n > 1 is uniformly integrable. But (X^l\t)f = X5'l(r)exp

du

<m(r)exp[A2rCj, where C„ = sup,>o.,e£ <^«(^ ql E[X^;l(t)] = 1, it follows that

a(t, q)e„(t, q)}.

sup E[{X^"^(t))^] < 00.

Since sup„ C„ < oo and

4.3. Ito Processes and Stochastic Integration

101

To remove the restriction that 6 be uniformly bounded, choose bounded ^„'s from (g)S(a) so that (3.2) holds and sup <0„(t, ql a(t, q)e„(u q)y < sup {6(1, q\ a(t, q)e(t, q)} t>0,qeE

t>0,qeE

for all n (cf. the proof of Lemma 4.3.2). One can then repeat the preceding argument to complete the proof. D There are a few matters that we still have to discuss before closing this section. In the first place, suppose that ^ ~ J^i^, b), where b^O. Let @d(a, b) be the class of progressively measurable 9: [s, oo) x E-^ R'^ such that 9 e @d{^) ^^^ J

\(9(u\b(u)y\du

< 00

for all T> s. Given 9 e @3(a, b), we can choose a version (unique up to a F-null set) of jj <^(w)» H^)y du which is right-continuous, almost surely continuous, progressively measurable and is equal to the ordinary Lebesgue integral js <^(w, q\ b{u^ q)y du if JJ I (fi(u, q\ b(u, q)} \ du < co for all r > s. We can therefore define \\9(ul d^u)} = \\9(u\ di'(u)} + \\9(u\ b(u)y du, ''s

''s

•'O

where i'(t) = ^(t) - fo b(u) du, The next matter concerns the problem of defining f \9(u), di(u)}

for s
i^ Jta, b) and 9 G ®5(a, b).

There are three possibihties, all of which give the same solution. First, one might note t h a t ^ ~ ^5(a, b) and 9 e @5(ti, b) means, in particular, that ^ ~ ^^d^(a, b) and 9 e (H)a^(a, b). This would give one definition of JfJ <0(M), di(u)}. A second possibihty is to define f\9(u)M(u))=

f\9(u)M(u))-

Jt\

Js

f\9{uUi(u)). Js

To see that these two coincide is quite elementary and is left to the reader. Finally, one might adopt f<XuutMO(u\dau)\ s

for some T > t2,3LS the definition of JJJ i9(u), di(u)}. In Exercise 4.6.9 below, the reader is asked to show that this definition is the same as the preceding one.

102

4. The Stochastic Calculus of Diffusion Theory

The final topic to be taken up is the stochastic integration of matrix valued integrands. Let ^ ~ J^5(«, b) and suppose that
(3.4) and

I kMOl

(3.5)

dt < 00

for all r > 5. Then (7*6 e (g)5(a, b) for all 6 e R\ and we define j ^ ^(M) ^{(M) SO that (3.6)

e,\c7(u)di(u))

=

\(c7*(u)e,d^{u)}

for all 0 e /?". It is easy to read off the properties of J^ (7(u) di(u) from those of js . In particular, Theorem 4.3.7 yields the following result. 4.3.8 Theorem. Let 1^ ^ J'J[a, b] and a: [s, oo) x E^ R" iS) R'^ be progressively measurable. Assume that sup Trace((7a(7*)(r, ^) < oo t>s

qe E

and sup \((jb)(t, q)\ < 00. t>s qe E

If^(t) = Js o-(w)
Proo/ Let 9 € R"he arbitrary. From (3.6)

= (<(7*(u)e^di(u)y •'s

= j<(r*{u)e, di'(u)} + j'ia*{u)0, = j'(c*(u)d, d4'(u)} + fie,

b(u)) du

(cb){u)y du

4.3. Ito Processes and Stochastic Integration

103

where i'(t) = ^(t) - J^ b(u) du and ^' - J'J[a, 0]. It follows from Theorem 4.3.7 that (3.7)

<0, lit)} - J\[e(7a(7*e, Oab].

From Theorem 4.2.1 (using i;), the validity of (3.7) for all 9 E R" is the same as f - J'„[(Taa*, ab]. D We close this section with the "chain rule" for stochastic integrals. 4.3.9 Theorem. Let ^ '^ -^|(a,^) and a : [s,co) x E -^ R^ ^ R'^ be a progressively measurable function with sup Trace ((Ta(T*){t, q) < oo t>s qeE

and sup \((Tb)(t, q)\ < 00. t>s qeE

Also, let p : [5, oo) X £ - • R*" ® J?" fee a progressively measurable function satisfying sup Trace (paacr*p*)(t, q) < 00 t>s qeE

and sup \(pab)(t,q)\ < 00. t>s qeE

Then if we take r,{t) =

(,T(u)dau)

and consider the stochastic integral with respect to tj, we have jy(u)dr,{u)=jy{uMu)d^{u). Proof. Consider the 2m dimensional process a(t) defined for f > s by (a,(t), a2(0) where ct,(t) =

j'p(u)a(u)di(u)

104

4. The Stochastic Calculus of Diffusion Theory

and ociit) = \ p(u) drj(u). We define y(t) for t > s as ad -h n dimensional process with components yi(t) and 72(0 where y,(t)^at)-i(s) and 72(0 = '/(O-

Clearly y(t:

1.1M)*'

and

'••>=i;rr'

;.,)^'H

We can now take a vector 6 e R"* and write <^, aj — a2> = <^, a) where 6 = (6, —9)e R^"". An elementary computation using Theorem 4.3.8 yields <^,ai(0-MO)-^1(0,0). Hence (Xi(t) - 0^2(1) = ai(0) - a2(0) = 0. This proves the theorem. D

4.4. Ito's Formula In Section 4.1 we saw that i( p(t) is a 1-dimensional Brownian motion, then

''0

The same argument as we used there yields the more general fact that

j'ii{s)dp{s)=mt)-mo)-{

4.4. Ito's Formula

105

As we indicated at the time, the appearance of the term r/2 on the right hand side results from the fact that "^j8(r)" is not a true differential, in the sense that (dp{t))^ i= 0. One can now ask how this fact manifests itself when one takes the " differential" of a more general function of a Brownian path. That is, suppose/is smooth. What is "i^/()5(r))"? When/(x) = x^, we have just seen that

df(m)=r{m)dm+¥"(m)dt. In other words, it is necessary to go out two terms in the Taylor's expansion of/in order to arrive at a true infinitesimal. This section is devoted to showing that the preceding is a general fact. A second way in which to view what we are about to do is " the identification of the martingale "

f(m)-f-2mp(s))ds as the stochastic integral

These preliminary remarks should become clearer as we proceed. 4.4.1 Theorem (Ito's Formula). Let ^ - ./^(a, b) on (£, ^,, P) with \i(s)\ bounded. Given a function fe C^-^([0, oo) x R"^) such thatf, df/dt, df/dxi, and d^f/dxidxj(l < i, j
(4.1)

f(t.m)-f(s,i(s)) = j\s/f{u,m)M'{u))

+ fil^-^

L^f(u,m)du,

t > s,

almost surely, where ^'(t) = ^(t) — \\ b(u) du. Proof Because of estimate (2.2), it is clear not only that V^/(-, (J(-)) e @d(a, b) but also that

[V./>.^(«))-v,/(«,?(«))pd« for all r > s as M -> 00, where

106

4. The Stochastic Calculus of Diffusion Theory

and

• R'"^^,and A:[0, (x>) x JE^Sj+i by: z{t) =

mj{t,m)),

B{t)=(Ht),[l+L,y{um)\

i
\la"{t)^{t,m) 1=1

if i=j=d + i if I
^-^/

if

l
Using (it;) of Theorem 4.2.1, one sees immediately that z ~ ^5+1(^, B). In particular, if 9(t) = {Vf(u (^(0), -1), then 0 is bounded; and so exp

A^ -' I {e(u\ dz'(u)} - - \ <0(u), A(u)e(u)y du

is a martingale for all k e R, where z'(t) = z{t) - \\ B(u) du. But {9(t), A(t)e(t)} = 0, and therefore j <^(w), ^Z'(M)> =0,

r > s,

almost surely. When this is written out, it is easily seen to be just equation (4.1). D Ito's formula in the form given by (4.1) above " identifies the martingale " in the definition of an Ito process. However, from a differential calculus point of view, (4.1) is not the most intuitive expression of Ito's formula. Indeed, if one wishes to emphasize the differential aspect of the formula, one should write: (4.2)

fit,i{t))-fis,m) = f'
where L^ = i Jj.J=i "'^(O S^/^^i ^^j 's the second order part of L,. Of course, (4.1) is equivalent to (4.2). The reason that (4.2) is more pleasing from a differential

4.5. Ito Processes as Stochastic Integrals

107

point of view is that it lends itself to the intuitively appealing differential formula:

(4.2') df{t, m=f((, m) dt+
dUt) dij(t) = ^'^(0 dt.

Equation (4.2') is, perhaps, the best and most concise statement of the "second order nature" of Ito processes. The reader may wish to compare this purely probabilistic statement with the analytic one given in Exercise 3.3.1. One more general comment about Ito's formula is in order. Equation (4.2') emphasizes the local nature of this formula and makes it seem unnatural that we have had to impose a global growth condition on/(cf. Theorem 4.4.1). The fact is that the growth condition is, in some sense, totally unnecessary from a path-bypath point of view; it is only there for reasons of integrability. The interested reader should consult the book of H.P. McKean Jr. [1969] or Exercise 4.6.10 below for a more expanded treatment of this point.

4.5. Ito Processes as Stochastic Integrals Let (P(t), ^t, P) be a ^/-dimensional s-Brownian motion and suppose that a: [s, co) X E^ R"^ (S)R'^ and b: [s, oo) x E-^ R'^ are bounded progressively measurable functions. An immediate consequence of Theorem 4.3.8 is the fact that any process i(t) satisfying (5.1)

i(t) - i(s) = I (7(u) dp(u) + j b(u) du,

t > 5,

is an Ito process having covariance a(') = (7a*(•) and drift b(•). In this section it is our purpose to prove the converse theorem. That is, suppose that ^ ~ ^d(^, b) on (£, J^,, P). What we want to show is that there exists an s-Brownian motion p(') for which (5.1) holds. Obviously, this will not be possible in general if we insist that P{-) live on {E,^t,P)' For instance, suppose that E consists of exactly one point q, ^ = J^^ = {(f), {q}}, and P = ^^. If {(r, q) = sin t, then ^ ~ J^?(0, b), where b(t, q) = cos t. On the other hand, there is no way in which one could put a 1-dimensional Brownian motion on (£, .i^,, P). Although this example may appear to be extreme, it demonstrates the necessity of distinguishing the situation in which a( •) may degenerate. For this reason, we are going to prove two versions of the converse theorem: one when a( •) is non-degenerate, and the other when it may degenerate.

108

4. The Stochastic Calculus of Diffusion Theory

4.5.1 Theorem. Suppose that a: [s, oo) x E-^S^ and b: [s, oo) x E -^ R'^ are bounded progressively measurable functions after time s on (£, ^) relative to { ^ , : r > 0 } . (Remember that S^ is the set of strictly positive definite d X d-symmetric matrices.) Let a: [s, oo) x E-^ R'^ x R'^ be a progressively measurable function for which a( •) = (T(7*( •). If^: [5, 00) x E-* R^isa progressively measurable function and P is a probability measure on (£, ^) such that ^ ~ ^%a, b) on (£, ^ , , P), then there exists a d-dimensional s-Brownian motion P(')on (£, ^^, P) for which (5.1) holds. Proof Note that o-(r, q) is non-singular for all (r, q) e [s, co) x E, and therefore ( Moreover, G~^a((j~^)* = I; and so, by Theorem 4.3.8, if P(t)= (c^-'(u)di'(u),

r>5,

where ^'(t) = ^(t) - J^ b(u) du, then p - J^5(/, 0). Thus, if p(t) = 0 for r < s, then p(') is an 5-Brownian motion on (£, J^,, P) after time 5. Finally, by Theorem 4.3.9, fcT(u) dp(u) = fa(u)a-'(u)

= i(t)-i(s)-

and this completes the proof.

di'{u) =

(b(u)du

fdi'{u)

a.s.,

D

We now turn to the case in which a( •) is allowed to degenerate. The idea is Doob's (cf. [1952]). Intuitively speaking, what is going on here is that we first have to ramify the original sample space and thereby create a space on which we can fit a Brownian motion which is independent of the original ltd process. We then hold this new independent Brownian in reserve and only call on it to fill in the gaps created by lapses in " randomness " of the original Ito process (lack of randomness corresponds to degeneracy in a( •)). In this way we get a full Brownian motion for which (5.1) holds, although we will, in general, have had to introduce an external source of randomness in order to do so. 4.5.2 Theorem. Suppose ^ ~ Jd(^, b) on (£, J^,, P), and suppose that a: [s, 00) x £ -> P*^ ® P*^ is a progressively measurable function such that a — ao*. Then there is a probability space (£, ?, P), a non-decreasing family of sub o-algebras {J^,: t > 0}, and progressively measurable functions:

4.5. Ito Processes as Stochastic Integrals

109

[5, 00) X £ - > S ' '

[5, oo)x E-^R'^i^R'^ [s, co)x E^R"^

I

[s, 00) X E^R*^

and P: [s, 00) X E^R'' such that a, a, B, and J are jointly distributed under P in the same way as a, a, b, and ^ are under P, ^(•) is a d-dimensional s-Brownian motion on (£, ^t, P), and Ut) - e(s) = j'5(u) d^u) + J'B{U) du,

t > s,

P-almost surely. Proof. Let Q = Q^ ^s in Chapter 1, define x('), J^ and {J^^: r > 0} accordingly, and let i^^f^ be ^/-dimensional Wiener measure starting at (5, 0). Set £ = £ x Q, ^ = ^ x'j^,^^ = ^^ X ^ , , and P = P X ir%. Extend a, a, b, and ^ from E to E and x from Q to £ in the obvious way, and denote these extensions by 5, a, B, I, and X, respectively. It is easy to check that all these extensions are progressively measurable after time s with respect to {^,: t > 0} and that

-©--((o;)'©)

on

(£,#i,P)

(cf. Exercise 4.6.4 below). Let n(r, q) be the orthogonal projection onto the range of 5(r, q) and n(r, q) the orthogonal projection onto the range of cr*(r, q)cr(ty q). Note that n(r, ^) = lima(r, q)(el + a(r, q))-' e^.0

and n is therefore progressively measurable. Similarly, fl is progressively measurable. Define a by a(f, q) = lim {el + a(r, ^))- 'n(t, q) and set f = a*a. The following relations are easily deduced from elementary linear algebra: (5.3)

ra = n

4. The Stochastic Calculus of Diffusion Theory

110 and

at = n.

(5.4)

We now define

£(M)

to be the d x 2^-matrix

(T(M),

/ — n(w)) and

Ht) = ft{u)d^-{u), where

-(,)==w-i;('<"')*. Note that this integral is defined, since, in fact,

t(^ ^jt* = tar + / - n = (ta)(tay + / - n = / by (5.3). In particular,^ ~ ^d{L 0) and is therefore a ^-dimensional s-Brownian motion. Finally, fd(u) d^(u) = \'d(u)l(u)

^S'(M)

= fdt(u) d^'(u) + fd(I - n)(u) dx(u)

where l'(t) = f (r) - J^ B(u) du. But, by (5.4), at = n, and therefore

ht)-^'(s)-(at(u)d^'(u) j Trace[(/ - n)a(/ - n)(M)] du = 0. On the other hand, a(I — fl) = 0 since the null space of a coincides with that of aa*. Thus \(a(I - n)(u) dx(u) \ s

We can therefore conclude that

(a(u) dm=i'(t) - r(s)=m - us) - ["%) du, t > s, P-almost surely, and this completes the proof.

D

4.6. Exercises

111

4.6. Exercises 4.6.1. Let (^ - J^5(0, b) on (E, ^,, P). A special case of Theorem 4.5.2 is the fact that at) - ^s) = fbiu) du,

t > 5,

P-almost surely. However, one doesn't really need any of the machinery developed in this chapter to prove it. In fact, it is instructive to derive it directly from Theorem 1.2.8 and Leibnitz's rule. 4.6.2. Let ^ ~ Jl(a, b) on (£, J^,, P), where P is defined on (£, ^). For each t > 0, define ^^ to be the set oi A £ ^ such that there exists a B e ^t satisfying P(AAB) = 0. Next, take ^t^o = f]d>o ^t-^s- Show that (^ - J^5(a, ft) on (£, ^,+o» P) (cf. Exercise 1.5.8). 4.6.3. Show that the condition that an Ito process be almost surely continuous is superfluous in the following sense. Let (E, JF, P) be a probability space, {^t'- t >0} a non-decreasing family of sub (j-algebras of ^, and a: [s, oo) X E -^ Sd and b: [s, oo) x E-^R*^ bounded progressively measurable function after time s. Suppose i: [s, oo) x E -* R^ is a. right-continuous progressively measurable function after time s which satisfies any one of the conditions of Theorem 4.2.1. Note that C = {q: i(', q)is continuous on [5, 00)} is an element of J^, and show that P{C) = 1. Perhaps the simplest approach is to first note that almost sure continuity was never used in the proof of Theorem 4.2.1, and therefore that i must satisfy (ii) of that theorem. Conclude that E[ \ ^12) —

^(rOn < A(t2 - ri)V<^2-'^>,

s
4.6.4. Let (£, J^, P) be a probability space and {J^,: r > 0} a non-decreasing family of sub c-algebras of J^. Suppose ^ ^ J\(a, b) on (£, ^ , , P). Given a second probabiHty space (£', J^', P'), extend c^, a, and b in the natural way (i.e., ^(r, {q, ({)) = (^(r, q) etc.) to E" — E x E' and call these extensions J, a, and h. Define P" = P x P' on (£", ^"\ where ^" ^ ^ x ^'. Show that | - >5(a, h) on (£", J^, X ^\ P"). Use this result to show that if ^ - J\{d, b') on (£', ^\, P'\ where {J^J: r > 0} is a non-decreasing family of sub c-algebras of ^\ and if P'lf

r."\

( ^^^^ ^) \

^^''^^U'(t.q')j a"(Uq") = a(t,q)®a'{Uq')

for r > s and q" = (q, q')eE\ then i" ^-J'Aa", b") on (£", J^;', P"), where d" = d^ d' and J^;' = J^, x 3F[.

112

4. The Stochastic Calculus of Diffusion Theory

4.6.5. The following is an interesting, and sometimes useful, representation theorem for one-dimensional Ito processes. Let (E, ^ , P) and {J^,: r > 0} be as in the preceding and suppose that ^ ^ -^d(a, 0). Assume, for the moment, that there is an a > 0 such that a(t, q)>(x for all (r, ^) e [s, oo) x E. Then for each q e E, r -• 5 + js a(u, q) du is a strictly increasing function from [s, oo) onto [s, oo). Let i(%q) be the inverse function. Check that {x{u •): t > s} is a non-decreasing family of bounded stopping times and that T(*, q) is continuous on [s, oo) onto [s, oo) for all qe E. Apply Doob's stopping time theorem (Corollary 1.2.6) to show that if P(t, q) = (J(T(r, q\ q) — (^(s, q\ t > s, then p(*) is a 1-dimensional s-Brownian motion on (£, <^t(r)> P)- ^^ particular, one will have shown that at) = as) + p\(1>M a(u)dui

(6.1)

t>s

where the distribution of j5(*) under P is if^i]\). In other words, the paths of such a 1-dimension Ito process are the same as those of a 1-dimensional Brownian motion; the difference being entirely in the rate at which the process follows these paths. When one puts it this way, it is clear that there should be no essential reason for the insistence on the uniform positivity of a. We will now outline how to proceed when this condition is dropped. As in Theorem 4.5.2, the difficulty caused by allowing a to be zero is that there may not be sufficient " randomness " to support a full-blown Brownian motion. The development outlined below is based on the ideas in H. P. McKean's book [1969]. Suppose that ^ ~ ^\(a, 0) on (£, i^,, P). Because of Exercise 4.6.22, we can and will assume that J^, = J^^+o. Define C = s + jf a(u) du. The first step is to prove that limt^^ ^(t) exists almost surely on {C < oo}. To see this, let T(r) = sup{M > s: s 4- J; a(v) dv U and set x(t) = oo if C < r. Note that {x(t)\ r > 5} is a non-decreasing, right continuous family of stopping times. Also, ^{') = ^(' Ax(T)) on {C < r}, and E[\^(t AX(T)) - ^{s)\^] < T for all t > s. Conclude from this, and Exercise 1.5.10, that lim,^<„ ^(t A X(T)) exists (a.s., P) for each T >s. Thus lim,^oo ^(t) exists (a.s., P) on {C < 00}. Let i^: E-^ Rhc SLU ^-measurable function such that ^^ = lim,^,^ <^(0 (a^., P) on {[, < 00}, and define TJ: [5, 00) X £ - > / ? by

n(t n\ = 1*^^^^^' ^)' ^) ~ ^('' ^^ ^^ ^(^) ^ ^

'^^'^^

\Uq)-as.q)

if

aq)
Note that r](', q) is right-continuous for all q e E. Using Corollary 1.2.6 and Exercise 1.5.10, prove that for all X e R: exp Ht)-j{{tAO-s)

-> ^x(t)^

P

is a martingale after time s. With Exercise 4.6.3, conclude from this that '/-^^l(X(..oo)(0,0)on(£,J^,(,),P).

4.6. Exercises

113

To complete the proof, let i^i]o t>e 1-dimensional Wiener measure on (Q, J/) starting at (s, 0). Let E = E x Q, ^ = ^ x J^, ^r =-^KO x ^ r . and P = P X irill .For q = (q, co) E E and t > s, define ^(^ q) = ri(t, q) + x(t, co) - x(tAi:(ql co). Using Exercise 4.6.4, show that j5 is a 1-dimensional s-Brownian motion on (JE, ^t, P). Finally, check that if ? and a are the natural extensions of i and a, respectively, to [s, oo) x £, then l(t)^^(s) + pl^s +

j\{u)du\

t > 5,

F-almost surely. This boils down to proving that ^(t) = ^(T(S + J^ «(«) ^M)) (a.s., P), which can be done most easily by calculating the second moment of the difference. 4.6.6. In Exercise 4.6.3, we pointed out that almost sure continuity is an unnecessary assumption in our definition of an Ito process. That is because we have adopted a very strong characterization of an Ito process. If, instead, we had chosen a characterization more in keeping with Levy's description of 1dimensional Brownian motion as the only ahnost surely continuous martingale p(t) such that P(0) = 0 and ^^{t) — t is a martingale, then the assumption of continuity would have been essential. Indeed, if rj(t) is a Poisson process with constant intensity 1, then tj(t) — t and (ri(t) — t)^ — t are martingales, but rj(t) — t is certainly not a Brownian motion. We will now outline how one can characterize Ito processes in a way which is consistent with Levy's idea. The procedure which we outline below is adopted from Kunita and Watenabe [1967]. Let (£, ^ , P) be a probability space, {J^,: r > 0} a nondecreasing family of sub (7-algebras of i^, and a: [s, oo) x E -^ S^ and /?: [5, oo) x £ -• /?** bounded progressively measurable functions. Let <^: [5, 00) x £-^P'^ be a right-continuous, almost surely continuous progressively measurable function. Then ^ ~ ^d(^, b) if and only if for each 6 e R^:

e,at)-as)-\Hu)du and

0, at) - as) - \HU) du\ - \\e, a(u)ey du are (£, ^t» P) martingales. The " only if " statement is immediate from Corollary 4.2.2. To prove " if ", it is certainly sufficient to treat the case when ^ = 0, since

114

4. The Stochastic Calculus of Diffusion Theory

Otherwise we can simply replace 4'(') by ^(•) - J^ ^(w) tiw. Giwen fe Co(R% s 1, TJ)"^ = t^ and Tr=(infjf>Tirli:

sup

\i(u)-^(Ti"\)\v\t-zi"l,\>-\] At-

Since (J( •) is almost surely continuous, for almost all ^ e £ there is a k(q) such that 4"^(^) = h for k > k„(q). Thus

Assuming that b = 0, show that

£[/(^(4"'))-/(^W"^)M] 1

''

2i.pl

ay (^(4«>0)

[Sx,dxj

a^^u)du,A tfc-iC)

+ o(;^)£[tr-Ti"»j. From this it is easy to conclude that:

£[m^2))-mri)M] = £ J"L„/(^H)^M,/I 4.6.7. Let ^ ~ J^d(a, ^) relative to (£, J^^, P). Given a progressively measurable function V:[s,oo)xE^R which is bounded above, show that for any fECt'([s,oo)xR<^) exp (J' V{u)du\ fit,m)

- f e x p Q " ^((7)^(7') (^^ + L J + K/")(«, (^(M))^^

is a martingale after time s. The proof is a simple application of Theorem 1.2.8. As simple as this fact is to prove, it is nevertheless extremely powerful and leads immediately to a general form of the Feynman-Kac formula. 4.6.8. Let (£, (f^) be a measurable space and {J^^: r > 0} a non-decreasing family of sub (T-algebras of ^. Suppose that >1 ^ [s, 00) x £ is a progressively measurable set after time s (cf. Exercises 1.5.11-1.5.13) such that (ti, q) ^ A whenever ri > s and there is a ^2 > t^ such that (^2, q) e A, Define A(t) = {q: (f, q) e A\ J(q) — [t\ (r, q) G A), and T(^) = sup{r > s: (t,q)e A}. Show that {x s. Next, let J(q) = [5, T(^)], the closure of the interval J(q), and show that A = {(r, q): t e J(q)} is again progressively measurable. Now suppose that/is a progressively measurable function on A into a metric space X. Choose XQ e X and define

4.6. Exercises

115

/(r, q) = XQ for {t, q) E A A a n d / = / o n A. Show that/is progressively measurable on A Finally, define/on [5, oo) x £ by/(r, q) =/(? AT(^), ^) and check that/is progressively measurable after time s. Observe that if/(s q) is right-continuous on J(q) for all q, then/(% q) is right-continuous on [s, 00) for all q. 4.6.9. Let (^ ~ J^5(a, 0) on (£, J^,, ?) and suppose that 0 £ @d(a). Given stopping times Tj and T2 such that 5 < TI < T2 < T, for some T > s, show that Zi.,...)(-W-)6@5(«)andthat

= f^fc,..»<e(uMI(«)>

(a.s.,P),

where fJ{e(u)Ji(u)) = |(T/) when | ( ) = X(0(W),^C^(M)). This can be done without ever resorting to simple approximations. The idea is to use nothing but the basic properties of stochastic integrals plus Doob's stopping time theorem. By the same sort of reasoning, show that if T is a stopping time and 0: E^ R*^ is an J*^^-measurable function satisfying: r''

1 <0; a(u)e} ^M < 00

.s

J

for all r > 5, then X[; oo)('^)0 e ®d(«. 0) and \\XIU, oo)(t)^, d^u)} = <^, i(t) - ^(t At)).

4.6.10. Kunita and Watanabe [1967] gave an elegant and easy derivation of Ito's formula based on the results of Exercise 4.6.9. Define the stopping time T^"^ as in Exercise 4.6.6 with t^ = s and t2 = t. Then, almost surely,

fit, at)) -f{s, as)) = i (/(tr, ^K')) -mi,, mi o))One now expands / in a Taylor's expansion, up to first order in time and second order in space, around (xi"! 1, (^(T["1 J). By Exercise 4.6.9, one can see that the terms corresponding to the first order spatial derivatives tends, as n -• 00, to

j. The other terms are easily recognized to be what they should be in the limit as n -• 00. The form of Ito's formula at which one arrives in this way is the one given

116

4. The Stochastic Calculus of Diffusion Theory

in (3.2). This is entirely natural, since the approach is based on a differential technique. 4.6.11. Suppose that ^ - J^5(a, 0) on (£, ^,, P) and 0: [s, co)x E^R" gressively measurable. Does one really need the condition

I (e(ui a(u)e(u)y du

< 00,

is pro-

T> s,

in order to define jj (^(w)^ d^(^)}l The answer is no, as we will now see. In fact, suppose that instead (6.1)

•||^<%), a{u)e{u)y du < 00

for all

T>s\=l.

Define, for / > 1, Oi(t. q) = X[o,n\\ <^("» du\e{u q) and check that 0, e (§)d(a). Set

Prove that f/+i(r) = J,(r) a.s. on {JJ <^(M), a(u)d(u)) du < /}. It then follows that there is a right-continuous, almost surely continuous progressively measurable J such that 1 = lim,_^ J, a.s. Of course I will not, in general, be a martingale, since l(t) need not even be integrable. Thus, the most suitable context in which to discuss this extension of Ito's integral is that of " local martingales ". To demonstrate that this extension is meaningful, the reader should use it to prove that Ito's formula holds for a l l / e C^'^([0, oo) x R**) without any growth conditions of/. Finally, it is important to notice that, in some sense, the condition (6.1) is the optimal one. In fact, one can show that the result proved in Exercise 4.6.4 can be extended to any stochastic integral i(') = is <^(")» ^^(w)>» where 9(') satisfies (6.1). That is, f(*) looks like a 1-dimensional s-Brownian motion run at the clock is ^0(u\ a(u)0(u)y du. Since a 1-dimensional Brownian motion has hmit superior equal + oo and limit inferior equal to — oo as time tends to oo, it is clear that our notion of stochastic integral breaks down completely if (6.1) doesn't hold. The interested reader is referred to the book of H. P. McKean [1969] for more details on this point, and also on the version of Ito's formula mentioned above. 4.6.12. Let (^ - J'a(a, 0) on (£, J^,, P) and assume that ^(s) = 0. Show that for each 2