ADVANCED TEXTBOOKS IN ECONOMICS VOLUME
17
Editors: C.J.BLISS M. D. INTRILIGATOR
Advisory Editors:
W.A.BROCK D. W.JORGENSON A.P.KIRMAN J.-J. LAFFONT
J.-F. RICHARD
ELSEVIER SCJENCE Amsterdam Lausanne •
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New York Oxford •
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Shannon
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Singapore
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S TOCHAS TIC METHODS IN ECONOMICS AND FINANCE
A. G. MALLIARIS Loyola University ofChicago
with a Foreword and Contributions by W.A.Brock University of Wisconsin, Madison
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INTRODUCTION TO THE SERIES
The aim of the series is to cover topics in economics, mathematical economics and econometrics, at a level suitable for graduate students or final year under graduates specializing in economics. There is at any time much material that has become well established in journal papers and discussion series which still awaits a clear, self-contained treatment that can easily be mastered by students without considerable preparation or extra reading. Leading specialists will be invited to contribute volumes to fill such gaps. Primary emphasis will be placed on clarity, comprehensive coverage of sensibly defined areas, and insight into fundamentals, but original ideas will not be excluded. Certain volumes will therefore add to existing knowledge, while others will serve as a means of communicating both known and new ideas in a way that will inspire and attract students not already familiar with the subject matter concerned. The Editors
CONTENTS
Foreword by W.A. Brock
xi
Preface
xiii
Acknowledgments
xvii
Chapter I .
Results from probability
1.
Introduction
2. 3.
Probability spaces
4.
5. 6. 7. 8. 9. 10.
Random variables Expectation
8
12
Martingales and applications
16
Stochastic processes
32
Optimal stopping
43 57 60
Miscellaneous applications and exercises Further remarks and references
2.
Stochastic calculus
1.
Introduction
2.
Modeling uncertainty
4.
1 1 4
Conditional probability
Chapter
3.
1
Stochastic integration ItO's lemma
5. 6. 7.
Stochastic differential equations
8.
Point equilibrium and stability
Examples Properties of solutions
65 65 65 69 80 89 92 96 102
Vlll
Contents Existence of stationary distribution
9. 10.
Stochastic control
II.
Bismut's approach
12. 13. 14. 15.
Jump processes Optimal stopping and free boundary problems Miscellaneous applications and exercises Further remarks and references
Chapter
1. 2. 3. 4. 5. 6. 7. 8. 9. *10. *11. 12. 13. 14. *15. 16. 17. 18. 19. 20.
Applications in economics
Introduction Neoclassical economic growth under uncertainty Growth in an open economy under uncertainty Growth under uncertainty: Properties of solutions Growth under uncertainty: Stationary distribution The stochastic Ramsey problem Bismut on optimal growth The rational expectations hypothesis Investment under uncertainty Competitive processes, transversality condition and convergence Rational expectations equilibrium
Linear quadratic obj ective function State valuation functions of exponential form Money, prices and inflation An N-sector discrete growth model Competitive firm under price uncertainty Stabilization in the presence of stochastic disturbances Stochastic capital theory in continuous time Miscellaneous applications and exercises Further remarks and references
Chapter
1. 2. 3. 4. 5.
3.
4.
Applications in finance
Introduction Stochastic rate of inflation The Black-·Scholes option pricing model Consumption and portfolio rules Hyperbolic absolute risk aversion functions
106 108 118 121 124 128 132
141 141 141 143 144 146 148 150 153 157 161 168 172 173 178 182 188 192 194 205 214
217 217 217 220 223 226
Conten ts
6. 7. 8. 9. 10. *II. *12. *13. *14. *IS. *16. 17. I8.
Portfolio jump processes The demand for index bonds Term structure in an efficient market Market risk adjustment in project valuation Demand for cash balances The price of systematic risk An asset pricing model Existence of an asset pricing function Certainty equivalance formulae
An example
A testable formula Miscellaneous applications and exercises Further remarks and references
ix
228 230 233 236 238 242 246 254 256 262 264 268 272
Selected bibliography
279
Author index
295
Subject index
299
•
Sections contributed by Professor W.A. Brock.
FOREWORD
In Stochastic Methods in Economics and Finance, A.G. Malliaris has undertaken the extraordinarily difficult task of assembling the relevant literature on stochastic methods used in recent articles in economics and fmance. Malliaris faces the fol lowing difficult tradeoffs. On the one hand he must present the mathematical literature on stochastic calculus, stochastic differential equations, optimal stochastic control, and opti mal stopping theory, among other topics, correctly and rigorously. On the other hand he must choose a low enough level of rigo.r so that the book is accessible to a wide audience. At the same time Malliaris must present an appropriate balance between the mathematical developments of the theory and a host of applications, so that the user is well prepared to read and contribute to frontier research in economics and fmance. Furthermore, since the mathematical background of the people who work in economics and fmance covers the entire spectrum, from those having skills equi valent to PhD's from the best mathematics departments to those with a rudimen tary understanding of fust-year analysis, Malliaris' task of constructing a book to appeal to such a heterogeneous audience becomes almost impossible: Nor is this all. The book must be of fmite size in order to keep the price fmite. These are important pedagogical difficulties. Dr. Malliaris has chosen the route of giving detailed citation to results that are further presented elsewhere. In short he has exposited what needs expositing and has told the reader where to go to fmd additional material. In this way he has prepared a "user's Guide" to an enormous mathematical-economic-financial lit erature. The hope is that a reader equiped with Malliaris' book will fmd a much easier path to the cutting edge of ongoing research using stochastic methods in econom ics and fmance over the alternative of hunting and pecking through the journals. Anastasios G. Malliaris is well-equipped to write such a novel work. Even though he already possesses a PhD in economics, he took several years of courses in the University of Chicago's economics and mathematics departments. He is a patient English language craftsman with an encyclopedic mind.
xii
Foreword
Let me give an example of Malliaris' expository skills that appears in his book. He was able to take raw material from my lecture notes, problem sets, examina tions, and oral delivery and machine this morass into readable material. Anyone who has attempted to write anything at all knows how much easier it is to com municate with the mouth than the pen. In conclusion, A.G. Malliaris has presented the profession with a bold and novel concept in the area of textbook and reference book writing. My personal opiillon is that he has solved a very hard constrained optimization problem admi rably well. I believe that the market will think so too. Chicago and Madison
W.A. Brock
PREFACE
Stochastic Methods in Economics and Finance
introduces the reader to certain
mathematical techniques by presenting both their theoretical elements and their applications. Topics such as martingale methods, stochastic processes, optimal stopping, the modeling of uncertainty using a Wiener process, Ito's lemma as a tool of stochastic calculus, basic facts about stochastic differential equations, the notion of stochastic stability and the methods of stochastic control are discussed and their usc i n economic theory and fmance is illustrated in numerous applica� tions. Among these applications we mention futures pricing, job search, stochastic a stochastic macroeconomic model, competitive fmn under price uncertainty,
capital theory, stochastic economic growth, the rational expectations hypothesis, the Black-Scholes option pricing theory, optimum consumption and portfolio rules, demand for index bonds, term structure of interest rate, the market risk adjustment in project valuation, demand for cash balances and an asset pricing model. Because the economics and fmance professions have accepted the superiority of dynamic deterministic methods over static analysis techniques, the time has come to further encourage the trend towards the application of dynamic stoch� astic methods. These methods are expected to capture the complexities, measurement errors and uncertainties associated with economic reality and to provide a way of mod� eling some of the researcher's pure ignorance about the future. Knowing the tech niques or the methods presented here will enhance the ability to attack success fully some of the difficult applied problems arising in economics and finance. This book alone will not provide sufficient intellectual capital for the economic researchers to be fully equipped to solve a wide range of problems demanding stochastic methods. It is almost impossible to produce a book meeting such a standard. However, if used as an introductory survey of advanced stochastic meth ods, this book could serve as an efficient and useful guide to an enormous math ematics, economics and fmance literature. Such, then is, the nature of this book: an introductory survey of advanced stochastic methods applied in economic anal ysis with detailed sections on further remarks and references to provide guidance for additional reading.
xiv
Preface There are pedagogical challenges in writing a book where no close substitutes
exist. For example, the content of the book, its mix between theory and applica tion, and its level of exposition are three of the difficult questions to be dealt with. Concerning content, we chose to present a broader rather than a narrower coverage by describing several methods. We should point out that some important methods, including econometric methods, stochastic difference equations techni
cellent sources on these subjects already exist. For the appropriate mix between
ques and discrete time stochastic control, are not included, primarily because ex theory and application, we chose to emphasize where possible each equally. Some
theoretical topics have been included for which there is no inunediate applica tion. Such is the case with several measure theoretic probability concepts and the notion of stochastic integration. The justification for including these topics lies in their usefulness for understanding subsequent theoretical developments with immediate applications. In illustrating the interplay between theory and applications we use two ap proaches. In Chapter
1, applications follow the presentation of martingale meth
ods and optimal stopping and the use of theory is immediately illustrated. In the rest of the book, mathematical methods are contained in Chapter
3
applications is natural in Chapter ters
2, while Chap
and 4 contain economics and fmance applications. Integrating theory and
1, but the separation of the two is more efficient
elsewhere to maintain the flow of exposition. Some methods such as Ito's lemma have been intensively discussed regarding both theory and application. Other methods, despite completion of their theoretical development, have not yet been widely applied and therefore must await the work of future researchers. Examples of these methods include continuous optimal stopping, stochastic differential equations and stochastic control. Furthermore, certain areas of applied research in economics and fmance, such as stochastic stability, have raised some difficult mathematical problems and appropriate mathematical methods are not yet fully developed. It is very difficult to exposit the methods in this book without making use of some of their measure theoretic probability underpinnings. For that reason we chose to make use of various fundamental notions such as a-field, probability space, measurable function, expectation and conditional probability. To make the book accessible to a wider audience, however, we kept the theoretical foun dations to a skeletal minimum, and we supplied the interested reader with refer ences where greater theoretical depth may be found. The primary audience of this book will include Ph.D. students in economics
but some MBA students specializing in fmancial theory or quantitative methods
and finance with aspirations of using stochastic methods as a tool of their research, may fmd this book helpful in their coursework or as a supplementary source. In
Preface
XV
addition, economic and fmancial theory researchers may use this book to teach themselves stochastic methods. Finally, some applied mathematicians who spe cialize in the methods described here may fmd the book useful because of the numerous applications it contains. An attempt has been made to keep mathematical prerequisites to a minimum. Many parts of this book could be understood by someone with a good background in analysis and basic probability theory. In the fust five sections ofChapter 1 we have collected some results from courses beyond analysis and probability that are needed forChapters 2, 3 and 4. Most exercises fall in two categories: they either infonn the reader of supple mentary facts with an indication of where to find an answer or are designed to develop computational skills. There are, however, some exercises, such as nos. (5), (9) and (l4)ofChapter 3, that move the reader into the area of model build ing. These are the hardest, and the reader is warned that they are suggested as possible illustrations rather than definitive contributions.
ACKNOWLEDGMENTS
A book does not exist without a content and in our case the content relies heavily on the pioneering developments of the many mathematicians and economists cited in detail in the pages that follow. I acknowledge my intellectual debt to all authors mentioned in the pages that follow and hope that this book will help usher the reader to some of the works cited. Professor W. Brock proposed the idea of writing this book about three years ago, and all during this period he generously supplied instruction, advice, encour agement and at the completion of the manuscript he offered to write the Fore word. He also gave permission to use his class, notes, to reproduce several of his exam questions and he contributed several sections which are starred in the table of contents. Insightful corrections and suggestions were offered by Jerry Bona (University of Chicago), F.R. Chang (University of Chicago), Patrick Brockett (University of California, Riverside), Mike Rothschild (University of Wisconsin, Madison), and
valuable instruction and clarifications on the subject matter of Chapter 4. Mike
anonymous referees. George Constantinides (University of Chicago) supplied
Intriligator provided encouragement with his interest in this book, and through his advice and support the writing converged to its completion. Several individuals showed interest in this work or helped in various ways: Boyan Jovanovic (Bell Laboratories),
George Kaufman (Loyola University of Chicago), Steve Magee
(University of Texas, Austin), Mike Magill (University of Southern California), John McCall (University of California, Los Angeles), Donald Meyer (Loyola Uni versity of Chicago), Ronald Michener (University of Virginia), Samuel Ramenof sky (Loyola University of Chicago), Scott Richard (Carnegie-Mellon University),
Brock and J. Ingersoll used portions of this book in courses they taught at the
Jose Scheink.man (University of Chicago), and Sam Wu (University of Iowa). W. University of Chicago and I presented parts of the book at a fmance seminar at
the University of Texas at Austin. From such a use several errors have been cor
rected. Barbara Novy and Jean Shenoha helped with the bibliographical research and did extensive proofreading. Mary Beth Allen and John Schmadeke provided editorial assistance and Carmela Perno typed various versions of the manuscript
xvili
Acknowledgments
accurately and efficiently. Dr. Ellen M. van Koten, economics editor, Mr. Leland K. Pierce, technical editor, and the staff at North-Holland Publishing Company were most coopetative and efficient during the production process. I am grateful to all, and for the errors that exist in the fmal product, almost surely, I take re sponsibility. I happily dedicate this book to my wife, Mary Elaine, for her many contribu tions in my life. Chicago
A.G. Malliari s
CHAFI'ER 1
RESULTS FROM PROBABILITY
The evolution of probability theory is due precisely to the consideration of more and more complicated observables. M. Loeve (1977, p. 7)
1 . Introduction In this chapter we present various ideas from modern probability theory in the form of definitions, theorems and examples. These ideas have been collected from a large number of sources and their presentation is intended to provide the reader with background material for the subsequent chapters of this book. Al though some sections in this chapter develop ideas at a rapid rate, there are sec tions on martingales and on optimal stopping problems which are more detailed. Several examples illustrate the applicability of probabilistic martingale theory and the theory of optimal stopping rules in economic analysis and finance. There fore, the reader may benefit from this chapter in at least two ways, firstly by de veloping an understanding of the theoretic underpinnings of the various applied probabilistic models, and secondly by being familiarized with martingale and op timal stopping theories which are actively used in applied research.
2. Probability spaces The occurrence or nonoccurrence of an experiment or a trial in which chance intervenes is usually called an outcome. The totality of outcomes of an experi ment are grouped together in a set denoted !by n. The elements of the set or space n are denoted by w and called sample points. A subset of n is called an
2
Stochastic methods ill economics and finance
event. As an example, for the number of unemployed as a proportion of the total U.S. labor force during next month, the space n is all rational numbers in the interval [0, 1], while w = 0.065 is an element or a sample point. Some events are: proportion of unemployed no more than 0.07, or, proportion of unemployed between 0.06 and 0.08. Intuitively, a probability is a valuation on some class of events. Let n be an arbitrary space consisting of points w. Certain classes of subsets of n are important in the study of probability. We now defme the class of sub sets called a-field or a-algebra, denoted by !F in this book. We say that ¥ is a a-field if the following three conditions are satisfied: (I) .Q E �, i.e. ¥" contains the space .Q itself; (2) A e.� implies A c E /F, i.e. if an event A which is a subset of n belongs to .tF, then the complement of A, denoted by A c, also belongs to .tF; and (3) A 1, A2, A3, ... E �implies A 1 U A2 U A3 . E :F, i.e. if a countable sequence of subsets of n belongs to �. then the countable union of these sub sets also belongs to .�. We call the elements of !F measurable sets. Note that for a given set n, the smallest a-field consists of the empty set, 0, and the set n itself, while the largest a-field consists of the power set of n, i.e. all the subsets of n. The largest a-field consists of 2 n subsets of n. The pair (.Q, §)is called a measurable space. Now let C§ be a class of subsets of n. The intersection of all a-fields con taining �I} is called the a-field generated by C!J and is denoted by a( t§ ). As an example we mention the a-field generated by the class of intervals (a, b] which are subsets of the real line R 1 This a-field is denoted by f/11 and its elements are called Borel sets. Note that�· is the smallest a-field containing the class of all intervals of R; in particular, it contains all open and all closed sets of R. Intui 1 tively, Yi is obtained by starting with intervals of R and forming repeated fmite and countable set-theoretic operations (unions, intersections and complements) in all possible ways. It was mentioned above that a probability is a valuation on a class of various events. We now make this notion precise. Let (.Q, .�) be a measurable space. A set function IJ. defined on :F is called a measure if it satisfies the following conditions: \ (I) JJ.(0) = 0; (2) A E §"implies 0 � JJ.(A) � oo; and (3) A1, A2, E.� and if An are pairwise disjoint, i.e. Ak nAm =0 for k -:l=nz, then .
..
•
•••
3
Results from probability
A measure of great importance is the Lebesgue measure, denoted by A, and defined on the class of Borel sets .o//1 of the real line R1 This measure assigns to every interval its length, i.e. •
=
A(a, b]
b- a.
The Lebesgue measure A can be extended in a straightforward manner to all Borel sets. For a detailed exposition see Ash (1972, ch. l ). Note that A(R 1) = oo and also that the Lebesgue measure of every countable set is zero. The Lebesgue measure A on ti/ 1 can be generalized to be defined on the Borel sets of the k-space Rk denoted by ::tk . A probability is a special kind of a measure denoted by P, where P(fl) = 1. Thus, for Ae :F we have 0 � P(A) � I. The triple (fl., !F, P) is called a proba bility space' where n is a noncmpty space of trials, :F is a a-field of subsets of n representing various events, andP is a probability measure defined on�. As an example consider as n all rational numbers in [0, I] denoting the unem ployed as a proportion of the total U.S. labor force in some future month. Let !F be the a-field of all subsets of this countable space n and let J.J.(w) be a non negative function defined on n such that �wen ll(w) = 1. Next, defineP(A) = �wEA J.J.(w) Then the triple (fl., .F,P) is a probability space. A special case of a probability space is that of a complete probability space. Let (fl., .tF, P) be a probability space and let N be a subset of this space. We say that N is negligible if there exists a set A E !F such that N is contained in A, and P(A) = 0. The probability space is complete if ff contains every negligible subset of n with respect toP. This concept is used in section 7 of this chapter. Consider a sequence {All} of events in a probability space (fl.,�' P). Define .
lim sup
n
An
and also lim inf
n
A ll
=
n 00
u 00
A n=l k=n k = {w: weAn u
for infinitely many n}
n A n=I k=n k = {w: weAn for all but finitely many n}. =
00
00
If all Ane.� then both lim sup11 An and lim inf11 An belong to !F. Next we need the defmition of independence before we close this section with the statement of an important lemma.
4
Stochastic methods in economics and finance
From elementary probability we recall that two events, A and B, are indepen dent if P(A nB) = P(A) P(B). We generalize this concept by stating that a finite collection of events A 1 , A2, ,An is independent if •
•••
P(A k n... nA k.) , I
=
P(A k,)
···
I
P(Ak.)
(2.1)
for each set k 1, , ki of distinct indices from 1, ... , n. An infmite collection of events is independent if each of its fmite subcollection is independent. The fol lowing lemma is useful and will be used later in this bQok. •••
Lemma 2.1 (Borel-Cantelli). Let {An} be a sequence of events in a probability space (n, /F, P). If :En P(An) < oo, then P(lim supn An)= 0. Also, if the sequence of events {An} is independent and �n P(An) = oo, then P(lim supn An) = 1. Proof. See Neveu {1965, pp. 128-129). 3. Random variables Let {n, :F) and (il', .Of"') denote two measurable spaces. A mapping X: n � n' is ( !F, !F' )-measurable if for each A'E
:F'
x-• (A')= {w: X(w)E A'}E.�. Intuitively, measurability of a mapping means that for every meaningful event in the image space n· there is a meaningful event in the domain space n. It turns out that an event is meaningful if it belongs to an appropriate a-field. The a-field generated by X and denoted by a(X) is the smallest a-field with respect to which X is measurable. A special case of importance is when the image space n' is the real line R 1 In this case we use as a a-field 9f 1 , i.e. the class of Borel sets. A function f: n R 1 is measurable if for each A E df 1 we have r-• (A')= {w: f(w) E A'} E §. A real function that is measurable in the sense just described is called a random variable. k Suppose that f: n-+ R is measurable. Then it is called a random vector. Note that f has the form •
I
f(w)= ift(w), .. , fk(w)) , .
-
where each component fj(w) is a real function. The mapping f is measurable if
Results from probability
5
and only if each component function f; is measurable. In many applications we assume that f is continuous, wluch implies that f is measurable. Let (n, !F, P) be a probability space, and (R 1 , Jf1, A) the measurable space with R 1 the real line, �1 the a-field of Borel sets, and A the Lebesgue measure, and then suppose that X is a random variable such that X: n � R 1 For all A E .'11 1 we define the distribution or law of X, denoted by P X' as •
P x(A)
=
P(X-1 (A))
= P{w:X(w) e A}.
(3.1)
The distribution of the random variable X assigns to a set A E ::if 1 in the target space a probability measure depending of course on X. The distribution function of X, denoted by F, is defmed by
F(x) = P{w: X(w) �x}
(3.2)
for x E R 1 We note that F is a nondecreasing function, right continuous such that •
F(x)�o as x�-oo, F(x)� 1 as x� oo.
There is a powerful theorem in probability theory that says that for a given dis tribution function Fhaving the stated properties,there can always be constructed a probability space (!1,.¥, P) and a random variable X such that F is the distri bution function of X. See Billingsley (1979, p. 15 9). The concept of a distribution function may be generalized for a random vec tor X such that X: n � R k. We then have for X E R k
F(x)
=
F(x1 ,x2, , xk) •••
=
P{w: xl (w) �Xt' , xk (w) �xk }. .••
(3.3)
Here X , , Xk are the k-components of X and we specify F as the joint distri 1 bution function. Let F be a joint distribution function of a k-random vector X = (X , , Xk). The marginal distribution function of a subcollection of m 1 components of X, where m �k, is obtained by replacing the unencountered argu ments with oo As an illustration,a two-dimensional, m = 2, marginal distribution can be written as •••
•••
.
6
Stochastic methods in economics and finance
F(x ,x2,oo, .. . 1
,co
)
A random variable X and its distribution Px have den sity f with respect to the Lebesgue measure iff is a non-negative rea) function such that for each A e ..If 1 ,
P{w: X(w)eA}= f f(x) d x .
(3.5)
A
Observe that the density function[ is determined only to within a set of Lebesgue measure zero. In the case where the random variable X has density f, then the density f and the distribution function Fare related by the equation
F(x)=
X
J
-oo
f(s)d s .
(3.6)
In many applications the distribution function F is continuously differentiable, in which case the derivative ofF can serve as the density f. There arc several examples of distributions that are well known from elemen tary probability, e.g. the binomial distribution (3.7) for x 0, 1, 2, ... , n. In this example we may take n = {0, 1, 2, ..., n} with.� consisting of all subsets of n , and define the random variable X as X(x) = x. Another example is the Poisson distribution with positive parameter m given by =
(3.8)
0, I , 2, .. . . Finally, we say that the random variable X has a normal dis tribulion with parameters J1 and a > 0 if
for x
=
1 f exp
Px(A) = --
...;2;0
A
r
-(x-JJ.) 2
2a
2
]
dx
(3.9)
for x eA and A e Jf1 Suppose that X1, .. . , Xn are random variables defined on a probability space (n, .�, P) such that X;: n R1, i = 1, 2, ... , n. We say that X1 , ... , Xn are inde pendent random variables if for all Borel sets A 1, .. ., A n, •
__,.
7
Results from probability
P{w: X1 (w)eA 1 ,
•••
, Xn(w)E: A n}
(3.10)
=P{w: X1(w)EA 1 } ···P{w: Xn(w)EAn}. There are two other equivalent definitions: X1 , variables if for x 1 , , x, real numbers
•••
, X, are independent random
..•
(3.1 1 ) Note the notation in {3.1 1 ) where the argument w has been suppressed for nota tional convenience. Alternatively, X1 , , X, arc independent if for x1 , , xn real numbers •••
F(x1 , •••,xn) =F 1 (x1 )
•••
•••
(3. 1 2)
F,, (xn),
where F is the joint distribution function of X1, , X, and F1 , ••• , Fn are the one-dimensional marginal distribution functions. The definitions above may be generalized to random vectors very simply. In (3 .1 0) we understand X; to be a random vector and A ; to be a k-dimcnsional Borel set. Furthermore, the definition of independence can be extended to an infinite collection of random variables X1 , X2 , simply by requiring that each fmite subcollection is independent according to {3.1 0) or the other equivalent definitions. For a sequence of random variables {Xn} defmed on (il, !F, P) and a random variable X, also defined on the same space, there arc three useful convergence concepts. First, if there is a set N E .
.•.
=
X"
�
X w .p J as
n � oo.
(3. 1 3)
Secondly, we say that the sequence of random variables {X,} converges probability to X if for each positive € we have
P{w: I Xn(w)- X(w) I > €} = P[ I X, - X I > E} � 0
in
(3.14)
as n � oo and we write p
X, �X as
n
� oo.
(3.15)
Stochastic methods in economics and finance
8
Finally, let {F,} and F denote the distributions of {X,} and X and suppose that as n � oo
f [(x)dFn (x)� f f(x) dF(x)
R
R
(3.16)
for every real-valued continuous bounded function f defmed on R. Then we say that {X, } converge in distribution to X and we write
Xn =>X as
n
� oo
.
(3.17)
Observe that (3.13) implies (3.15), and that (3.15) implies (3.17).
4. Expectation If X is a random variable defmed on a probability space (!1, IF, P) , the expecta tion of X is defined by E(X) = f XdP, n
(4.1)
provided the integral exists, i.e. fn XdP < oo. A question naturally arises: What is the meaning of the right-hand side of (4.1)? We immediately proceed to answer this question by defming the integral f X(w) dP(w) f X(w)P(dw) = f XdP. n n n =
(4.2)
First, assume that X is a non-negative random variable. For each finite decom position of the space n into sets A ;. i.e. a collection {A;} such that A ; E .�, A ; n A i = 0 fori =I= j and U;A; n, consider the sum =
� [inf l
wEA ;
X(w) )P(A ;).
(4.3)
Using (4.3) we now define the meaning of(4.2) as follows: f XdP =sup� [inf X(w) ]P(A ;) . l wEA·I n
(4.4)
In (4.4) the supremum extends over all finite decompositions {A;} of n into �-sets.
Results from probability
9
Secondly, observe that an arbitrary random variable X may be written as
{4.5)
X = X+- x-
'
where x+ denotes the positive part of X dcfmed by x+(w) =
{
X(w)
if 0 �X(w) �oo,
0
otherwise,
(4.6)
and where x-denotes the negative part of X defmed by X1w) =
{
-
X(w) if -oo �X(w) �0,
0
otherwise.
(4.7)
Note that x+ and x-are non-negative and measurable and therefore (4.4) applies to both of them. Thus, the integral of an arbitrary random variable is defmed by f XdP = f X+dP- f x-dP.
n
n
n
(4.8)
If each term of the right-hand side of (4.8) is finite then we say that X is integ rable. Since I X I= X'" + x-. note that X is integrable if and only iffn I X I dP
n
n
(2) If f and h are integrable and o: and f3 arc finite real numbers, then o:f + (3h
is integrable and also
10
Stochastic methods in economics and finance
f (a[+ {3h)dP= a ffdP+ {3 f h dP.
n
n
n
(3) Iff and h are integrable and iff �h w .p.l then ffdP � f lzdP. n
n
(4) Iff is integrable then
f[dP � f lfl dP.
n
n
(5) Iff is integrable then for A E .'F, fA fdP < oo
.
For a proof of thls theorem sec Ash (1972, pp. 41-42). For sequences of random variables we state three basic theorems. Theorem 4.2. (Monotone convergence). Let {X,} be an increasing sequence of non-negative random variables on (U, :F, P) and let X, (w)-+ X(w) w.p.l. Then
f X, dP-+ f XdP.
n
n
Observe that this theorem justifies interchanging the limit and expectation opera tions, i.e. lim,-+oo E(X,) = E(lim,-+ .... X,). Next we define lim sup, X, and lim inf, Xn for a sequence of real random variables as follows:
( (
) )
lim inf X, (w) = sup inf X k (w), n n k �n lim sup X, (w) n
=
inf sup Xk(w). n k>n
Theorem 4.3. (Fatou). For a non-negative sequence of random variables {Xn} on en, !F, P),
f lim inf X, dP �lim inf f X, dP. n n
n
Theorem 4.4. (Dominated convergence). Let {X,} be a sequence of random variables and Y an integrable random variable all on (U, .tF, P) such that I Xn I � Y w.p.l, for all n . If Xn -+ X w.pJ, then X and X,1 are integrable and
f X,. dP -+ f XdP.
n
n
Results from probability
11
Proofs of these theorems may be found in Ash (1972, pp. 44-50). With the above background on integration we now state some fundamental defmitions. Let X be a random variable on (n, !!F, P) and let k > 0. We say that E(Xk) is the kth moment of X and that E ((X - E (X)l) is the kth central moment. When k I , E(X) is usually called the mean of X and when k = 2, the second central moment is called the variance of X, written as var X and denoted by a2 , t.e. =
•
a2
=
var X= E((X- E (X))2 ),
(4 .9)
provided E(X) < oo. The positive square root a is called the standard deviation. Note that if k > 0 and E(XK} < oo , then E(x2) < oo for 0 < Q < k. Also, if X1, , Xn arc independent random variables on (n, .¥, P) such that E (X;) < oo for all i = 1 , 2, ... , n, then E(X1 , , Xn ) < oo and also •••
•••
E (X1 , ... , Xn ) = E(X1 )
···
E(Xn );
(4 .1 0)
furthermore
n var (X1 + ... + X,) = _L var X;. t=l
(4.11)
For two random variables, X and Y, each having a finite expectation, the co variance of X and Y is defined by cov (X, Y)
=
E((X- E (X)) ( Y - E(Y)})
=
E (XY)- E(X) E ( Y).
(4.12)
From (4.1 0) observe that if X and Y are independent, then cov ( X, Y) = 0; however, the converse is not true. Consider the random variables X and Y and suppose that their variances, de noted by a,i and a}, respectively, are nonzero and finite. We define the co"ela tion coefficient between X and Y, denoted by p (X, Y) as p (X, Y) = cov (X, Y)faxay.
(4.13)
We close this section by stating a useful fact. Let X be a random variable on (n, Y, P) with distribution function F. Assume that g: R -+ R is a measurable function and let Y = g (X). Then E(Y) = f g (x} d F (x). R
(4.14}
12
Stochastic methods in economics and finance
In various applications it becomes easier to compute the expectation of a ran dom variable by integrating over R instead of over n. In such cases we may use (4.14} by letting g(x) = x. S.
Conditional probability
Recall from elementary probability that the conditional probability of a set A given a set B, denoted by P(A 1 B), is given by
P(A I B) = P(A nB)/P(B),
(5.1)
provided P(B) -:1= 0. A conditional probability is associated with events in a given subset of the space n. Intuitively ,a conditional probability represents a re�valua tion of the probability of A occurring in light of the information that B has al ready occurred. In this section we study conditional probability in a general con text for a space (!2, !F, P), where the conditional probability of a set A is defined with respect to a a-field � in .'F. The notation used is P [A I f/J ] and the intuitive interpretation is analogous to (5 .l ), in the sense that the conditional probability of the set A is being evaluated in light of the information available in the a-field c!J , with •!J contained in .OF. At this point a digression is necessary to present two definitions and the Radon-Nikodym theorem which will establish the existence of conditional probability. Consider two measures v and P on a measurable space (!2, §). We say that the measure v is absolutely continuous with respect to the measure P, or equivalently that v is dominated by P, if for each A E .
Theorem S . l . (Radon-Nikodym). Let (!2, -� be a measurable space. Suppose that v and P are two a-finite measures on (!2, .'F) such that v is dominated by P. Then, there is a non-negative measurable function f such that
v(A) = f fdP A
(5.2)
for all A E :F.
For a proof see Ash (1972, pp. 63-65). The function f in (5.2) is called the Radon-Nikodym respect to P and is denoted as dv/dP.
derivative of v with
Results from probability
13
With the above brief background we motivate the definition of conditional probability of a set A given a a-field �IJ in .
(5.3)
for all G E �/}. Restrict the probability measure P to the a-field './J and note that if P (G) = 0 then v (G) = 0 , which means that the measure v, as defmed in (5.3), is dominated by P. For the measurable space (n, (.§), the facts that v and P are a fmite and also that v is dominated by P imply that by the Radon-Nikodym theo rem there is a non-negative random variablefsuch that (5.2) holds. Combining (5.2) and (5.3) we obtain that v (G) = P (A n G) = ff dP
(5.4)
G
for all G E (/}. The random variable f is precisely the conditional probability P[A I 0' ] which has, by the preceding analysis, two properties: {I) P[A I �/}] is measurable with respect to
f P[A I �IJ] dP = P(A n G).
(5.5)
G
There are many random variables P[A I •§ ) which are equal w.p.l. For a given P[A I �/}] we say that this random variable is a version of conditional probability. When the conditional probability of A is with respect to a a-field generated by a random variable X we write P[A I a (X)], or simply P[A I X], which can be gen eralized to the conditional probability of A given an arbitrary sequence of random variables X1 , X2 , written as P[A I a (X1 , X2 , ...)] or P[A I X1 , X2 , ]. Some of the basic properties of conditional probability are summarized in the next theorem. •••
•••
Theorem 5.2. Let (n, .
P(UAn I c'fJ ]= � P[An I �]. n n
14
Stochastic methods in economics and finance
(3) If A E .� and BE 9'-- such that A CB, then
P[B - A I � ] = P[B I � ] - P[A I �4 ] and also P[A I (1 ] �P[B I <§ ] . (4) If {A n} is an increasing (decreasing) sequence with A as its limit, then P[An I q; ] increases (decreases) to P[A I C§ ] as its limit as n -+ oo .
Having briefly discussed conditional probability we next present the notion o f
conditional expectation.
Let X be a non-negative integrable random variable on (n, .¥, P) and defme the measure v on '!J by v(G) = f XdP
(5.6)
v(G) = f fdP.
(5.7)
G
for G E <§, where �1 is a a-field in .� As earlier in this section, if P is restricted to rJ then v, as defined in (5.6), is dominated by P and an application of the Radon Nikodym theorem yields the existence of a random function f such that G
This random variable is denoted by E [X I �] and called the conditional expected value of the random variable X given the a-field q;_ Note that E{X I �1] satisfies the two properties: (1) E[X 1 0' ] is integrable and �t; measurable. (2) From (5 .6) and (5 .7) we obtain that for G E 'd f E[X I � ] dP= f XdP.
G
G
(5.8 )
For a general random variable X, not necessarily non-negative, its conditional expected value can be obtained by using (4.5) and defining E[X I I§ ] = E [x• I <:4 ] - E[X- I '!J ] . Since there are many E[X I <§] which vary on a set o f probability zero, one such given conditional expected value is called a version of the conditional expectation of the random variable X. It is worth mentioning that E[X I <5] is a random vari-
Results from probability
15
able whose value depends on w , i.e. E[X I ��] (w), and we interpret this to mean the expected value of X given the information in the a-field �IJ. Conditional expectation satisfies various properties reported in the theorems below. In theorem 5.3 we assume that X and Y are integrable.
Theorem 5.3. Let .r:F, P) be a probability space and �IJ a a-field in .'F. (I) If X is a constant random variable, i.e. X = c, c E R, then E LX I �t} 1 =
(n,
c
w .p.l. (2) If X and Y are random variables and a, b E R, then
E [aX ± b Y l �'7 ] = aE[X I �IJ] ± b E [ Y I �4 ] . (3) If X and Y arc random variables such that X � Y w .pJ , then
E [X I �tJ ] � E[ Y I ·§ ] . (4) For X a random variable For a
I E[X I •.tJ ] I � E[ I X I I ��] . proof sec Billingsley (1 979, p . 397).
Theorem 5.4. Let
X be a random variable on (n, ;F, P) and suppose that it is
integrable. Suppose that
E[E[X I �IJ2 1
'!J 1
and
arc a-fields in � such that
1 �4 1 ] = E[X I �IJ1 ]
'!J 1 C
�IJ2 • Then
w.p.l.
For a proof see Tucker ( 1967, p. 2 1 2).
X and Y be random variables on (n, .¥, P), and suppose Y and XY are integrable and X is �IJ-measurablc. Then Theorem 5.5. Let
E[XY I � ] = XE( Y I �1 ]
for ·.tl a a-field in ."F.
w.p.l
For a proof sec Tucker (1 967, pp. 2 1 3-2 1 4). Finally we state Theorem 5.6. (Conditional form of Jensen's inequality). Let
h be a convex
16
Stochastic methods in economics and finance
X a random variable on (n, !IF, P) with t'!J a a-field in .'F. Sup X lz (X) arc both integrable. Then lz(E[X I � ]) � E [h(X) I �] w.p.l.
function on R and pose that and
For a proof see Tucker (1967, p. 2 1 7). We close thls section by stating two important facts. Suppose that is a random variable on .�, and let C!J be a a-field in !F. In our discussion thus far we have explained the meaning of conditional proba bility whlch readily extends to the case of a random variable since 1 I t§] = for We also explained the mean I (§ ], with = ing of conditional expectation, writ ten as I 1J ] . For both concepts of con ditional probability and conditional expectation we would like to associate a probability distribution function. In this analysis we follow Billingsley (1979, p. 390) and we state
X
P[A
(n, P)
A {w: X(w)eA'}
A ' e�1 • E [X
P[X E A
X on (n, !F, P) with conditional probability P[A I �4] = P[X eA' I rg ) P[w: X(w)eA' I � ] for A IE 9f 1 there exists a function p (A'' w) defined for A ' E !Jf 1 and wE n satisfying two properties: ( I ) for each w E n, p(A ', w) is, as a function of A', a probability measure on 9f 1 , and (2) for each A 1 e 9f 1 , p (A', w) is, as a function of w, a version of P[XeA I �1 ]. The probability measure p ( · , w) is called the conditional distribution of X given
'
·,
R
6. Martingales and applications
Having presented the elements of conditional probability we now study martin gale theory and some of its applications in economics and finance. Martingale theory uses conditional probability extensively so the usefulness of the previous section will shortly become apparent. Let us also mention that martingale theory,
Results from probability
17
like various other areas of probability theory, has its origins in notions of gambling in the following sense: let X1 , X2 , be a sequence of random variables defmed on a common probability space Within the gambling context these random variables denote the gambler's total winnings after 1 , 2 , ... , n, ... trials in a succession of games. The gambler's expected fortune after trial n + 1 , given that he has completed n trials already, is denoted as E[Xn + 1 1 X1 , , Xn ]. As in the previous section E [Xn + 1 I X1 , , Xn ] denotes the expectation of Xn+ 1 condi tioned upon the a-field generated by the random variables X1 , , Xn . Intuitively, the a-field generated by x. ' .. , xn contains all the past information of the gam bler's fortune up to and including the nth trial. If a game is fair then the gambler after the (n + 1 )st trial will expect on the average to be neither wealthier nor poorer than he was before this trial, i.e. •••
(n, �' P).
•••
•••
•••
.
(6.1) Stated differently, eq. (6.1 ) , called the martingale property, states that in a fair game the gambler's fortune on the next play is on the average his current fortune and is not otherwise affected by the previous history. If instead of = in (6.1) we put � or �' then the game is favorable or un favorable, respectively. With this motivation we state the defmition of a martin gale.
6.1
Definitions
Let X1 , X2 , be a sequence of random variables defined on a common proba bility space (U, !:!', and let !F1 , !F2 , be a sequence of a-fields all belonging to !F. The sequence {(Xn , .'Fn), n = 1 , 2 , ... } is a martingale if for each n it satis fies the four conditions below: •••
P)
•••
( 1 ) .
Condition ( I ) states that { .�n }, n = 1 , 2, ... is an increasing sequence of a fields in .�. Intuitively, the requirement of an increasing {SFn } implies that the :tmount of information contained in the sequence of a-fields { ffn } is increasing. This is called the monotoneity property of the a-fields { .'Fn } in.¥" and it attempts
Stothastic methods in economics and finance
18
to capture the practical idea that the past to time n + 1 includes more events, information or history than the past to time n. The overall informational struc ture represented by the monotonically increasing sequence { ::/'11 } captures the concept of learning without forgetting. In some applications .
n .
and X1 , ... , X, continue to be measurable with respect to the new a-field a(Y, X1 , ... , Xn). It is theoretically natural and in many applications appropriate to allow a-fields { .�:, } to be larger than the minimal ones a(X 1 , ... , Xn ). Fama (1 970) has used various sizes of a-fields to denote various degrees of information. Condition (3), called the integrability property says simply that X, is integrable, i.e. the expectation of Xn is finite. Finally, the martingale property expressed in condition (4) says that Xn is a version of E [X,.+ 1 I :F,, ] and in a gan1bling context this condition indicates that the game is fair. Note that condition (4) is equivalent to (6.2)
for A E .
for A E .
A
A
(6.4)
Results from probability
19
for A E :�, . Therefore condition (4) and (6.4)are equivalent. The same reasoning used inductively yields
f x, dP = f x,
A
A
..
1 dP = ... = f x,+k dP A
for A E :F, C .�r+ 1 C ... C .,;;;r + k and k � 1 . This means that X, is a version of E [Xn +k 1 -�, ] and therefore the martingale property in condition (4) can also be written as
Some additional definitions arc appropriate. Condition (4) still makes sense for non-negative X, which docs not satisfy condition (3). For such non-negative X, which satisfy conditions (1 ), (2) and (4) but not necessarily condition (3), we say that {(X, , :�, ), 11 = 1 , 2 , . . } is a generalized martingale. The sequence {(X, , .'F, ), 11 = 1 , 2, ... } is a submartingale ifit satisfies conditions (1 ), (2), (3) and .
(4*)
E[X,+ 1 j :F,,] � X, w.p.l.
If instead of > in (4*) we put �, then {(X, , .'F, ), n = 1 , 2, ... } is called a supermartingale. Using (6.4) we state that {(X, , -�,), n = I , 2 , ... } is a submartingale if and only if I xn+ l d P � I X, dP A A
for A E .OF, and
A E .-F,
{(Xn , .� ), 11 = -,
I x,+ 1 dP� f x, d P. A A 6.2.
(6.5) 1 , 2 , ... } is a supermartingalc if and only if for
(6.6)
hxamplcs
With the notions stated above we now give some examples of a probabilistic na ture following Billingsley ( 1 979, p. 408) and Doob ( 1 97 1 ), before we illustrate the application of the martingale concept in economics and finance. (1) Let (n, .'¥', P) be a probability space, let v be a finite measure on .�, and let -�1 , -�2 , . be a nondecreasing sequence of a-fields in .'F. Suppose that v is dominated by P when both arc restricted to .� , i.e. suppose that A E -�, and ..
Stochastic methods in economics and finance
20
P(A) = 0 imply v(A) = 0. By theorem 5.1 there exists a Radon-Nikodym deriva tive of v with respect to P, when both are restricted to � , and denote it by Xn . Note that Xn is measurable � , integrable with respect to P and by (5 .2) it satis fies for A E � v(A) =
f Xn dP.
(6.7)
A
But A E g;;, implies A E -�1+ 1 so that v(A) = fA xn + 1 dP. Therefore (6.4) holds and {(Xn , -�), n = 1 , 2 , ... } is a martingale. (2) Another example is this; suppose Y is an integrable random variable on (n, � P) and { .� } is a non decreasing sequence of a-fields in .
i.e. Xn is the conditional expectation of Y conditioned on the a-field � - From the very definition of conditional expectation in the previous section X, is both measurable and integrable. Furthermore, condition (4) holds since E [Xn + 1 I :F, ]
=
E [E [ Y 1 �� + 1 ] 1 � ]
= E[Y I�] = xn by theorem 5.4. Thus {(Xn, .� ), n = I , 2, ... } is a martingale, obtained from suc cessive conditional expectations of Y as we know more and more. (3) This example supposes that Y0 , Y1 , Y2 , ... are independept random vari ables with Y0 = 0, E( I Yn I) < oo and E ( Yn ) = 0 for all n ;a: 1 . Defme X0 = Y0 and Xn = Y1 + ... + Yn for n ;a: 1 . Then Xn is a martingale with respect to the a-field generated by Yn. Condition (l) holds because the a-field generated by Y. , ... , Yn , denoted by a(Y1 , ... , Yn ), is contained in the a-field generated by Y1 , ... , Yn + 1 , denoted by a(Y1 , ... , Y,+ 1 ), i.e.
Condition (2) holds since measurability is preserved by addition. Condition (3) follows from
Finally, condition (4) holds since
Results from probability
21
E[Xn +t l Yo , ... , Yn] = E[Xn + Yn + t l Yo , ..., Yn ] E[Xn I Yo , ..., Yn ] + E[Yn + t l Yo , ..., Yn ] Xn + E (Yn + t ) = Xn. =
=
In establishing condition (4) note that theorem 5 .3 is used. Also note that E[ Yn + 1 I Y0, , Yn ] = E(Yn + 1 ) follows from the assumption that Y0 , Y1 , are independent random variables. (4) As an example of a submartingale assume that {(Xn , .�), n 1 , 2, ... } is a martingale. We claim that {( I Xn I , �), n = I , 2, ... } is a submartingale. Condi tion ( 1 ) holds trivially since we consider the same family of a-fields, while condi tions (2) and (3) hold because measurability and integrability of Xn imply the same for I Xn 1 . Condition (4*) follows from theorem 5.3 because •••
•••
=
6.3. Applications in economics and finance The concept of a martingale has found several applications in economics and fmance. Below we present some such applications. ( 1 ) In this application we follow Samuelson { 1965), with minor modifica tions, to show that futures pricing, under certain conditions, is a martingale. Let . .. , X1, Xt+ 1 , , Xt+ T ' . .. represent a sequence of bounded random vari ables defined on a probability space For example, this sequence may represent a time sequence of prices, say spot prices of wheat or gold. X, denotes the present spot price and x,+ T denotes the spot price that is to prevail T units of time from now. The assumption of boundedness of the random variables is not too restrictive because commodity prices are always finite. An economic agent is assumed to know at least today's price as well as the past prices. In the language of probability we assume that the economic agent knows all the available infor mation generated by the process which is in the a-fields �, where •••
�
=
(n, ff, P).
o(X0 , X1 , , X,).
(6.8 )
•••
Note that .'F, contains in particular the past price realizations of the process. These prices, denoted by x0 , x 1 , , x1_ 1 , x1, are specific values of the process for a Specific W E fl, where, •••
Stochastic methods in economics and finance
22
Xo(w) :: Xo ,
.• .
, x, _ l (w) = x , _ J ' X,(w) x,. =
The economic agent cannot know with certainty tomorrow's price, X,+ 1 , or any future price, xt+ T ' However, as time goes on his information increases because he observes additional price realizations. Needless to say .� is a monotone in creasing sequence by the definition in (6.8). Parenthetically we mention that Fama ( 1 970) introduced the terminology weak, semi-strong, and strong informa tion sets to describe information sets .'F, which include past values of the process, all publicly available past information, and all past information both public and internal to an economic agent, respectively. In this present example .� contains at least weak information which guarantees the measurability of X,. Next consider tuday's futures price quotation for the actual spot price that will prevail T periods from now. We use the notation Y(T, t) to write the futures price that will prevail T periods from period t and quoted at t. Let another period pass; then the new quotation for the same futures price is written as Y(T- 1 , t + 1 ). Thus, we have a sequence Y(T, t ), Y(T- l , t + 1), ... , Y(T- n, t + n), ... , Y(l , t + T- 1 ).
(6.9)
Samuelson's ( 1 965) fundamental assumption is that a futures price is to be set by competitive bidding at the now-expected level of tlze terminal spot price. This is similar to Muth 's ( 1 96 J ) rational expectations hypothesis and can be writ ten as (6. 1 0) for T = I , 2, ... Note that (6.1 0) makes sense because the conditional expecta tion of Xt+ T with respect to .� exists and this is SO because Xt+ T is integrable. The integrability of X,+ T follows from the assumption of the boundedness of Xt+ T made at the beginning of this application. Now we are ready to establish that (6.9) is a martingale. The a-fields associated with (6.9) are .�,.c;;;+ l , ... ,.�+n' .. , .-F,+ T- 1 . Such a-fields defined in (6.8) form a monotone increasing sequence whkh satisfies the first condition of a martin gale. The measurability and integrability conditions of (6.9) follow from eq. (6.1 0) and the properties of conditional expectation of x,+ T ' More precisely, E[Xt+ T 1 .�] is measurable .� and by (6.10) so is Y(T, t). Furthermore, E [X,+ T I .-¥, 1 is integrable and .
.
(6.1 1 )
Results from probability
23
for A E .� and so is Y(T, t) by (6.1 0). The right-hand side of (6.1 1 ) follows from the boundedness assumption of Xt + T · Thus, we only need to establish the mar tingale property,
E [ Y(T
-
l , t + 1 ) i .�] = Y(T, t).
(6.12)
Eq. (6.12) is easy to establish by using (6.1 0) and theorem 5.5. More specifically, E [ Y(T - l , t +
I) 1 :� )
= = =
E[E[Xt+ T I �+ t ] 1 .� ] E(Xt+ T l.q;;; ] Y(T, t).
Thus, futures pricing under the assumptions stated is a martingale. The theoretical paper of Samuelson ( 1 965) summarized in tllis application and the work of Mandelbrot (I 966) generated great interest in econometric test ing of the properties of stock prices. Although Samuelson's paper establishes the martingale property for futures pricing rather than for an equity asset, a share of a stock may be regarded as a sequence of futures claims due to mature at succes sive intervals. Thus, the martingale property properly applied to stock prices may be used as a measure of capital market efficiency. A capital market is efficient if the prices of the securities incorporate all the available information. Here we briefly report some findings of Jensen ( 1 969) and refer the interested reader to Fama (1 970) for a detailed survey of the theory and empirical findings on effi cient capital markets. Jensen distinguishes between weak and strong information and he proceeds to test the strong martingale hypothesis where the expectation of prices T periods from /, denoted by xt+ T ' is conditioned upon all information available at t. As stated earlier such information includes the past values of the process and information both public and internal to the firm. Notationally, Jen sen (1 969) writes e, for a-fields that contain all such information. Then, the strung martingale hypothesis is written as (6. 1 3) where, according to Jensen ( 1 969),[(1) represents the norm11l accumulation rate which depends on the length of the period T. Note that (6.13) actually represents a submartingale for the sequence X1, Xt+ 1 , , Xt+ T since, •••
24
Stochastic methods in economics and finance
holds for f(T) � 1 . Jensen's ( 1 969) empirical analysis of 1 1 5 mutual funds shows that current prices of securities incorporate all available information and there fore the best forecast of future prices is the present prices plus a normal expected return over the period T. We close this application with a remark on Samuelson's ( 1 965) sequences . .. , Xtt ... , Xt+ T • ... , and their unconditional expectations in (6.9). Note that these sequences are assumed to be given exogenously. LeRoy ( 1 973) has attempted to derive the martingale property when the assumption of exogenously given se quences is relaxed. He analyzes the relation between the riskiness of a stock and the risk aversion of investors to formally derive endogenously probability distri butions of rates of return. LeRoy (1 973, p. 437) concludes "not that any par ticular systematic departure from the martingale property is to be expected, but only that under risk aversion no rigorous theoretical justification for an exact martingale property is available". Ohlson (I977), in commenting on LeRoy's (1 973) paper, shows that the martingale property holds when investors have con stant relative risk aversion and the percentage change in dividends is stationary. (2) The application of futures' pricing as a martingale can be generalized fol lowing Samuelson (1 965). Let o: = ( I + r)-1 , where r is a measure of forgone safe interest and postulate (6.14) instead of (6.1 0), i.e. we allow the conditional expectation of the spot price, T periods from now, to be appropriately discounted. Then the sequence Y(T, t), . .. , Y(T - n, t + n), ... , which satisfies the axiom of expected present discounted value in (6.14), is a submartingale with respect to the a-fields g;; , ... , g;; + n' . . . . Here we assume that the a-fields include at least weak information. To establish that the sequence is a submartingale we only check the submar tingale property. Conditions (1 ), (2) and (3) are easily established, as in the pre vious application. Note that in the economics and fmance literature conditions (1 ), (2) and (3) are seldom checked carefully. The reader, however, may use the analysis in the previous example as a model in establishing the monotoneity, measurability and integrability conditions. The submartingale condition is obtained by using theorem 5.5 and eq. (6.14) as follows:
E [ Y(T - I , t + 1 ) I.�] T = E [o: - 1 E [Xt+T I §,+ ] I _:F, ] 1 T = o: - 1 E[E[Xt+ T I .�+ ] 1 .::?', ] I
Results from probability
25
exT - 1 E[Xt+ T I ff, ] = exT- l ex- T Y(T, t) = ex- 1 Y(T, t) =
+
r) Y(T, t) � Y(T, t). = (1
ln this application the discounted futures price will rise in each period by a percentage equal to
r.
Samuelson (I 965) uses this application to provide a ra
tional explanation of the doctrine of
normal backwardation. The moral is this:
the martingale property of futures pricing establishes that all methods used to read out of the past sequence of known prices any profitable pattern of predic tion are doomed to failure. {3) In this application we follow Samuelson {1 973) to show that under cer tain conditions stocks that are capitalized at their expected present discounted values satisfy the martingale property. Although our purpose is to establish the martingale property, before we do so in this application we generalize the fa miliar
present discounted-value rule ofcapitalization.
Let
x,, ..., xt+ T be a sequence of dividends of a given stock paid out at time
t, ... , t + T, on each dollar invested at time t, ... , t + T. Suppose that the discount rate is r and remains constant. Allowing r to change per unit of time does not add any new insights into the analysis; it just complicates the notation. Initially, we assume that
x,, ..., xt+T are nonrandom and w e write the familiar equation
(6.1 5)
where
V, is the value of the stock at time t obtained from the present discounted
value rule of capitalization. Using (6. 1 5 ) we can obtain the value of the stock
V,+ 1 , as follows: xt+ l + T � � = � - L1 T=l (1 + r)T T=2
next period,
v.1+ 1
_
Using {6. 1 5 ) and (6.16) it follows that
(6.16)
Stochastic methods in economics and finance
26
which can be written as (6.17) Eq. (6.17) is useful because it expresses the value of the stock next period as a function of its current price, V,., the discount rate r, and the stock's dividend at the end of next period, x1+ 1 • As a special case, if r V,. = xt+ l , then V,.+1 = V,. , which means that if the discount rate is equal to the rate o f return, x1 + 1/V,, then the value of the stock will remain the same. With the above review we now generalize eqs. (6 . 1 5) and (6.17) to make them stochastic. Let (n, .�, P) be a probability space and xt(w), ... , xt+ T (w), be a sequence of random variables which denotes stock dividends. In what follows we delete the w's to simplify the notation. For each random variable xt+ T ' T 1 , 2 , ... , we assume the existence of a distribution and of a conditional expectation. The sequence of a-fields in !F is denoted by .¥,+ r and it contains at least weak information , i.e. a(xt, ... , xt+ T) C .�+ T · This means that the investor knows at least the dividend history of the stock. At this point we can immediately generalize (6 .15) by writing it as =
vt
=
=
=
E [ V, I .� ]
E[ 00
�
T=1
-
xt+ T
�
(E
T::J (1 + r)T
L
xt+ T
T { 1 + r)
.-J ]) .
.'?',
·�
(6.18)
Note that the stochastic generalization of (6.17) is also easy. As a first step, compute vt+ 1 using (6.16) and (6.17)
vt+ 1 - E [ V,+ 1 I ff,+ t ] =
E[ �
xt+ T
T=2 (1 + r)T - 1
+
·� .
J
Next, observe that from (6.18), (6.19) and theorem 5.5 we get
(6. 1 9)
Results from probability
=E
=
[[
E
E
27
f
T=2
(
� ± xr+ l �
+
�
xt+ T
T=2 ( I + r)T- 1
) ·�J .
(6.20)
Finally, multiplying both sides by 1 /( 1 + r) and using the definition of v1 in {6.18), we conclude
E [vt+ 1 I g;=; ]
[
oo
xt+ T ( I + r)T
=
( 1 + r) E
=
{I + r) v1 - E [xt+ 1 1 .� ] .
�
T=l
(6.2 1 )
Eq. (6.2 1 ) is the stochastic generalization of (6. 1 7) and it can be used to help us decide under what conditions the sequence v1, , vt+ T is a martingale. In other words we ask: Under what conditions does E[vt+ 1 I .¥, ] = v1? The answer is when rvr = E [xt+ 1 1 .� ] . This means that the martingale condition holds when the discount rate is equal to the conditional expected return of the stock. We conclude therefore that the sequence vr vt+ T ' T = l , 2 , ... of discounted expected values of a stock is a martingale provided •••
,
(6.22) From this last equation we at once decide that v1, v1+ T ' T = 1 , 2 , ... , is a sub martingale if in (6.22) instead of = we have � - Tltis is so because (6.2 1 ) with (6.22) having � instead of= yields
The submartingale property says that the conditional expected value of the stock next period is greater or equal to its current value. This is so because the condi-
28
Stochastic metlrods in economics and finance
tional expected rate of return is smaller than or equal to the discount rate. We close this application with an important remark. Recall that in this appli cation we assumed the existence of conditional expectation for the sequence x,, ... , x,+T but we did not explain the way the individual investor forms his ex pectations .. A more complete analysis would require an individual investor to form his subjective expectations y , , . . , Yt+T as a sequence of random variables where y t+ T• T = 1 , 2, ..., denotes the investor's expected rate of return T periods from now. Having done so, a relationship needs to be established between Yt+ T and xt+T· At this point, Samuelson's (1965) axiom of expectation formation or Muth's (1961) rational expectation hypothesis can be used to postulate .
(6.23) Eq. (6.23) links the investor's subjective expectations to the markets' objective conditional expectations. Thus, we may start with an investor and his subjective expectations yt+ T• use (6.23) to link the subjective expectations to the objective ones and proceed from there using xt+T as was done above. (4) In this application we follow Hall (1978) to study a simple model of intertemporal stochastic optimilization to obtain a martingale property for the marginal utility of consumption. Consider an individual with a strictly concave one-period utility function u ( ) whose life-cycle consumption problem under uncertainty is given by •
max E
[£
,
(6.24)
T=O
subject to the condition
N
Tl:=O
(6.25)
•
In (6.24) and (6.25) the notation used is: ct+T T = 0, I , . . . , N is consumption, S is the rate of subjective time preference, r is the real rate of interest, wt+T is earnings and A , is the value of assets at time t. The individual considers the prob lem today, at time t, and his length of economic life is t + N periods. The a-fields each include information, at least, about the sequence of con F,, �+ 1 , .•• :F,+N sumption and the sequence of marginal utility of such consumption, respectively, at t, ... , t + N. Hall ( 1978) derives a necessary condition for this ma.x.intization problem of the form ,
Results from probability
29
(6.26) where u' ( ) denotes marginal utility. Assuming B = r, (6.26) says that the se quence of marginal utilities satisfies the martingale property. If we assume that r < B , then (6.26) says that the sequence of marginal utilities is a submartingale. Hall (1978) also shows that if the stochastic change in marginal utility from one period to the next is small, then consumption is a submartingale, written as ·
(6.27)
where X, is given by 'A1
- ( 11 + B )
u ' (c1)Jc1 u " (c1)
_
+r
•
(6.28)
From (6.28) we obtain that X, � 1 because u'' < 0 and r is assumed to be greater than B . Before we close this application we mention that Foldes (1978) has obtained martingale conditions for a dynamic discrete time model of stochastic optimal saving. 6.4.
Basic theorems on martingales
We conclude the section on martingales by stating some useful facts in the form of theorems. These facts have not yet been widely used in the economics and fi nance literature on martingales and their presentation is motivated by probabilistic interest. There are several excellent sources where these and additional results on martingales may be found, such as Ash (1972), Billingsley (1979), Doob (1953), Meyer (1966), Neveu (1975) and Tucker (1967). Here we follow Ash (1972) and Billingsley (1979).
.
Theorem 6.1. ( 1 ) Let X1 , X2 , . . be a martingale relative to .�1 , F2 , . . . on (n, !F, P). If 4>
.
is a convex function and if l/>(Xn) are integrable , then, 4>(X1 ), 4>(X2 ), ... is a sub martingale relative to �1 , !F2 , . . . (2) Let X1 , X2 , ... be a submartingale relative to !F1 , !F2 , ... on (n, §", P). If 4> is a nondecreasingand convex function, and if the l/>(Xn) are integrable, then l/>(X 1 ), 4>(X2 ) is a submartingale relative to !F1 , �2 , ... . , •••
Stochastic methods in economics and finance
30
To prove (1) we only need to show that E[ct>(Xn + t ) 1 .� ) � ct>(Xn)· Since Xn is a martingale we have that E [Xn+ 1 1.¥, ] = X,, . So, cf>{E[X,+ 1 I �) } = cf>(Xn). Since cf> is convex, Xn and ct>(X,) are integrable and we may apply Jensen's inequality for conditional expectations in theorem 5.6 to obtain E[cp(Xn + 1 ) I :� ] � cp{E[Xn + t 1 .� }} = ct>(Xn). To prove (2) we need only to show again that E[cp(Xn+ l ) I � ] � cf>(Xn). Since Xn is a submartingale E[Xn + t 1 .¥,, ] � X, . By hypothesis ct> is nondecreas ing so cp{E [Xn+ 1 1 .� ]} � cf>(Xn)· Now apply again Jensen's inequality to con clude Proof.
Theorem 6.2. (Kolmogorov's inequality). Let X1 , X2 , gale on .
0. Then,
(n,
•••
,
Xn be a submartin
Proof. Let A > 0 be given and define A 1 , ... , An, A as follows:
Note that the A k 's as defined above are disjoint. Also note that since X 1 , X2 , , X, is a submartingale one inductively may obtain E [Xn +k I .� ] � Xn for k � 1 Then n f Xn dP = L f X, cJP k=l Ak A n = L f E [Xn l � ] d P k = l Ak n � L f Xk dP k=l A k n � A L P(A k ) AP(A), k=l •••
.
.
=
Results from probability with A
31
k E .�k = a(X1 , , Xk). Therefore, •••
AP(A)
=
AP,w: �ax X; (w) � A l
l
j
t �n
� J X, dP � J X� d P � E( I Xn I ). A
This concludes the proof.
!�
Next we present the notion of an
upcrossing which is fundamental
for the Up
crossing Theorem. This upcrossing theorem is used by probabilists in the proof of the important result known as the martingale convergence theorem. However,
we present the notion of upcrossing here because it may be useful to researchers in finance and economics also. Let
[a, In
be an interval with
a < (3, and let X 1 ,
The number o f u pcrossings of [a, (3] by X1
•••
, Xn
be random variables.
(w), ... , Xn (w) is the number of times the sequence passes from below a to above {3 (see fig. 6.1 ) .
0
0
0
0
0
0
Figure 6 . 1 .
In the figure above there arc two upcrossings, for
to the strings of consecutive
follows:
yl
= 0,
2 �k �n + l 0
yk
=
1 's above
0
1 1
if
if
if
if
yk - l yk yk - 1 yk-1 J
=
1
=
0
=
1
=
0
and
and
and
and
the graph for
xk - l � (3, xk- l > a, xk - l < {3, xk - 1 � a.
16. These correspond the variable Y defined as
n =
32
Stochastic methods in economics and finance According to the definition an upcrossing corresponds in
1
0 on either side. Now define
an unbroken string of 's with a if
Yk =
1
and
Y2 , Y3 ,
•••
, Yn
to
Yk + 1 = 0,
otherwise, then the number of upcrossings is
Let g;-0
2, . . , n + I .
=
{0 , n} and !fie = a(X1 , , Xk). Then Yk is measurable � - 1 , k = 1 , •••
and U is measurable.
X1 , Xn be a submartingale (n, !F, P). Then the number of upcrossings, U of [a, J3], satisfies
Theorem 6.3.
(Martingale upcrossings).
E ( I Xn I) + I a l
E(U) �
J3 - o:
For a proof see Billingsley
Let
•••
,
on
.
(1979, p. 415) or Ash (1972, p. 291).
The fmal result establishes convergence of martingales.
Theorem 6.4.
(Martingale convergence).
Let
X 1 X2 , ,
•••
be a submartingale on
(fl, fF, P) and assume that K = supn E( I Xn I ) < oo . Then Xn � X w.pl, where X is a random variable such that E( I X I) � K. For a proof see Billingsley
(1979, p. 416).
7. Stochastic processes
stochastic process is a collection o f random variables {Xt, tE T} on the same probability space (n, �, P). Note that X,(w) = X (t, w) has as its domain the k product space T x n and as its target space R or R . The points of the index or parameter set T are thought of as representing time. If T is countable, especially if T {0, I , 2, 3, ... } N, i.e. the set of non-negative integers, then the process is called a discrete parameter process. If T = R or T = [a, b 1 for a and b real numbers or T = [0, oo) i.e. if Tis uncountable, then we have a continuous parameter proA
=
=
,
33
Results from probability
cess. Although the index set T can be rather arbitrary, in this section and the rest of this book the most often used index set is T = [0, oo). n denotes the random or sample space and for fiXed w E n, Xt (w) = X ( w) for t E T, is called a sample path or sample function corresponding to w. Other terms used in various texts to describe Xt(w) for a fiXed w are realization or trajectory of the process. For ·,
fiXed w E n the usual notation of the process is Xt; however, X(t) is also used in this text as well as in the literature. Obviously for a fiXed
t E T, Xt(w) = X(t,
)
is a random variable. The space in which all the possible values of Xt lie is called ·
state space. Usually the state space is the real line R and Xt is called a real
the
valued stochastic process or just a stochastic process. It is also possible for the
state space to be
k R , in
which case we say that
Xt
is a k-vector stochastic pro
cess. The various martingales presented in the previous sections are good examples
of stochastic processes. Additional examples will be discussed in this section; in
particular we briefly plan to discuss the
Markov process, and
the
Brownian motion or Wiener process, the Poisson process. However, before we do so we state
some basic notions in the form of definitions and theorems of stochastic processes.
7. 1.
Basic notions
An important feature of a stochastic process
the random variables of the
{Xt , t E T} is the relationship among This relaprocess, say X, , .. . , Xtn for t 1 , , tn
E T.
tionship is specified by the joint distribution function of these variables given by
Px for
t
•
•
...•
x
t n
(H) = P [w
•
:
•••
(7
(Xt. (w), ... , Xtn (w)) eH]
.I)
HE {jf n . It must be pointed out a t the outset that a system o ffinite-dimen
sional distributions of the form of (7 . I ) does not completely determine the prop erties of the process in the case of an arbitrary index set
T.
However, the fust
step in the general theory of stochastic processes is to construct processes for given fmite-dimensional distributions.
Suppose that we are given a stochastic process having (7
.I) as a fmite-dimen sional system. Note that (7 .I) necessarily satisfies two consistency properties. The first property is the condition ofsymmetry. Let p be a permutation of (I , 2, ... , n) and define fp :
R
n
-+ R n by
(7.2)
The random vector
(Xt
•
,
... , X, ) n
=
f.p (Xtp l , .. . , X,pn )
(7.3)
Stochastic methods in economics and finance
34
must have distribution Px 1
l
• . .• • x
from the left-hand side of (7 .3) and (7 . 1 ) 1, from the right-hand side of (7 .3). This leads to the
and also Px 1 • •• • • x 1 1; 1 pi pn condition of symmetry written as px,
t
.
- px , . .. . . x , r.p... . x 1 piZ t1 pl
l
(7.4)
·
The second consistency property is called the is written as pX t
t
•
... , X t (H) = pX t n
•
• ... , X t • X t n+ 1 11
condition of compatability; it
(H x R I )
(7.5)
for HE .'Jf " . The conclusion of the above analysis is this: given a stochastic process {X1 , t E T} then its finite-dimensional distributions satisfy properties (7 .4) and (7 .5). Naturally, the mathematical question arises: Does the converse hold true? That is to say, given finite-dimensional distributions having properties (7 .4) and (7 .5), does there exist a stochastic process having these finite-dimensional distribu tions? The question is answered affirmatively by the famous Kolmogorov theorem. Theorem 7 .1. (Kolmogorov). Given a system of finite-dimensional distributions satisfying the symmetry and compatability consistency conditions, then there exists a probability space (n, � P) and a stochastic process {X1, tE T} defmed on this space, such that the process has the given finite-dimensional distributions as its distributions.
For a proof see Billingsley ( 1 979, section 36). Kolmogorov's existence theorem puts the theory of stochastic processes on a firm foundation. Next, we proceed to state some useful definitions. Consider two stochastic processes {X,, tE T} and { Y,, tE T}, both defined on the same probability space (n, fF, P). These two processes are said to be stochastically equivalent if for every tE T, P[w: X1 (w) -:/=- Y,(w)] = 0. Alterna tively, the two processes are equivalent if for every t E T, X1(w) = Y1 (w) w.p.l. If two processes are equivalent we say that one is a version of the other and we conclude that their finite-dimensional distributions coincide. However, it is not always the case that equivalent processes have sample paths with the same prop erties. This can be illustrated with a simple example which also substantiates the remark made earlier that finite-dimensional distributions do not completely de termine all the properties of the process. The example considers two processes
Results from probability
35
{X,, t � 0} and { Y,, t � 0 } on the same probability space (n,.¥, P). Define first X, for t � 0 as X,(w) = 0
(7.6)
for all W E n,
and secondly define Y, for t � 0 as Y,(w)
=
{�
if V(w) = t,
(7.7)
if V(w) * t,
where V is a positive random variable on (n, !!F, P) with a continuous distribu tion given by P[w: V(w) =x) = 0 for eachx > 0. For X1 and Y, defined as above, P[w: X1 (w) * Y1(w)) = 0 for each t � 0 and therefore they arc stochastically equivalent. Also, X1 and Y1 have the same finite-dimensional distribution which, for t 1 , , tn say, are given by •••
pX t
I
' . . . , Xt
tr
(H) = Py t
I
' ... , y t
n
(H) =
{
1
if II contains the origin of R " ,
0 otherwise,
for HE ;% n . However, note that the equivalence of the two processes is not suf ficient to guarantee the same sample paths for x, and Y,. For w E n, from (7 .6) we know that X1 (w) = 0 wlillc from (7.7) Y1 (w) = 0 with a discontinuity at t V(w), where it obtains the value I. Thus, the sample paths of these two pro cesses arc not the same. To correct irregularities of this form probabilists have in troduced the concept of a separable process. Consider a process {X1, t E [ 0, oo)} defined on a complete probability space (n, :F, P). This process is separable if there is a countable, dense subset of T = [0, oo) denoted by S = {t1 , t2 , } such that for every interval (a, b) C [0, oo ) and every closed set A C R it holds that =
•••
P[w: X1 (w)E A for all I E (a, b) () S) = P[w: X,(w)E A for all ! E (a, b)) .
(7.8)
Observe that the definition requires that the probability of the set where [X1 E A for all t E (a, b) n S] and [X1 E A for all t E (a, b)] differ is zero. The motivation of this definition is to make a countable set of time points serve to characterize the sample paths of a process. The important question is: Given a
Stochastic methods in economics and finance
36
stochastic process having a system of consistent finite-dimensional distributions does there exist a separable version with the same distributions? Fortunately the answer is yes and we are therefore allowed to consider separable versions of a given process. For a detailed statement of the existence theorem and its proof see Billingsley (1 979, section 38) or Tucker (1967, section 8 .2). We next present some simple facts about three important processes. 7.2.
The Wiener or Brownian motion process
A Wiener process or a Brownian motion process {z t , t E [0, oo)} is a stochastic process on a probability space (!1, .� P) with the following properties: ( I ) z0 (w) 0 w .pJ, i.e. by convention we assume that the process starts at 0. (2) lf O � t0 � t 1 � •• • � tn are time points then for HE � � , =
P[zt - z,.
,
1- 1
.
E H; for
i � n] = i
E H;l -
This means that the increments of the process z1. - z1.1- 1 , i � k, are independent I random variables. (3) For 0 � s < t, the increment zt - zs has distribution P[z ,
-
zs E H ] = - -1 - J exp -
-
V2 n(t - s)
H
I_ L
J dx
x2 2(t - s)
This means that every increment z, - zs is normally distributed with mean 0 and variance o2 (t - s); here we assume that a = 1 , i.e. we standardize it. (4) For each w E n, z1(w) is continuous in t, for t � 0. Note that condition (2) reflects a kind of lack of memory. This means that the displacements zt' - zto , ... , z1n - 1 - z tn -2 of the process during the intervals [t0, !1 ] , ••• , [tn _2 , tn - J ] in no way influence the displacementz,n - z,n l of the process during [tn - 1 ' tn] . The past history of the process does not in fluence its future position. The future behavior of the process depends on its present position but it does not depend on how the process got there. Formally, ifO � t0 < t 1 < t2 < ... < tn < t then for real x,x0, ••• , xn -
(7.9) Eq. (7 .9) is called the Markov property and it will play an important role in de fming Markov processes. Here we need to emphasize that condition (2) of the
Results from probability
37
Wiener process requires independent increments which is actually more restric tive than the Markov property. To understand condition (3) suppose that zt denotes the height at time t of a particle above a fued horizontal plane. Then the fact that zt - zs is assumed to have mean 0 says that the particle is as likely to go up as to go down, i.e. there is no drift. The fact that we assume that the variance grows as the length of the interval [s, t] means that the particle tends to wander away from its position at time s and having done so no force exists to restore it to its original position. The increments of a Wiener process are stationary in the sense that the dis tribution of z, - zs depends only on the difference t - s. From property ( 1 ) we have that z0 0 which enables us to describe the behavior of increments by say ing that z, is normally distributed with E(z1) = 0 and E(z; ) = t . To compute its covariance note for 0 � s < t that =
cov(zsz,)
=
E(zsz,) = E(zsz t - zszs + zs zs)
=
E(zs [z, - z9] + z;)
= E(zs [zt - zs]) + E(z;)
= E(zs) E(zt - zs) + E(z; ) = E(z;) = s minimum {t, s } .
=
Finally, condition (4) is added because in many applications such continuity is essential. However, we immediately state an important theorem about the dif ferentiability properties of the Wiener sample paths. Theorem 7.2. (Nondifferentiability of the Wiener process). Let {zt , t � 0} be a Wiener process in (.Q, !F, P). Then for w outside some set of probability 0, the sample path z,(w), t � 0, is nowhere differentiable. For a proof see Billingsley ( 1 979, section 3 7). Intuitively, a nowhere differentiable sample path represents the motion of a particle which at no time has a velocity. Thus, although the sample paths are con tinuous, theorem 7.2 suggests that they are very kinky, and their derivatives exist nowhere. A complete analysis of the structure of the Brownian motion sample paths is found in Ito and McKean (1 974). Intuitively, a simple way to illustrate the nondifferentiability of the Wiener process is to observe from property (3) that for 0 � s < t,
E
(
zt - zs
t-s
)
2
-
a2
t-s
-
-t-s 1
Stochastic methods in economics and finance
38
assuming
a = l � taking limits,
2 z, -zs) ( t-.s t - =
lim E
S
(7. 1 0)
oo .
Suppose, however, that the Wiener process were differentiable; then, if we denote
2 ( z, - zs - z; )
its derivative by lim E
z� we have
t-s
t-s
= 0.
2 ) (z, -zs = E(z�)2 • E
This last equation implies Jim t-s
(7 . I I )
t-s
Note that (7 . 1 1 ) contradicts (7 . 1 0).
I n the engineering literature, the derivative of a Wiener process is called
noise. Further remarks will be made about it in the next chapter. Let {z1, t � 0 } be a Wiener process and use it to construct a new {w1, t � 0} defined by w1 = z1 + pt, t � O , where J1 is a constant. Then we say that
white
process (7 . 1 2)
{ w1, t � 0} is a Wiener process or
ian motion process with drift and J1 is called the
Brown
drift pararneter. In this case the
only modification that occurs in the definition of a Wiener process is in property (3) where
w1 - w9
is normally distributed with mean
a2 (t - s), assuming a = 1 . Finally, let
w1
p(t - s)
and variance
be a Wiener process with drift as defined in (7 . 1 2). Consider
the new process given by
y1 = e Then {y
cess.
xp
,
w ), t � 0.
(
(7 . 1 3)
1, t � 0 } is called a geometric Brownian motion or geometric Wiener pro
39
Results from probability
7.3.
The Afarkov chain and the kfarkov process
A kfarkov chain is a stochastic process {XI ' t 0, 1 , 2, ... } with a countable state space E defined on a probability space (Q, .
.
(7 . 1 4) Eq. (7 .14) is called the kfarkov property and it says that the probability that the random variable X1+ 1 will be at state j, conditioned on the past behavior of the random variables X0 , , X1 (i.e. conditioned on the a-field generated by X0 , ... , X1 and denoted by a(X0 , , X1) = a(Xn , n � t)) , is equal to the probability that X1+ 1 will be at state j, conditioned only on the current or present informa tion supplied by X1 (i.e. conditioned on the a-field generated by X1 and denoted by a(X1)). Put differently, the Markov property says that the past and future arc statistically independent when the present is known. All that matters in de termining the next state X1+ 1 of the process is its current state X, and it does not matter how the process got to X1. The probability of X1+ 1 being in state j, given that X1 is in state i, is called the one-step transition probability and is written as •..
•••
t
P;j
t+ I - P[X1+ =
I
t
= I. I X = I•J
(7 . 1 5)
for i,j E £. Observe that (7 . 1 5 ) denotes the dependence of the probability on the initial and final state as well as the time. If the one-step transition probabilities are in dependent of time, i.e. if independent of t E N,
(7 1 6) .
holds, then we say that the Markov chain is time-homogeneous or that the Mar kov chain has stationary transition probabilities. Such probabilities can be placed in a matrLx such as Pu P2 1
P1 2 P22
P1 3 Pn
(7 . 1 7)
Stochastic methods in economics and finance
40
The matrix P in (7 .17) is called the transition matrix of the Markov chain. The matrix in (7 . 1 7) satisfies , first, the condition P;; � 0 for i, j E E and, secondly, the condition '1:-iEE P;; = I for i E E. The probability that the Markov chain moves from state i to state j in s steps or periods of time is the (i,j) th entry of the sth power of the transition matrix P. Tills can be written as P[Xt S = j I x, +
=
i] = Pf;.
(7 .18)
In (7 .18) Pf; denotes the (i, j) th entry of the sth power of P. Using the fact from matrix algebra that ps+r = ps P' we obtain for i,j E £, (7 . 1 9) � pf,l k pkrI ' kEE which is called the Chapman·-Kolmogorov equation. Eq. (7 .19) says that if the Markov chain starts at state i, in order for it to be in state j after s + r periods or steps, it must be in some intermediate state k after the sth step and then move �.+r = lJ
·
from there into state j during the remaining r periods. The function 1r0(i) for i E £, defined by 1To(i) = P[X0
=
i],
(7.20)
is called the initial distribution of the Markov chain and it satisfies , first, the con dition 1r0(l) � 0 for i e E and, secondly, the condition �lEE 1r0 (l) = 1 . Finally, for a Markov chain X,, t E N, with transition probability P;; we say that the func tion 1r(i), i E E, is a stationary distribution if1r(l) � 0 for i E E,and 'E-;EE 1r(i) = I and also
� iEE
1r(1)P'.1. = 1r(j)'
j E £.
(7 .21)
Suppose that the stationary distribution 1T exists and that lim P' (i, j) __.. 1r(j),
j e £,
(7.22)
as t __.. oo. Then , it may be proved that, regardless of the initial distribution of the chain 1To, the distribution of X, approaches 1T as t -+ oo. If this occurs we say that 1T is a steady state distribution. For details sec Hocl, Port and Stone ( 1 972, ch. 2). Having briefly discussed some notions about Markov chains, we move on to present various defmitions about Markov processes. A stochastic process {X,, t E [0 , oo)} defined on a probability space (fl, .'F, P)
41
Results from probability
with state space the real line R, is called a Markov process if for 0 � s � t < oo and
A E �1
P[X, E A I a(Xu , u � s)] = P[X, E A I a(Xs)l
(7.23)
holds w .p.l. Equation (7 .23) is a generalization of eq. (7 .14) and it is also called the Mar kov property. In (7 .23) the left-hand side is a conditional probability with con ditioning on the a-field generated by all the random variables Xu for u E [0, s], while the right-hand side indicates the conditioning on the a-field generated just by the random variable Xs . As before, what matters in a Markov process in de termining its future behavior is not its past but only its current position. A con dition that is equivalent to (7.23) is this: for n � 1 , A E �· and 0 � t0 < t 1
< ... < tn < t < oo,
P[X, E A I x, ' ..., x, l = P[X, E A I x, l 0
,
(7.24)
,
holds w .p.l. For a Markov process X1 , t � 0, we defme the transition probability, denoted by P(s,x, t, A ), as follows: for 0 � s < t, A E Yf 1 and xER,
P(s,x, t, A ) = P[X1 E A I Xs = x]
(7.25)
w .p.l. The transition probability exists by fact 1 of section 5 of this chapter and has two properties. First, for 0 � s < t and A E Y£ 1 , P(s, t, A ) is a measurable function with respect to Yf 1 , where s, t and A are fiXed, and secondly, P(s, x, t, ) is a probability measure on .'// 1 for fiXed s, t and x E R 1 In a manner analogous to (7 .19) the Ozapman-Kolmogorov equation, for a Markov process for 0 � s � u � t
·
•
P(s, x, t, A ) = f P(u,y, t,A )P(s,x, u, dy). R
(7.26)
The Markov process is time-homogeneous if its transition probability P(s, x, t, A ) is time independent or stationary, i.e. for any u > 0,
P(s + u, x, t + u, A ) = P(s, x, t, A ),
(7.27)
which means that the transition probability is a function of only three arguments, x, t - s, and A . In such a case the Chapman- Kolmogorov equation is written as
42
Stochastic methods in economics and finance
(7 .28)
P(s + t,x, A ) = I P(s,y, A ) P (t, x, dy). R
standard example of a homogeneous Markov process is the Wiener process
z1, t � 0, with stationary transition probability for t � 0 and A E .Jf 1 , A
P(t, x,A) = P[zs+ t E A l z9 = x] =
1
I exp
Vf;/ A
(
- (y 2
xt )2 ) dy.
We conclude this section by discussing the Poisson process.
7.4
17le Poisson process
A Posson i process, with parameter 'A., is a collection of random variables {X1, t E [0, oo) } defined on (!"2, .�-. P) having as state space N = {0, 1 , 2 , ... } and satisfy ing the following three properties: {1)
X0 = 0 w.p.l. For each 0 < t < t2 < ... < t,, the increments X, , X1 - X, , ... , X, n
(2)
- X1,
1
1
are mdependent. •
•
:s
1
0 � s < t < oo the increment X1 - Xs has a Poisson distribution with parameter 'A.(t - s), i.e. the distribution of the increments is given by (3)
_
For
P[X - X = k ] = for k e N.
I
S
['A.(t - s)] k k!
exp
[- 'A.(t - s)]
Note that property ( 1 ) is just a convention while property (2) simply states
that the number of events in disjoint time intervals arc independent. Property (3)
may be derived from two postulates as is done in Karlin and Taylor ( 1 97 5 , pp. 23-26). These two postulates are stated below.
First, the probability of at least one event happening in a time period of du
ration
b.t is
P(b.t) = 'A.b.t + o(b.t),
(7.29)
'A > 0 and b.t -+ 0. The notation o(b.t) means that as At approaches O,o(b.t) approaches 0 also, but at a rate faster than that of At.
with
43
Results from probability
Secondly, the probability of two or more events happening in ll.t is o(ll.t). This probability is very small and it essentially says that the simultaneous occur rence of more than one event during a time interval of length ll.t is almost 0. In Chapter 2, section 1 2, we will have an occasion to use (7 2 9) .
.
8. Optimal stopping
Suppose that a fair coin is tossed repeatedly and after each toss we have to make the decision of stopping or going on to the next toss. Let Y1 , Y2 , be indepen dent random variables denoting successive tosses with common probability distri bution P( Y; = J ) = P( Y; = J ) = t , where Y; = 1 represents heads on the ith toss and Y; = I represents tails. If we stop after the nth toss we are to receive a reward denoted by X, which is a function of the first n tosses, i.e. X, = [,, ( Y1 , Y2 , , Y,). The mathematical question is when we should stop so as to maxi mize our expected reward. A stopping nile or stopping tirnc or stopping variable is a random variable r with values in { I , 2, 3 � . } and such that the event {r = n} depends only on the past values of Y1 , , Y11 and not on the future values Y,+ 1 , Y, +2, If 1 is a stopping rule then E(X.,) measures the average reward asso ciated with the stopping rule r. Denote by C the class of all stopping rules for which E(X.,) < oo . The V, where •••
-
-
•••
.••
.
•••
•
.
(8. 1 ) is called the
r
l'alue of the sequence {X, } and i f a stopping rule exists such that (8.2)
then r is said to be an optimal stopping nile. To illustrate the concepts presented we discuss two examples from Robbins ( 1 970, pp. 334--336). In the first example we consider the reward function given by X, = min { 1 , Y1 + . . + Y, } - n/(n + .
1)
(8.3)
for n � I , where min denotes the minimum or smallest value between I and Y1 + ... + Yn . As a stopping rule we consider r
= first integer 11 � I such that Y1 + .. + Yn = 1 . .
(8.4)
Stochastic methods in economics and finance
44
For a stopping rule as in (8.4) it is difficult to compute E(X7) exactly, but we know that E(X7) > 0, since •• •
{ 1 , Y1 , , Y7 }) - E
E(X7) = E(min
( ) 7 -
r+1
=
1
-
(T+1 ) 7
E
> 0.
(8.5)
To show that the stopping rule in (8.4) is optimal we next demonstrate that for any stopping rule other than the one in (8.4), its associated E(X7) is negative. To do this we need a famous result. Lemma 8 . 1 . (Wald). Suppose that Y1 , Y2 , distributed random variables such that
•••
are independent and identically
E(Y;) = p. < oo . Suppose also that r is any stopping time of the sequence Y1 , Y2 , E(r) < oo Then E(L'[: 1 Y;) always exists and .
(f )
•••
such that
(8.6)
E . Y; = p. · E(r). l=l
For a proof see Wald (1944) or Shiryayev (1978, p. 1 75). To apply Wald's lemma in our case note that Y1 , Y2 , ••• representing succes sive tosses of a fair coin, are independent identically distributed random variables with ,
ll =
E ( Y;) = 1
t +(
-
•
1)
•
� 0. =
Furthermore, suppose that r is any stopping rule for which E(r) < oo and hence, in particular, if T E C, then E(Y1 + . . + Y7) = 0. This implies that .
E(XT ) � E(YI
+
( )
... + YT) - E _!_ � - � . T+ 1
(8.7)
Thus, for all stopping rules other than (8.4), eq. (8.7) shows that E(X7) < 0. The supremum of all E(X7) for r E C is therefore (8.5) and by (8.1) for r as in
(8.4)
V = E(X7) > 0. Robbins (1 970; p. 335) concludes that (8.4) is optimal.
45
Results from probability
As a second example suppose that instead of {8.3) the reward function is given by X"
=
n
n2 n n+1
i=l
(
Y; + 1 2
)
(8.8)
for n = 1 , 2, ... . Note that (8.8) says that if we stop after the nth toss with all heads we receive an award of n2" /(n + 1 ), while otherwise the award is 0. Sup pose that we are at a stage n in which all heads have appeared and our reward is n2 n /(n + 1 ). What is the conditional expected reward of making one more tossing before we stop? The answer is
[
n2 n E Xn + l I Xn = n+1
--
]
I {n + 1 ) 2 n + l 2 " (n + 1 ) =-· >X . = n 2 n+2 n +2
(8.9)
Eq. (8.9) suggests that we should not stop because our expected reward after one more toss, Xn + 1 , conditioned on X, , is greater than our reward without any further tossing. But, suppose that we act wisely and have another tossing. Even tually, a tail will occur and our final reward will be 0. Thus, acting wisely at each stage does not imply the best long-run policy. This suggests that for the reward function (8.8) no optimal stopping rule exists. However, stopping rules do exist. Consider, for example, the class of stopping rules {Tk }, k 1 , 2, ... , where Tk stops after the kth toss no matter what sequence of heads and tails has appeared. For such stopping rules we have =
E(X k ) r
=
I 2k
-
·
k2k + k+J
(
I
-
)
1 - ·0 2k
=
k k+I
From (8 .1 0) as k � oo we obtain using (8 .1) that V = exist but no optimal rule exists. 8. 1.
(8.10)
--
l.
Thus, stopping rules
Mathematical results
Having motivated the concept of optimal stopping we proceed to establish some basic mathematical results and then give some illustrations from the theory of job search and stochastic capital theory. Our analysis draws heavily from the two classic books of Chow, Robbins and Siegmund ( 1 97 1 ), and DeGroot ( 1 970). Let (D., fF, P) be a probability space and let {,:;;;, , n = 1 , 2, .. . } be a sequence of increasing a-fields belonging to .¥. Let Y1 , Y2 , be random variables having •••
46
Stochastic methods in economics and finance
a known joint probability distribution function F and defined on (!1, .�, P). We assume that we can observe sequentially Y1 , Y2 , • •• and we denote by X 1 , X2 , •• • the sequence of rewards. If we stop at the n th stage, Xn = [,, (Y1 , Y2 , ••• , Y,). It is assumed that X 1 , X2 , ••• are measurable with respect to .?'1 , .tF"2 , •• • , i.e. X, is measurable � for n = I , 2 , . .. . A stopping rule or stopping variable or stopping time is a random variable T = T(w) defined on (!1, .�, P) with target space the positive integers 1 , 2 , . . whlch satisfies two conditions. First, .
P[w: T(w) < oo ) = 1 ,
(8. 1 1 )
and secondly
{w: T(w) n} e .�1 =
(8.12)
for each n .
Eq. (8. 1 1 ) says that the stopping rule takes a finite value w.p.l and (8.12) says that the decision to stop at time n depends only on past information included in the a-field .�1 • Put differently, (8.12) indicates that no future information is available to influence the decision to stop at time n. In general � is quite arbitrary, although in some applications we may take .� = a(Y1 , Y2 , ... , Y, ). The collection of all sets A E § such that A n {r = n} e �� for all integers n is a a-field in.¥ and is denoted by .�. We note that T and X7 are measurable .� . For any stopping rule T, the reward at time T is denoted by X7 whlch is a ran dom variable of the form on
{T n}, =
otherwise,
(8.13)
for 11 = I , 2, . . As in (8 .1) the value of the reward sequence, V, is the supremum of E(X7) for T e C. Note that l
00
E( I X7 l ) = L E [ I X, I I {T = n}) P[T = n] < oo. n=l
(8.14)
With the above notions available to us we now ask the question: Under what conditions does an optimal rule exist? The answer is given by the ne}.;.t theorem.
Results from probability
Theorem 8 . I . (Existence of optimal rule). Suppose that X1 , X2 , of rewards on as described above such that w .p.l
(n, .�, P)
E( I sup n Xn I ) < oo,
47 •••
is a sequence
(8.1 5)
and also as n � oo lim xn
� - 00
w.p.l.
(8.16)
Then there exists an optimal stopping rule. For a proof see DeGroot ( 1 970, pp. 347-348). The two sufficient conditions of theorem 8.1 have an intuitive explanation. Condition (8.15) says that even if we could observe the entire sequence of the random variables Y1 , Y2 , and then select to stop so as to maximize the re ward, the expected reward would still be finite. In other words, even with per fect foresight the payoff is limited. But even with a finite expected reward as in (8.15), it might be beneficial not to stop. Condition (8.16) makes certain that w .p.l we stop at a finite time. Although theorem 8.1 is very useful because it gives us sufficient conditions for the existence of an optimal rule, it does not, however, tell us anything about the nature of the optimal rule. Mathematical research has obtained results on the nature of optimal rule in some special cases. We follow DeGroot ( 1 970) to discuss such an important case. Consider the reward function which has the form •••
(8.17)
for n 1 , 2, . . . , where c > 0 denotes the fiXed cost of every observation. Eq. (8.1 7) says that our reward at period n is the difference between the largest value ob served among the random variables Y1 , , Yn and the cost of such observations or sampling. We then have the following theorem. :::
•••
Theorem 8.2. Suppose that Y1 , Y2 , is a sequence of independent and iden tically distributed random variables on with a distribution function F and let Xn , n = l , 2, ... be a sequence of rewards as in (8.17). If •••
(D, .07, P)
E(Y� ) < oo
(8. 1 8)
for n = 1 , 2, ... , then there exists a stopping rule which maximizes E(X7) and which has the form: stop as soon as some observed value Y � V and continue if
Stochastic methods in economics and finance
48
Y < V, where V is the value of the reward sequence obtained as a unique solu tion of the equation 00
J
v
( Y - V) d F(Y) = c.
(8.19)
For a proof see DeGroot (1 970, p. 352).
8.2. Job search Before we state additional mathematical theorems on optimal stopping we move to apply the results stated so far to the theory of job search. In this application we follow Lippman and McCall (1 976a). Consider an individual who is seeking employment and who searches daily, until he accepts a job and who receives exactly one job offer every day. The cost of generating each offer is c > 0. There are two possibilities: sampling with recall, i.e. when all offers are retained, and sampling without recall, i.e. when offers are made and not accepted are lost. Notationally, the random variables Yn represent the job offers at periods n = I , 2 , .. and we assume that the job searcher knows the parameters of the wage distribution F from which his wage offers Y, are ran domly generated. To keep the analysis simple we assume the participant in the job search to be risk neutral and seeking to maximize his expected net benefit. The searcher's decision is when to stop searching and accept an offer. Note that his sequence of rewards has the simple form .
Xn = max { Y1 ,
• • •,
Yn }
-
nc
as in (8.17), for n = 1 , 2, ... , in the case of sampling with recall, and the form
Xn
=
Yn
-
nc
in the case of sampling without recall. This last equation simply says that in the case of sampling without recall the searcher's reward is the difference between the current offer and his total cost of the search. Below we discuss the case of sampling with recall for independent and identically distributed Y1 , , Y, . This job search problem can be analyzed using the results from the theory of optimal stopping. We proceed to discuss the existence and nature of optimal stopping and then we obtain some additional insights from the analysis. By theorem 8 .l , to establish the existence of an optimal stopping rule for the •••
Results from probability
49
job search problem, we need to establish the two conditions in (8 . 1 5 ) and (8.16). This is accomplished in the following lemma in which independence of Y1 , Y2 , is not needed.
•••
Lemma 8.2. Let Y1 , Y2 , be a sequence of identically distributed random variables having a distribution function F. Let c > 0 be a given number and define •••
Z
= sup X, , ,
(8.20)
where X, is as in eq. (8.1 7). If the mean ofF exists, then lim X, --+ - oo as n � oo , w.p.l. If the variance of F is finite then E( I Z I ) < oo . For a proof see DeGroot ( 1 970, pp. 350-352). This lemma is useful because it helps us establish the existence of an optimal rule for the job search problem. Note that this lemma is a purely mathematical result which states sufficient conditions for (8.15) and (8 . 1 6) to hold. More specifically, if the mean and variance of the common distribution F exist then 2 (8.15) and (8.16) hold. Thus, assuming that E ( Y ) < oo , n = 1 , 2, ... , we can use lemma 8.2 and theorem 8.1 to conclude the existence of an optimal rule. The nature of the optimal rule in the job search model is described by theorem 8.2. More specifically, for any wage offer Y, the optimal stopping rule for the job searcher is of the nature or form ,
accept job if continue search if
y�
V,
Y<
V.
(8.2 1 )
I n the job search literature the critical number V in (8.2 1 ) is called the reserva tion wage and any policy of the form of (8.2 1 ) is said to possess the reservation
wage property.
Consider the first observation Y 1 from a sequence of independent and iden tically distributed random variables Y1 , Y2 , .. . The expected return from fol lowing the optimal policy in (8 .2 1 ) is E (max { V, Y1 }) - c. From the definition of V, the optimal expected return from the optimal stopping rule satisfies .
V = E max ( V, Yt l - c.
(8.22)
50
Stochastic methods in economics and finance
Note that E max (V, Y1 )
=
= =
V
V
V
v
J
0
v
J
0
J
=
V
00
J
0
dF(Y) + 00
J
v
YdF(Y) 00
J
d F( Y) +
J
d F( Y) +
v
v
00
00
J
YdF(Y)
J
YdF( Y) -
v
00
v
dF(Y)
v
J
V+
v
dF(Y) + V
0
00
J
d F( Y) ± V
v
- V =
d F(Y) +
00
00
J
v
( Y - V)dF(Y)
( Y - V) dF(Y).
(8.23)
Putting the result of (8.23) into (8.22) we obtain eq. (8.19) of theorem 8.2. Let us go a step further in analyzing eq. (8.19) which has just been derived. Define H(V) =
00
J
v
(Y - V) dF( Y).
The function H is convex, non-negative, strictly decreasing and satisfies the fol lowing properties: lim H(V) -+ 0 as V -+ oo, lim H(V) -+ E(Y1 ) as V -+ 0, dl/(V)/ d V = - [ 1 - F(V)], d2H(V)/d V2 � 0. Graphically, we may illustrate H as in fig. 8 . 1 .
v
figure 8 . 1 .
51
Results from probability
From fig. 8.1 we see that the lower the cost of search c > 0 is, the higher the reservation wage V and the longer the duration of search will be. Studying the equation H(V) c, which is (8.19), we obtain a simple economic interpretation, i.e. the value V is chosen so that equality holds between the expected marginal return from one more observation, H(V), and the marginal cost of obtaining one more job offer, c. In this application the job searcher behaves myopically by com paring his wage from accepting a job with the expected wage from exactly one more job offer. We conclude by remarking that in the infinite time horizon case with F( ) known there is no difference in the analysis between sampling with recall and sampling without recall. In the latter case search always continues until the reser vation wage is exceeded by the last offer. Note, however, that if the time horizon is finite, or if F( · ) is not known, these two assumptions cause the results to be different, as is shown in Lippman and McCall (1 976a). =
·
8.3. Additional mathematical results We continue the analysis on optimal stopping by illustrating the role of martin gale theory in problems of optimal stopping. Let X1 , X2 , ••• be a sequence of random variables on (n, !F, P), denoting re wards and let F1 , .fF2 , ••• be an increasing sequence of a-fields in .¥. We assume, as earlier, that X, is measurable with respect to �" n I , 2, ... , and that X, = fn ( Y1 , .•• , Y, ). The pair {(X, , �� ), n = 1 , 2, ... } is called a stochastic sequence. If E( I X, I ) < oo, n = I , 2, .. ., then we say that {(Xn , .�), n = I , 2, . .. } is an in tegrable stochastic sequence. If we interpret a stochastic sequence as a martin gale it is natural to inquire whether E(X1 ) = E(X7), with a stopping time. The reader will recall that the martingale property may be viewed as representing the notion of a fair gamble and therefore asking whether E(X1 ) = E(X7) means that we are inquiring whether the property of fairness is preserved under any stop ping time r. The results that are stated next explore this question and are obtain ed from Chow, Robbins and Siegmund (1971) and DeGroot (1 970). =
T
Theorem 8.3. Suppose that {(Xn , !F,,), n = I , 2, ... } is a submartingale on (il, �, P), r is a stopping time and n is a positive integer. (1) If P[r � n]
(2) If P[r < oo]
=
=
1 , then E [X, 1 .�7 ] � X7• I and
E(X7) < oo and also
(8.24)
52
Stochastic methods in economics and finance
f lim inf 1Z {T > tt x,: dP = 0, }
(8.25)
then for each n, (8.26) For a proof of this theorem see Chow, Robbins and Siegmund (1971 , p. 2 1 ). From eq. (8.24) we conclude that the unconditional expectations satisfy the relation (8.27) provided P[r � n] = 1 . For supermartingales the conclusion is as in (8.27) with the inequalities reversed and for martingales we have (8 .27) with equality signs. Concerning the condition in (8.25), we note that it is satisfied for a sequence of random variables X 1 , X2 , which are uniformly integrable. The defmition is this: a sequence of random variables {X" , n = I , 2 , ... } on (!1, !F, P) is wzi fomzly integrable if, as a -+ oo, then •••
lim sup n
f I X11 I d P { I X11 1 > a}
-+
0.
(8.28)
Note that (8 .28) implies that sup11 E( I X11 I ) < oo , which allows us to conclude in particular that E(X7) < oo for a stopping time, and therefore (8.26) holds. The last result of this subsection is the monotone case theorem. For a stochastic sequence {(X11 , �,), n = 1 , 2, . . } assumed to be integrable let
T
.
(8.29) for n = 1 , 2, ... . If
A 1 C A 2 C ... CA, C ...
(8.30)
and 00
u
n= I
A
n
=n
(8.31)
both hold, we say that the monotone case holds. In this case the next theorem tells us the nature of the optimal stopping rule.
Results from probability
53
Theorem 8.4. (Monotone case). Suppose that {(X, p .�1 ), n � 1 , 2, ... } is a sto chastic sequence wl1ich satisfies the monotone case. Let the stopping variable s be defined by
s = first n ;;;?; l such that xtl ;;;?; E [XPI+ i :�, J ' provided E(X-;) < oo. Then if lim inf "
f x,: {s > " }
�
I
0
holds, we obtain that
for all T such that E(XT ) < oo and lim inf n
f X,� = 0. {T > n }
For a proof see Chow, Robbins and Siegmund ( 1 97 1 , p. 55).
8.4. Stochastic capital theory We conclude this section by presenting a brief analysis of stochastic capital theory following Brock, Rothschild and Stiglitz (1 979). Introductory lectures on capital theory often begin by analyzing the follow ing problem: you have a tree which will be worth X(t) if cut down at time t, where t = 0, 1 , 2, ... . If the discount rate is r, when should the tree be cut down? Also, what is the present value of such a tree? The answers to these questions are straightforward. Suppose we choose the cutting date T to maximize e-rT X(T). Note that at t < T a tree is worth crt e-n Most of capital theory can be built on this simple foundation. It is our purpose to analyze how these simple questions of timing and evaluation change when the tree's growth is stochastic rather than deterministic. Suppose a tree will be worth X(t, w) if cut down at time t, where X(t, w) is a stochastic process. When should it be cut down? What is its present value? We ask these questions because, as in the certainty case, one can usc such an analysis to answer many other questions of valuation and timing. Before we can analyze the problem of when to cut down a tree which grows stochastically, we must specify both the stochastic process which governs the
X(T).
Stochastic methods in economics and finance
54
tree's growth and the valuation principle used. Here we analyze a discrete time model because for such models analysis of some problems can be done both more easily and in greater generality. We assume the tree's value follows a discrete time real-valued Markov process which we write as , X1• To complete the specification of our problem, we must describe what the person who owns the tree is trying to maximize. The simplest assumption is that he is maximizing expected present discounted value. Thus, if we let C be the set of stopping times for X1, then the problem is to find r E C to maximize e- rr. An apparently more general approach would involve the maximization of expected utility, is a strictly e-rr). This is only apparently more general because if increasing function, w)) is a Markov process with essentially the same properties as X w) and there is no analytical difference between maximizing E(e- rr X(r)) and maximizing Of course, the interpretation of the stochastic properties of is thought of as being depends on whether measured in dollars or in utiles. If the discount rate is
X1 , X2 ,
•••
EX(r)
E(U[X(r)]
U(X(t,
(t,
(3
=
1
1
+r
U(X)
E(U[X(r)] e-n). X(t)
X(t)
'
(8.32)
E{f37 X7).
our problem is to choose a stopping time r to maximize Let us choose the simplest possible specification for the X1 process, i.e. suppose that (8.33) where €1 are independent and identically distributed random variables with ex pected value p.. It is clear that in thi� case the optimal rule will be of a particu larly s!mple form: pick a tree size X and cut the tree down the ·first time that
x, � x.
If the process xt is deterministic, x,+ I = x, + p., it is easy to fmd the optimal cutting size, which we denote by X. X must satisfy
X = (3(X + p.),
(8.34)
since the l.h.s. of (8.34) is the value of cutting a tree down now and the r.h.s. is the present discounted value of the tree next period. If X satisfies (8.34), then the tree owner is indifferent as to whether he sells the tree now or keeps it for a period. For small the value of the r.h.s. (keeping the tree for a period) exceeds the value of the l.h.s. (harvesting the tree now). Note that if X is a solution to (8.34), then p.(X = (1 - (3)/(3 r, or X = J.L/r, i.e. the growth rate equals the inter est rate. How is tllis analysis changed if the tree's growth is uncertain? The answer
X,
=
Results from probability
55
depends on whether the stochastic process Xt is strictly increasing or not. Let X be the optimal cutting size for the random process in (8 .33). Then if € is positive, which implies that X, is an increasing process, uncertainty has no effect on the cutting size. Theorem 8.5. If
(8.35)
P[Xt + 1 > Xr] = 1 , then X = X .
We give a heuristic argument rather than a rigorous proof. This result is implied by the more general theorem 8.7 below. Let V(X) be the value of having a tree oi size X assuming it will be cut down when it reaches the optimal size. Then if X is the cutting size it must be that A
V(X) = X
for X � X
V(X) > X
for X < X.
(8.36)
and A
(8.37)
i,
Also at the tree owner must be indifferent be!ween cutting the tree down now and letting it grow for a period. That is to say, X must satisfy
X = {3 E(max {X + E, V(X + €)}).
(8.38)
However, since € is non-negative, V(X + e) = X + € and (8.38) becomes X = f3 E(X + €) = {3(X + p.) which is the same as (8 .34). Since the solution to (8.34) is unique, X = X. If € is negative, tltis argument does not go through. A
-
Theorem 8.6. If P[E < 0] > 0, then X > X. This theorem says that trees will be harvested when they are larger under uncer tainty rather than under certainty. Again we give a heuristic rather than a rigorous proof. Suppose for simplicity that € has a density function f( ·) with support on 1 , + 1 ]. Then (8.38) becomes
[-
Stochastic methods in economics and finance
56
X= A
( 0 ( J (i 0
(3 f V(i -1
>P
-1
+ e) [(e) d e +
+ e) [(e) d e +
1
J (i + e)[(e) d e
0 1
J (i + e) [(e) de
0
= f3(X + p.). Thus,
i>
1
J.l(3 -
(3
=
11
r
)
)
= x.
We note here two implications of these simple propositions: that uncertainty can increase the value of a tree and that strictly increasing processes behave dif ferently from processes which can decrease. Theorem 8.6 implies that uncertainty can in some cases increase the value of a tree. Suppose you have a tree of size X. Then if its future growth is certain, its value is just X. If its evolution is uncertain and if its size may decrease, then you will not cut it down; its value exceeds X. Writing (X) as the value of a tree of size X in the deterministic case and ve (X) as the value of a tree when the incre ments in the tree's growth are the (nondegenerate) random variable €, we have shown that if e were negative
Vd
(8.39)
for some X. (Continuity implies that (8.39) holds for X E [X - o, �) for some o .) It is natural to ask whether (8.39) holds under more general circumstances. Another implication of theorems 8.5 and 8.6 is that strictly increasing pro cesses are different from processes which may decrease. We show that this is gen erally true by showing in the next theorem below that theorem 8.5 holds for a very wide class of increasing processes. Theorem 8.7. Let X, be a Markov process such that
P[Xt+ t � X, ] = 1 and that
E [.BX,+ 1 - X, I X, � X, ] � O, E [.8Xt+ 1 - X, I x, � X, ] � 0,
57
Results from probability
t where xt is a nonincreasing sequence. Let yt = [3 Xt. lf xt � Xt , then Yt, yt+ I ' Yt+ 2 , ... is a supermartingale. This proposition implies that uncertainty does not affect the time at which the tree is cut down. By theorem 8.3 we know that if T is any stopping time, then E [ YT I xt � Xt ] � Yt ; it is optimal to stop when the tree's height first exceeds xt . To establish this theorem we must show that for all t
However, since
it suffices to observe that E(.Bxt+T+ 1
-
xt+T 1 xr � xt 1 = E(E(.Bxt+ T+ 1
= E(E[.8xt+T+ I
-
xt+T 1 xt+T J 1 xt � Xt) xt+T I xt+T
;;;. xt +T] I xt :>: Xt) � 0. -
9. Miscellaneous applications and exercises ( 1 ) Suppose that {An} is a countable sequence of sets belonging to the a field !F. Show that n, A, E :':!'. 1hls fact establishes that a a-field is closed under the formation of countable intersections. Next, suppose that A E ff and BE.¥', where .� is a a-field. Show that A - B E .�. (2) Let be the set of all positive integers and let .91 be the class of subsets A of such that A or A c is fmite. Is sl a a-field? Explain. (3) Let R 1 be the set of real numbers and let !!A 1 be the a-field of Borel sets. The class Bf 1 contains all the open sets and all the closed sets on the real line. This shows that the class fA 1 of Borel sets is sufficiently large. However, the reader must be warned: there do exist sets in R 1 not belonging to For such an example see Billingsley (1979, section 3). (4) Suppose that n = { 1 , 2, 3 , ... }, i.e. n is the class of positive integers and !F is the a-field consisting of all subsets of n. Define J.l. (A) as the number of points, i.e. integers, in A and �-t(A) = oo if A is an infinite set. l11en 11 is a measure on�; it is called the counting measure. Next, consjder (!2, .�) as above and let p 1 , p2 , ... be non-negative numbers corresponding to the set of positive integers such that 'J:,;P; = I , i = 1 , 2, ... . Define
Z Z
:'11 1
•
Stochastic methods in economics and finance
58
�-t(A) = L P;· x;EA Then 11 is a probability measure. (5) Probability as a special kind of a measure satisfies several useful proper ties. Here are some such properties. Let (n. !F, P) be a probability space. (a) If A E !F and B E .� and A C B then P(A) � P(B). This is the monoto
nicity property. (b) If A E .�then P(Ac) 1 - P(A). (c) If {A, } is a countable sequence of sets in !F then =
( )
P U , A, � :r , P(A,).
This is called the countable subadditivity property
or Boole 's inequality. (d) If {A, } is an increasing sequence of sets in !F having A as its limit then P(A,) increases to P(A). For proofs of these statements see Billingsley (1979, section 2) and Tucker ( 1 967, pp. 6 -7). (6) Suppose X and Y are random variables defined on the same space (il, !F, P). Then (a) eX is a random variable for c e R; (b) X + Y is a random variable provided X (w) + Y(w) =1= oo - oo for each w; (c) X Y is a random variable provided X(w) Y (w) =I= 0 • oo for each w; and (d) XJY is a random variable provided X(w)/Y(w) =1= oo I oo for each w. be a random variable on (il, /F, P) with finite expectation E (X) (7) Let and finite variance var X. Then for k such that 0 < k < oo we have
X
P[w : I X(w) - E(X(w)) l � k ] �
var X
k
2
•
This is called Chebyshev's inequality and it indicates that a random variable with small variance is likely to take its values close to its mean. (8) Let X be a random variable om (il, .'F, P) with finite expectation and sup pose that is a convex real-valued function such that E((X)) < oo. Then (E(X)) � E ((X)). This is called Jensen's inequality and it has found several applica tions in economic theory. If a consumer prefers risk then his utility function is convex, and Jensen's inequality says that the expected value of his random utility function is greater or equal to the utility of the expected random variable X. In this case the consumer wH: pay to participate in a fair gamble. Suppose that is a concave real-valued function with E(X) < oo and E((X)) < oo. Then (E(X)) � E ((X)), which is also called Jensen's inequality, and in economics it may be
Results from probability
59
used to describe a risk-averse consumer who will avoid a fair gamble by purchasing insurance. Friedman and Savage (1 948) have suggested a utility function com posed of both concave and convex segments. For further applications of Jensen's inequality in economics see Rothschild and Stiglitz (1 970). (9) Let en,� P) be a probability space with A E !F such that P(A) > 0. Let {Bn } be a finite or countable sequence of disjoint events such that P(Un Bn) = 1 and P(Bn) > 0 for all n. Then for every k,
P[Bk I A ] = P[A I Bk ] P[Bk ] I :r n P[A I B, ] P[Bn ]. This is called Bayes ' theorem. Using conditional probability this theorem may be stated as follows:
P[B I A )
=
f
B
P[A I .19] dP I f P[A I 9i ] dP n
for B E .'M, where EM is a a-field in .'F. ( 1 0) Theorems 4.2, 4.3 and 4.4 can be extended for conditional expecta tion. They are as follows. (a) Conditional form of monotone convergence theorem. Let {X, } be an in creasing sequence of non-negative random variables on (n, .� P) and let Xn (w) -+ X(w) w.p.l and assume that E(X) < oo . Then for rJ a a-field in !F, w.p.l as n � oo . (b) Conditional form of Fatou's lemma. If {Xn } is a sequence of non-negative random variables on (n, .�, P) with finite expectations and if E(lim infn Xn) < oo as n -+ oo , then E [lim inf n Xn I <.1 ] �lim inf n E [Xn I w.p.l as n � oo, for
C§
a a-field in :F.
q; ]
(c) Conditional form of dominated convergence theorem. Let {X11 } be a se quence of random variables and Y an integrable random variable, all on (n, §", P) such that I Xn I � Y w.p.l for all n . If Xn -+ X w.p.l and if tfJ is a a-field in �, then
60
Stochastic methods in economcs i and finance
E [X11 I
'!J ] � E[X I (jJ ]
w.p.l as n � oo . Proofs of these three theorems may be found in Tucker ( I 967, pp. 2 1 5-21 6). (1 1) Consider the sequence of discounted expected values of a stock, vt+ T ' T = I , 2, ... , described in application (3) of section 6 and assume that
It is natural for a stockholder to want to know the probability that the maxi mum value of vt+ T will exceed $ X, where X is some positive number. Find con ditions on vt+ T which will allow the investor to compute P[w: maxT vt+ T ;?: X]. Also, find conditions on vt+ T that will allow vt+ T to converge to, say, v as T � oo , i.e. some random variable denoting a discounted expected value of the stock. (I 2) Consider the Wiener process with drift {wr, t � 0} given by wt = zr + Jll, Jl =1= 0, where {z, t ;?: 0} is a standard Wiener process. Compute that the incre ment wt-l-vs for 0 � s < t has mean and variance ( 1 3) Suppose that {z t , t ;?: 0} is a Wiener process. Show that {(zr,.�), t = 0, 1 , 2, . .. } is a martingale, where .?, = o (z0 , z 1 , Z2 , , z r) . (14) The job search application of section 8 can be extended to allow for discounting. Arguing as in the case with no discounting, when the job searcher seeks one more job offer, his expected return is JJ[E(max { V, Y1 } - c)], which implies a reservation wage V whkh is the solution to
J.L(t-s)
t-s.
•••
V = JJ [E(max { V, Y1 } - c)], where t3 = 1/(1 + r), with r being an interest rate. Observe that as the last equa tion illustrates, we assume that the search cost is incurred at the end of the period and that the wage offer is also received at the end of the period. Study this appli cation to conclude that as the interest rate increases the reservation wage rate will decrease. For details see Lippman and McCall (l976a).
10. Further remarks and references There are several good textbooks on probability. The material presented in this chapter, and particularly in sections 2 , 3 , 4 and 5 , may be found in Ash (1 972), Billingsley ( 1 979), Chung ( I 974), Loeve {1 977), Neveu ( 1 965), Papoulis ( 1 965) and Tucker (1 967) among other textbooks. Loeve is a classic textbook now in its fourth edition. Tucker and Neveu are concise with Neveu having more advanced
Results from probability
61
material than Tucker. Ash develops measure and integration theory, functional analysis, and topology in the first part of his book. He then applies these con cepts in probability theory. On the other hand, Billingsley ( 1 979) emphasizes the interplay between analysis and probability with the analytical issues being motivated by probabilistic questions. The standard reference on the various re sults on convergence is Billingsley (1 968). For a brief introduction to some prob abilistic concepts with applications in microeconomic theory, see McCall ( 1 97 1 ). It may be noted that probability theory began during the seventeenth century with Pascal and Fermat among others, when problems which arose in games of chance were formulated in mathematical terms. Modern probability theory, as it is essentially known today, was given a solid mathematical foundation by Kol mogorov in 1 933. His path-breaking work was translated from Russian into English in 1950. See Kolmogorov (1 950). Section 6 on martingales reports definitions, examples and theorems from Ash {1972, section 7 .3), Billingsley (1 979, section 35), Doob ( 1 95 3 , ch. VI), Meyer ( 1 966, part B) and Neveu (1975). The two classic sources for martingales used to be Doob (1953) and Meyer (1 966) with the· latter book making extensive use of the first book. To these two classics we must add Neveu ( 1 975) and the updated Meyer and Dellacherie (1 978). For a brief introduction to the subject see Doob's descriptive article in the
American Mathematical Monthly, i.e. see Doob ( 1 9 7 1 )
o r Feller ( 1 97 1 ). Karlin and Taylor ( 1 975, ch. 6) also have a nice discussion on martingales. Note that Doob ( 1 953) uses the terminology scmimartingale and lower semimartingale in lieu of submartingale and supermartingale, respectively. The economics and finance applications in section 6 are based on Samuelson's (1965, 1 973) two papers and Hall's (1 978) paper. See also MaCurdy ( 1978) and Malliaris (1981). The reader who is interested in a more detailed survey is re ferred to Fama ( 1 970). Grossman and Stiglitz ( 1980) offer some constructive comments on the subject of efficient markets and proceed to redefine the notion of efficiency. A nontechnical presentation of the concepts of information, mar tingales, and prices is Alchian ( 1 974) and a recent survey of macroeconomic mar tingales and econometric tests of the martingale property is O'Neill ( 1 978). Although in the application of futures pricing as a martingale we followed Samuelson {1 965), another paper of significance is Mandelbrot ( 1 966). Also, see the earlier paper by Houthakker (1961 ). It is safe to argue that the papers by Samuelson and Mandelbrot are among the primary sources which generated the recent interest in the martingale concept. Mandelbrot credits Bacheller's ( 1 900) doctoral dissertation in mathematics on "Theorie de la Speculation" as the first work in this area. Actually , Bachelier discovered the theory of Brownian motion five years before Einstein. For historical interest we mention that the name mar tingale is due to Ville ( 1 939). The martingale property has been tested not only for stock market prices
62
Stochastic methods in economics and finance
( 1970), Sargent { 1 972, 1 976), and Modigliani and Shiller {1 973); for exchange rates see Cornell (1977); for price expectations see Mullineaux ( 1 978) and McNees ( 1 978). In the theory of continu ous trading? Harrison and Pliska (1981) show that a security market is complete but also elsewhere; for interest rates see Roll
if and only if its vector price process has a certain martingale representation property. The economics and fmance literature related to the martingale property demonstrates an important interplay of several concepts: martingale property, random walk, market efficiency, rational expectations, arbitraged prices, sta bilizing speculation, and statistical dependence among prices. Several theoretical papers attempt to clarify the precise relationship among these concepts. Among
(1 970), 1978), Shiller {1978) and Lucas (I 978).
these papers we mention Fama
Mandelbrot
( 1 97 1 ),
The basic reference on stochastic processes is Doob's
Dan thine
(1 953)
( 1 977,
classic book.
Also, an exhaustive analysis of stochastic processes may be found in Gihman and Skorohod's
(1 974, 1 975, 1 979)
three volumes. These are advanced books and
for the reader who is interested in an introduction to this subject we suggest Kar lin and Taylor ( 1 975), Cox and Miller (1 965), Prabhu ( 1 965), vinlar ( 1 975), and Hoel, Port and Stone Tucker
(1 972) among others. The books of Billingsley (1 979) and
( 1 967) among several
other textbooks have a chapter on stochastic proa
cesses. Historically, the stochastic process which was first investigated in some detail is the Brownian motion. The English botanist R. Brown observed in
1 827
that
small particles immersed in a liquid exhlbit ceaseless irregular motions. Much later, in
1905, Einstein
described the proc,ess mathematically by postulating that the
particles under observation are subject to perpetual collision with the molecules of the surrounding medium. The concise mathematical formulation of the theory of Brownian motion was given by Wiener in
1918.
In this book and in many
others the terms Brownian motion process and Wiener process are used inter changeably. In the past, a distinction was made between a Brownian motion pro cess, which was one satisfying properties ( 1 ), Wiener process which was one
(2) and (3) of the defmition, and a satisfying conditions {1 ), (2), (3) and (4) of the
defmition. The Markov chain is due to the Russian mathematician A.A. Markov who de veloped his ideas in Bernoulli in
1 769.
1 907
in an effort to solve a problem originally posed by D.
Other mathematicians who contributed to the development
of Markov chains and Markov processes are A. Kolmogorov and W. Feller. For some basic references the reader is referred to Karlin and Taylor
( 1 975),
Bha
(1 960), Cox and Miller ( 1 965), vinlar ( 1975), Hoel, Port and Stone (1 972) and Feller ( 1 968). For a more advanced and complete treatment see Dynkin ( 1965). rucha-Reid
Results from probability
63
For an introductory exposition on optimal stopping the reader is encouraged to consult Robbins' {1 970) descriptive paper in the American Mathematical Monthly and also Breiman ( 1 964). As already mentioned in section 8 , we used extensively the two books by DeGroot {1 970) and by Chow, Robbins and Sieg mund {1971). In these books several important results are presented which were first discovered by Chow and Robbins ( 1 967), DeGroot ( 1 968), Dvoretzy ( 1 967), Yahav ( I 966) and several other mathematicians. In section 8 we did not present problems of optimal stopping related to Markov chains and processes. This im portant subject is covered in �inlar {1975) and the classic book by Dynkin and Yush.kevich (I 969). At a more advanced level we recommend Shiryayev {1 978) and Ruiz {1 968). Actually, Shiryayev {1 978, ch. 2) can be used to fully justify the optimal rule stated immediately after eq. (8.33). A useful book on optimal stopping, written essentially for students interested in business applications, is Leonardz {1974). For many applications of optimal stopping in job search models the standard reference is Lippman and McCall {1 976a, 1 976b, 1 979). We conclude by remarking that our original intentions were to present a much smaller subset of this chapter as an appendix. As the material converged to its current size we decided that a chapter is more appropriate organizationally than a long appendix. We make this remark to encourage the reader to use Chapter 1 as a reference or as a foundation chapter. Clearly, the main subject matter of this book is treated in Chapters 2, 3 and 4.
CHAYIER 2
STOCHASTIC CALCULUS
The doctrine that knowledge of matters of fact is only probable is one of the cen tral theses of contemporary analysis of scientific method. Nagel (1 969, p. 5)
1 . Introduction
This chapter presents various mathematical methods of stochastic calculus that are useful in modeling and analyzing the behavior of economic and fmancial phe nomena under uncertainty. Such methods include Ito's lemma, stochastic dif ferential equations, stochastic stability and stochastic control. The study of such methods necessitates the presentation of various mathe matical concepts and results, such as the notion of a stochastic integral, the prop erties of solutions of stochastic differential equations, various approaches to stochastic stability, and so on. These concepts and results are also presented in this chapter to accustom the reader to the theoretical underpinnings of stochastic calculus. 2. Modeling uncertainty
In modeling, analyzing and predicting aspects of economic reality, researchers are placing greater and greater emphasis upon stochastic methods. Such methods are expected to capture the various complexities, measurement errors and uncer tainties that are associated with economic reality. The question that arises na turally is: How can combinations of complexity, uncertainty and ignorance,
Stochastic methods in economics and finance
66
which are present in the process of economic theorizing, be incorporated into dynamic analysis? An answer to this question is described in this section, first for the discrete time case and next for the continuous time case. This analysis follows .A.strom (1 970).
2.1.
Discrete time
Consider the discrete time index set T consisting of the set of positive integers {0, I , 2, 3, .. . } and let x(t) denote a real state variable at time t, t e T. A dynamic deterministic system may be described by the following difference equation:
x(t + l ) = f(t, x (t)), t E T,
(2.1)
with initial condition x (0) = x0 • Needless to say, such deterministic difference equation models abound in economic theory. Consider, for example, the three sector macroeconomic model, yt = Ct + It + Gt, ct = ao
+ a l yt- l .
For given values of It and Gt, denoted as It and Gt, the difference equation ob tained is Yt = (a0 + It + Gt) + a1 Yt- l .
(2.2)
Eq. (2.2) describes a special case of eq. (2.1), namely a linear autonomous dif ference equation. In a way similar to the one used by econometricians in modeling uncertainty, we proceed to make the deterministic model of eq. {2.1) into a stochastic model. Recall that a standard way of introducing uncertainty into eq. (2.2) is to add a random variable Ut to (2.2). It is often assumed that Ut is a sequence of indepen dent random variables, normally distributed with mean zero and finite variance. Thus (2.2) becomes
(2.3) Eq. (2.3) is an example of a linear stochastic difference equation. Returning to eq. (2.1 ) , one way it can be converted to incorporate uncertainty is to assume that x (t + 1 ) is a random variable having the expression
Stochastic calculus
x(t + 1 ) = f(t, x(t)) + v(t, x(t)), t e
T,
67
(2.4)
where f is the conditional mean of the random variable x(t + 1 ), conditioned upon x(t), and v is a random variable with mean zero and fmite variance, denoted by a2 (t, x) . We assume that the conditional distribution of the random variable v, given x(t), is independent of x(s) for s < t. Eq. (2.4) describes a stochastic difference equation and on the basis of the assumption of independence we con clude that the process {x(t), t e T} is a Markov process. Next, suppose that the conditional distribution of v, given x(t), is normal so that the random variable u (t), where u(t) = v(t)/a(t,x), is normally distributed with mean zero and unit variance. Note that u (t) is independent of x and {u(t), t e T} can be taken to be a sequence of independent, identically distributed nor mal variables with mean 0 and unit variance. Eq. (2.4) thus becomes
x(t + l ) = f(t,x(t)) + a(t,x(t))u(t), t e T.
(2.5)
Eq. (2.5) is a stochastic difference equation. I t is studied in several books such as Astrom ( 1 970) and Chow (197 5). 2. 2.
Continuous time
Consider a dynamic deterministic model described by a differential equation dx
- = f(t, x(t)) dt
(2.6)
for t e T, where T = [0, T] or T = [0, oo), and with initial condition x(O) = x0• Again, there are many examples of economic models described by differential equations such as (2.6); for example, consider dk - = sf(k(t)) - nk(t), k(O) = k0 , dt
(2.7)
i.e. Solow's differential equation of neoclassical growth analyzed in Solow (1 956). Note that (2.7) is a special case of (2.6) because it is autonomous or, in other words, time-dependent. To introduce uncertainty into (2.6) we proceed as follows. Consider the model at time t and at time t + At; the deterministic equation (2.6) is obtained from
x(t + At) - x(t) = f(t, x(t)) At + o(At)
(2.8)
Stochastic methods in economics and finance
68
by dividing both sides by 6.t and taking limits as 6.t � 0. In (2.8) note that o(6.t) means that o(6.t)/6.t � 0 as 6.t � 0. Now observe that (2.8) is a differ ence equation and we may repeat the procedure followed in the discrete case. Let {v(t), t e T} be a stochastic process with independent increments, i.e. as in section 1 of Chapter 1 we assume that if H; e Eli and if 0 � t0 � t 1 � � tk , then .••
P[v(t;) - v(t;_ 1 ) e H;, i � k 1 = i!Jk P[v(t;) - v(t;_ 1 ) e H;] .
Eq. (2.8) now becomes
x(t + 6.t) - x(t) = f(t,x(t)) 6..t + v(t + 6.t) - v(t) + o(6.t). Assume that the conditional distribution of the increment of normally distributed and write
(2.9)
v, given x(t), is
v(t + 6.t) - v(t) = a(t, x(t)) [z (t + 6.t) - z(t)] ,
(2.10)
where {z (t), t E T} is a Wiener process with mean zero and unit variance. Sub stituting (2.1 0) into (2.9) we obtain
x(t + 6.t) - x(t) = f(t, x(t)) 6.t + a(t, x(t)) [z(t + 6t) - z(t)] + o(6.t).
(2.1 1 )
In this last equation the term o(6.t) is a random variable and usually denotes that E I o(6.t) l 2 /6.t � 0 as 6.t � 0. At this point observe that in (2.1 1 ) we are not able to divide both sides by 6.t and take limits as we did in (2.8). This is so
.
lim
z(t + 6.t) - z(t) 6.t
does not exist. Recall theorem 7.2 of Chapter 1 which says that a Wiener process is nowhere differentiable outside some set of probability zero. Thus, instead of dividing (2.1 1 ) by 6.t and taking limits in the usual sense we only take limits as 6.t � 0 in a sense to be explained in the next section. In conclusion (2.1 1 ) can be written in the formal expression
dx = f(t,x) d t + a(t,x) dz,
(2.12)
which is called Ito 's stochastic differential equation with initial condition x (0) = x0 and t e T.
Stochastic calculus
69
Equation (2.12) plays an important role in applied mathematics. See the books by Schuss ( 1980), Soong (1973) and Tsokos and Padgett (1974). For a brief historical evolution of eq. (2.12) since 1908, see Wenham (1970). It also plays an important role in economic theory and fmance , as the next two chapters will illustrate. Because of the significance of eq. (2.12) in continuous stochastic state models in economics and finance, we proceed to give meaning to the formal ex pression in (2.1 2). 3. Stochastic integration Consider a probability space (n, !F, P) on which both a stochastic process x(t, w) and a Wiener process z(t, w) are defined for w E n and t E T. Eq. (2.12) is a shorthand notation for
dx(t, w) =f(t,x(t, w)) d t + a(t,x(t, w)) dz(t, w),
(3.1)
which can also be written as
dx(t) =f(t, x(t))dt + a{t,x(t))dz(t)
(3.2)
dx1 = f(t, x1) d t + a(t, x1) dz1,
(3.3)
or
where w is being suppressed in (3.2) and (3.3). Transforming eq. (3.2) into an integral equation we obtain
x(t) = x (O) +
t
t
J f(s,x(s))ds + J a(s,x(s)) dz(s). 0
(3.4)
0
Note that x(O) is a random variable since x (O) = x(O, w) for w E n, which could degenerate into a constant. As a rule, the first integral in the right-hand side of (3.4) can be understood as a Riemann integral. The second integral is more of a problem because dz (t) does not exist. In other words, although z (t) is continuous, it is a function of unbounded variation and the second integral cannot be inter preted as a Riemann-Stieljes integral. We need, therefore, to present a theory to make the second integral meaningful. Stochastic integration was developed by Ito (1944) as he generalized a sto chastic integral first introduced by Wiener in 1 923. Parts of Ito's original work were presented, initially by Doob (1953), and later by the Russian mathemati-
Stochastic methods in economcs i and finance
70
dans Gihman and Skorohod ( 1 969). However, it is only recently that the ideas of stochastic integration, and stochastic calculus in general, have become acces sible to the applied researcher by the publication of the books by Arnold (1974), Astrom (I 970) Gihman and Skorohod (I 972), Bharucha-Reid (1972), and Ladde and Lakshmikantham (1980) among others. In the analysis below we first give an intuitive analysis; we then proceed to a detailed exposition on stochastic integration.
3. 1.
Intuitive analysis
Consider the meaning of the ordinary differential equation form dx =f(t , x) d t; x (O) = x0 ;
t E T= [O, T],
(2.6) written in
the
(3.5)
we say that y(s) is a solution of (3.5) if dy (s) ds
-
-
!(
(3.6)
s,y (s))
for each s E T. Or, what is the same thing, upon integrating both sides of (3.6) we say that y (s) solves (3 .5) if, for all times t, s, s � t, t
y (t) - y (s) =
J f(r,y (r)) dr. s
(3.7)
Now note that the right-hand side of (3.7) is an integral which can be approxi mated by dividing the time interval [s, t ], which is a subset of [0, T], into small subintervals. Let E > 0 be given and consider the partition
(3.8) We can then write: y (t) - y (s) =
n-1 .1:
t=O
f(t; , y (t;)) (t;+ 1 - t;) + O(E) ,
Stochastic calculus
71
where O (e) -+ 0 as e -+ 0. l11is is a standard approximation procedure for calcu lating integrals in calculus. Now we follow a similar approach for stochastic cal culus. We say that the stochastic process y (t) solves (3.2) for all times t, s, for s � t, if t
t
y(t) - y(s) = J [(r,y(r)) dr + J a(r,y(r)) dz(r), s
(3.9)
s
where the right-hand side of (3.9) is defined by t
t
11 - 1
J f(r,y(r)) dr + J a(r,y(r)) dz(r) = � s
s
f(t;,y(t;)) (t;+ 1 - t;) + {3.10)
In {3.10), O(e) is a random variable and O(e) -+ 0 as e -+ 0 means O(e) 1 2 -+ 0 as e -+ 0. Recall that denotes unconditional expectation. There are some tech nical mathematical problems in justifying (3.1 0) and in proving the existence of stochastic processesy(t) that satisfy
EI
E
y(t) -y(s) = +
n- 1
� 0
a(t;,y(t;)) [z(t1+ 1 ) - z (t;)] + O(e)
for partitions of the form of (3.8). These technical aspects are treated in some detail in the next section. Let us proceed, however, to illustrate one possible dif ficulty. Given functions g(t, z) and z (t) when max I !;+ 1 l; 1 -+ 0, then the two limits -
and
are the same in the certainty case. Thls is not so in the stochastic case. By way of illustration let us consider the simple case where g(t, z) = z (t), where z (t) is a Wiener process with unit variance, and write as e -+ 0:
Stochastic methods in economics and finance
72
and
In the certainty case, i.e. when z(t) is deterministic rather than stochastic, A and B arc ordinary Ricmann- Sticljcs integrals, and if we assume that z(t) is integ rable then B --A � 0 as e � 0. Thus, in the deterministic case there is no ambiguity concerning the concept of convergence. However, dealing with z (t, w) as a ran dom variable one must specify which limit concept is being used because there arc several concepts of convergence. In the present case the appropriate interpre tation of these limits is the following:
(3.1 1)
e �
0, where E denotes expectation. The limit defined in (3.1 1 ) is called the limit in mean square and describes one more convergence concept in addition to the other three convergence concepts defined in section 3 of Chapter 1 . For A as
and B defined as in (3.1 1 ), it can be shown, using the fact that the Wiener pro cess has independent stationary increments z(ti+ 1 ) - z (t;) whose variance is ti+ l - t;. that B - A = t-s =F O. TI1creforc we can get a multitude of stochastic integrals depending on whether we take limit expressions of type A or B or a convex combination of these two types. Ito's analysis uses limit expressions of type A and this is the type of stochastic integral that we consider next. For another type of stochastic integral sec section 1 5 at the end of this chapter.
3.2.
Rigorous exposition
This analysis follows Arnold (1 974, pp. 57-75), Bharucha-Rcid (1972, pp. 221 -228}, Gihman and Skorohod ( 1 972, pp. 1 1 -32) and Ladde and Laksh mikantham (1980, pp. 1 14- 1 22). Our purpose now is to define carefully Ito's stochastic integral and to present its basic properties. Let z, be a Wiener process for t � 0, defined on some probability space (i1, �, P). A family of a-fields .
1
73
Stochastic calculus
(2) .-F, contains the a-field generated by z3, 0 �s � t, i.c. .-F, -�a(z3,0�s�t)
= !!r1 ; and
(3) .� is independent of the a-field generated by the increment zu - z1, t � u < oo , and denoted by a(zu - z,, t � u < oo) Note that condition (2) means that z, is measurable with respect to :F, for every t � 0. Condition (3) means that for h = u-t, t � u < oo , the process z(t+h) .
- z (h) is independent of any of the events of the a-field .� . In particular, con dition (3) means that ff0 can contain only events that arc independent of the entire process z,, t � 0.
As an example consider the family g;; = �,. It is the smallest possible non anticipating family of a-fields. It is often desirable to augment .2� with other t u < oo) for example events de events that are independent of scribing initial conditions. In the case of stochastic differential equations we usually take -� = c), where c is a random variable independent of
a(zu - z,, �
a(,q',, z,). Consider now a(t, w): [0, T]
,
a(zu -
x
n -+ R which is assumed to be measurable in (t, w). It is said to be nonanticipating with respect to the family of a-fields � if it satisfies the two conditions: T] , and ( I ) the sample path t, ) is.�-measurable for all (2) the integral
a(
tE [0,
·
T
J I a(t, w) 12 d t
(3.12)
0
is fmitc w .p.l. Note that for a function w) with continuous sample paths w.p.l, the last integral is an ordinary Riemann integral. In more general cases (3.12) is taken to be a Lebesgue integral. This defmition generalizes to matrix-valued functions, in which case the norm in (3. 1 2) is defined as follows:
a(t,
I a(t, w) I
=
( f f a� )'12
=
(tr
a a' ) 1/2 , •
where tr denotes trace and the prime denotes transpose. A special class of nonanticipating functions is the class of nonanticipating step functions. The nonanticipating function w) is called a tep function if there exists a partition of the interval T � say
[0, ]
0 to < t =
1
< ... < t
n
=
T,
a(t,
s
(3.13)
Stochastic methods in economics and finance
74
a(t, w) a(t;, w)
lt;, t;+ 1 ) 0, a(t, w): [0, T] stochastic integral a
such that for t E for i = 1 , 2, ... , n - l . For such = step functions we now define Ito's stochastic integral. x n � R a nonantici Let (n, .'F, P) be a probability space, pating step function for a partition of the form of (3.13) and x n � R a Wiener process. The of with respect to over the interval is a real-valued random variable denoted by and defined as
[0, T]
/(a)
z(t, w): [0, T] z
T
/(a) = J(a(w)) = J a(t, w) dz(t, w) n = J adz = .� a(t; _ 1 , w) [z(t,. , w) - z(t; _ 1 , w)] t=l 0 n = � a(ti - l) [z(t;) - z(ti - 1 )] 1 0
T
(3. 14)
w
The presence of E n emphasizes the fact that Ito's integral is a random vari able and the omission of is sometimes done for notational convenience. It is important to remark that in the defmition of the step function, and consequently in the definition of stochastic integral, the left-hand side endpoints of the sub intervals are used for evaluation. Next, consider an arbitrary nonanticipating function Suppose that is a sequence of nonanticipating step functions. It is natural to ask if the function could be approximated in some sense by An affirmative answer is provided by the next lemma.
w
a(t, w).
{an }
a
{an}.
a(t, w)
Lemma 3.1 . Let be a nonanticipating function. Then there exists a se quence of nonanticipating step functions such that as n --+ oo
J l an (t) - a(t) l 2 dt --+ 0
{an }
T
w.p.l.
(3.15)
0
For a proof of this lemma see Arnold ( I 974 , p. 67). The result o f tlus lemma is strong and it implies the weaker result, i.e. as n --+ oo ,
(3 . 1 6)
Stochastic calculus
75
This follows from the fact that convergence w .p.l implies convergence in proba bility, as is indicated in section
3
of Chapter
I.
The general existence of Ito's
stochastic integral will be established by showing that for a sequence of non
{an } satisfying (3.16) an dz converges in probability to J[ adz, i.e.
anticipating step functions
J
T
an dz
0
p
the sequence of integrals
T
J
�
Jl
(3.17)
adz.
0
{an } is a sequence of nonanticipating step functions approximating a nonanticipating function a in probability as in (3.16) and observe that
Suppose that
J
0
l an - am l 2 d t � 2
From our assumption about
J
as n, m
0
J
T
l a - an l 2 d t + 2
0
l a - am l 2 dt.
{an } satisfying (3.16), we conclude that
P
T
0
J
T
T
{3.18)
l an - am l 2 d t � O,
� oo. If we show that Jr an dz is a Cauchy sequence converging in proba
bility, then we will conclude that there exists a random variable to which it con verges
in probability. A technical lemma is needed in establishing that J[ an dz is
a stochastic Cauchy sequence. and cS
Lemma 3.2.
> 0,
P
Let
f
a(t, w) be a nonanticipating step function. Then for all > 0
[ I o(t) dz(t) > ] 6
.;;
: +P
6
For a proof of this lemma see Arnold Applying this lemma to
lim sup P
n,m-oo
l
J
an - am
o(t) 1 1 d t >
(I 974, pp. 68-69).
j.
we obtain
T
0
[! I
T
an (t) dz{t) -
J
0
am (t) dz(t) > cS
J (3.19)
76
Stochastic methods in economics and finance
From (3.1 8) and (3.19) we conclude that i.e.
IJ' an dz is a stochastic Cauchy sequence,
J an (t)dz(t) - J am (t)dz(t) T
T
0
0
J
> e -+ 0,
as m -+ Therefore (3.17) holds and the limit is unique w .p .l and independent of the choice of the sequence of nonanticipating step functions satisfying (3.1 6). We summarize our results in
n, {an}
I[ adz
oo.
a(t, w)
Lemma 3.3. Let be a nonanticipating function and suppose that the se quence of nonanticipating step functions approximates in the sense of (3.16). For each n, suppose that is defrned by (3.14). Then, as there exists a random variable such that
{a,} /(an) [ I a(t, w)dz(t, w) T
T
J an (t, w) dz(t, w) - J a(t, w) dz(t, w)
0
0
a
n -+
J
>e
oo
-+ 0.
(3.20)
The above analysis leads to a definition of Ito's stochastic integral. It is a defini tion based on the concept of convergence in probability as in (3.20). Other defi nitions can be given also based on a different concept of convergence. Shortly we will give such a definition based on convergence in the mean square and refer the reader to further sources on the subject of stochastic integration.
Defmition. Let (Q, .OF, P) be a probability space and consider the Wiener pro cess and the nonanticipating function both defined on [0, T] x n. of with respect to over the interval [0, T] , i.e.
z(t, w) Ito's stochastic integral a
z
a(t, w),
J a(t, w)dz(t, w) T
0
denoted by
/(a), is a random variable, unique w.p.l and defrned as the lmit in
probability, of the stochastic Cauchy sequence
i
T
J an(t, w)dz(t, w) � J a(t,w)dz(t, w) =l(a) T
0
0
{a, }
as lemma 3.3 indicates. Here is a sequence of nonanticipating step func tions that approximates in the sense of convergence in probability, i.e.
a
77
Stochastic calculus
We repeat that /(a) is unique w.p.l and independent of the choice of the se quence {a11 }. Uniqueness is proved in friedman (1975, p. 66). An alternative definition. As we remarked earlier in this section, an alternative definition of Ito's stochastic integral can be given using the concept of conver gence in the mean square instead of convergence in probability. Since conver gence in the mean square implies convergence in probability , the definition we arc about to give is more general and it yields as a special case the dcfmition al ready given. The alternative definition is based on the following Lemma 3.4. Let a(t, w) be a nonanticipating function such that
J
T
0
E l a(t, w) l 2 dt < oo.
(3.21)
Then there exists a sequence of nonanticipating step functions {an } satisfying for each n
j E I a, (t, w) 1 2 d t < oo, T
0
and approximating a in the mean square limit, i.e. as n 4 oo T
J
E l a,1 (t) - a(t) l 2 d t 4 0.
0
Furthermore, as n 4 oo, there exists a random variable J[ a(t) dz(t) such that
E
J
T
0
an (t) dz(t) -
J
T
0
a(t) dz(t)
2
4 0.
(3.22)
For a proof sec Arnold (1 974, pp. 71 - 73) and Gihman and Skorohod (1972, pp. 1 1 - 1 5). Note that in order to obtain this convergence in the mean square we have to assume (3 .21) which is more than is assumed in lemma 3 . 1 . The al ternative definition based on lemma 3.4 is now similar to the earlier dcfmition
78
Stochastic methods in economics and finance
with the only difference being the concept of convergence. Specifically, we de as the mean square limit of the Cauchy sequence J'[ n fine JJ' as in (3 .22). Such a limit exists, is unique and independent of the choice of because J'[ d is a Cauchy sequence satisfying
a
a(t, w)dz(t, w) (t, w) dz (t, w) {an } an z E
T
T
J an (t) dz(t) - J am (t) dz(t)
2
�
0,
0
0
as m , n � oo. This concludes our discussion on the concept of a stochastic integral. Before we state some of the basic properties of stochastic integrals we suggest some ad ditional references for the reader with an interest in this area. They are Meyer ( I 967), McKean (1969), Metivier and Pellaumail ( 1 980) and Friedman (1975). Having given a meaning to f adz we state some of its properties.
3.3.
Properties ofIto's stochastic integral
Ito's stochastic integral has several useful properties. We summarize some of these properties in the following theorems.
z(t, w): [
Theorem 3.1. Let (n, �, P) be a probability space and 0 , T] x n � R a Wiener process. and are real-valued, nonanticipating functions de ( I ) If fined on [0, T] x n and E R and E R, then
a2 (t, w) a1 a2
a 1 (t, w)
T
T
T
J (a 1 a1 + a2 a2)dz = a 1 J a1 dz + a2 J a2 dz. 0
0
0
a1
(2)
a2
( 1 ) and let a 1 (w) and a2 (w) be real a 1 a 1 + a2 a2 is a nonanticipating function on
are as in Assume that and valued random variables such that [0, T]. Then T
T
T
J (a1 a1 + a2 a2)dz = a1 J a1 dz + a2 J a2 dz. 0
(3) Let function of
0
0
[s, u ] be a subset of [0, T] and Jet X(s, [s, u ] . Then
u1
denote the characteristic
Stochastic calculus
79
J X(s.u 1 dz z(u) - z(s). T
=
0
{4) Suppose that a(t, w) is a nonanticipating function on If E I a(s) 1 2 ds < oo. Then E If a(t) dz(t) = 0 and T
[0, T] such that
T
f a(t}dz(t) 1 2 = f E I a(t) 1 2 dt.
EI
0
0
For proofs of these propositions see Gihman and Skorohod (1972, pp. 1 1 -1 3). Note that properties (I )-(3) are similar to the properties of the ordinary Rie mann-Stieljes integral. Convergence in probability is established next for a sequence of nonantici approx.imating in probability. pating arbitrary as opposed to step functions
a
{an}
z(t,
]
Theorem 3.2. Let (Q, :F, P) be a probability space, w): [0, T X n ... R a Wiener process, a sequence of arbitrary real-valued nonanticipating func tions defmed on [0, T] x n, and w): [0, T] x n --. R a nonanticipating func tion. Suppose that as n --. oo
{an}
a(t,
T
J 1 an(t) - a(t) 1 2 ds --. 0. p
(3.23)
0
Then T
T
J an dz --. J p
0
(3.24)
adz,
0
where in both (3.23) and (3.24) we have convergence in probability. For a proof see Arnold (1974, p. 74). The last theorem we want to present is about Ito's stochastic integral as a function of its upper limit. Suppose that X( o , ) is the characteristic function of t [0, with [0, C [0, T) . Suppose that is a nonanticipating function on [0, ] for each such that 0 T, where T is arbitrarily large. Define the process
t]
t,
t]
a
�t � t x(t) = x(t, w) = J a(s , w) dz(s, w) = J a(s) X(o, t) dz(s). T
0
0
t
(3.25}
Stochastic methods in economics and finance
80
Note that x(t) is a real-valued stochastic process defined uniquely up to stochastic equivalence for t E [0, T], with x (O) = 0 w.p.l. The following theorem states some of the properties of x(t) under the assumption that we have selected a separable version of x (t). (See section 7 of Chapter 1.) Theorem 3.3. Let {Q, :F, P) be a probability space, z(t, w): [0, T] x n -+ R a Wiener process, a(t, w): [0, T] x n -+ R a nonanticipating function and define x (t), t E [0, T] as in (3.25). Then: (1) x(t, w) is �-measurable and thus nonanticipating. (2) If J� E I a (s, w) 1 2 d s < oo for all t E [0, T], then {x,, §;) is a real-valued martingale. Also,
Ex, = 0
and
E I x1 1 2
=
t
J E I a(s , w) 1 2 ds.
(3.26)
0
(3) x(t, w) has continuous sample paths w.p.l. See Arnold (1 974, pp. 8 1 -84). Note that since (x,, .�) is a martingale, (x1-xr , �) is a martingale also for 0 < r � t < T and ( I x, - xr I , .¥,) is a submartingale. Therefore we may apply the various theorems on martingales and submartin gales available to us to obtain useful results about (x,, .� ) and (x1 -xr , F,). The above analysis concludes our discussion on stochastic integration. For a more detailed exposition of this topic the reader is referred to McKean {1969) and the more recent book by Metivier and Pellaumail (I 980). The reader should also be informed that at a more advanced level a new theory of stochastic integration has been developed called the Strasbourg approach. The primary reference here is Meyer ( 1976). For a fascinating application of the Strasbourg approach to economic analysis see Harrison and Pliska (1981 ) . They use the Strasbourg approach in developing a theory of a frictionless security market with continuous trading.
4. Ito's lemma We are now ready to state the definition of stochastic differential and then state and prove Ito's lemma which is the basic stochastic calculus rule for computing stochastic differentials of composite random functions. Consider a probability space (n, !F, P), a stochastic process x(t, w) : [0, T] x n -+ R that is measurable for each t E [0, T] with respect to �, and a Wiener pro cess z(t, w) : [0, T] x n -+ R. Assume that a(t, w): [0, T] x n -+ R is a nonanti-
Stochastic calculus
81
cipating function on [0, T] and also that f(t, w): [0, T ] x n � R is measurable for each t E [0, T] with respect to .� and also that J[l f(t, w) I d t < oo w.p.l. Let 0 � r � s � Tand suppose
x (s) - x (r) =
s
s
J f(t, w) d t + J a(t, w) dz(t, w). r
r
The stochastic differentialoftheprocess x (t) is defined to be the quantity f(t)d t + a(t) dz(t) and is denoted as dx (t), i.e.
dx(t) = f(t) dt + a(t) dz(t). Next we state and prove Ito's lemma which originally appeared in Ito (1951a) and later in Ito (1961). In our presentation below we follow Arnold (1974, pp. 96-99), Gihman and Skorohod (1969, pp. 387 - 389) and Ladde and Lakshmi kantham (1980, pp. 1 22 - 126). Lemma 4.1 (Ito). Let u (t, x): [0, T] x R � R be a continuous nonrandom func tion with continuous partial derivatives u1 , ux and uxx · Suppose that x(t) = x(t, w): [0, T] x n � R is a process with stochastic differential
dx(t) = f(t) d t + a(t) dz(t). Let y(t) = u(t, x (t)). Then the proccssy(t) has also a differential on [0, T] given by
dy(t)
=
[u1(t,x(t)) + ux(t,x(t))f(t) + i uxx (t,x(t)) a2 {t)] d t + ux(t,x(t)) a(t) dz(t).
It is sufficient to prove the lemma for step functions f and a because the general case may be obtained by taking limits. Also, it is sufficient to prove the lemma for a subinterval where f and a arc constant as functions of t, i.e . f(t, w)
Proof.
= [(v;) and a(t, w) = a(w). Consider the partition 1
0 < I < t < ... < tn = t � T. Since y(t)
=
2
u(t, x(t)) withy (O) = u(O, x(O)), then n
y(t) -y(O) = k=L [u(tk , x (tk)) - u(tk - 1 , x(tk - I ))] . l
(4. 1)
Stochastic methods in economics andfinance
82
Using Taylor's theorem we can write u(tk , x (tk)) - u (tk - l ' x(tk- I ))
=
u , (tk _ 1 + 8 k (tk- tk _ 1 ), x (tk _ 1 )) (tk-tk _ 1 ) + ux (tk_ 1 , x(tk_ 1 )) (x (tk) - x(tk_ 1 )) -
I
+ 2 uxx(tk - 1 , x(tk- 1 ) + f)k [x (tk ) - x(tk- I )]) (x (tk) - x(tk_ 1 )? ,
(4.2)
where 0 < fJ k < 1 and also 0 < 7fk < 1 . The first observation we want to estab lish is that the limit of y (t) - y (O) does not change in (4.1) as l) n � 0. From the continuity assumption of x(t) , u, and uxx , and the boundedness of x (t), we con clude that there are random variables a, and {3n that approach 0 w p l as l) n = max (tk - tk- 1 ) � 0, satisfying the two inequalities .
.
and
Taking summations in the fust inequality above we obtain
I 'Lu,(tk - 1
+ flk (tk - tk- l ), x(tk- 1 )) (tk - tk - 1 ) -
- 'Lu,(tk - t • x (tk_ 1 )) (tk - tk _ 1 ) I � � a, (tk - tk _ 1 ).
For the second inequality we remark that
and also that
Therefore , taking summations and using the results from the last two equations we conclude
I LUXX (tk - 1 , x (tk- 1 ) + ek [x(tk) - x(tk- I )]) [x(tk) - x(tk - 1 )]2 -
'Luxx (tk _ 1 , x (tk
_
� {3n � [x(tk) - x(tk
_
1
)) [x (tk ) - x (tk
1 )]2
.!; 0,
_
1 )P
I
83
Stochastic calculus
as
o, -+ 0. Thus, the limit ofy (t) - y(O) in (4. 1 ) does not change as on -+ 0. Putting (4.2) into (4.1) our proof will be completed if we show that
L {u, (tk - l , x (tk _ 1 )) (tk - tk _ 1 ) + ux (tk _ 1 , x(tk _ 1 )) [x(tk ) -x(tk _ 1 )1 + � uxx (tk - 1 , x(tk - 1 )) [x(tk ) -x(tk - 1 )]2 } .!; t
J [u, (s,x(s)) + ux (s, x(s))f+ � uxx (s,x(s)) a2 ] ds + J ux (s, x(s)) adz(s). t
0
(4.3)
0
Observe that as
o, -+ 0,
t
I: u,(tk - P x(tk_ 1 ))(tk - tk _ 1 ) -+
J u,(s,x(s)) ds
w.p.l
(4.4)
0
and also
p
'L ux (tk _ 1 ,x(tk - 1 )) [x(tk ) - x(tk_ 1 )] -+ t
t
J ux (s,x(s)[ds + J ux (s,x(s)) adz(s). 0
(4.5)
0
Both (4.4) and (4.5) hold because of the continuity assumption. Tltis means that in (4.3) it remains to study
Lux x (tk - 1 , x (tk - )) [x (tk ) - x (tk - )]2 =[2 LUxx (tk - 1 ' x(tk- 1 ))(tk - tk - 1 )2 + 2[a I:uxx (tk - l , x(tk _ 1 )) (tk - tk _ 1 ) [z(tk ) - z(tk _ 1 )] + a2 �uxx (tk _ 1 , x(tk _ 1 )) [z(tk ) - z (tk _ 1 )]2 . t
t
(4 .6)
By the assumption of the continuity of uxx and the continuity of z(t), the first two terms in (4.6) converge to w.p.l. That leaves only to show that as
0
'Luxx (tk - 1 ' x(tk - ))[z (tk ) - z(tk _ 1 )F .!; 1
To prove (4.7) first note that as
on -+ 0
LUxx (tk - l ' x(tk _ 1 ))(tk - tk _ 1 ) -+
t
t
J uxx (s, x (s)) ds.
0
J uxx (s, x (s)) ds
0
on -+ 0,
(4.7)
)
Stochastic methods in economics and finance
84
w.p.l. Therefore (4.7) will be established if it is shown that as Sn -+ 0
�uxx(tk- l ' x(tk - l ))([z(tk ) - z(tk - l )]2 - (tk - tk _ 1 )) � 0.
(4.8)
For notational convenience denote the left-hand side of (4.8) by Sn . Thus (4.8) can be written as p sn � o
for Sn
-+
(4.9)
0. For (4.9) we use a truncation technique. Define if I x(t;) I � N for all
i � k,
otherwise, and let
and finally,
Observe that Es: = 0 and E(S:)2 -+ 0 as Sn -+ 0 because the ek are independent of e'}fh other and of uxx (tk _ I ' x (tk _ 1 )) ��- 1 • Furthermore, this implies that s: -+ 0 as Sn -+ 0 for each N. Also, the error of the truncation is given by
and such error can be made arbitrarily small by choosing N large because x (s) is finite w .p.l. In conclusion,
P[ I Sn I > e ] � P[ I S� I > e] + P[Sn -:1= s: 1
(4.10)
with both terms of the right-hand side of (4.10) being arbitrarily small. Thus (4.9) holds. Thls proves the theorem in the special case. Turning next to the gen eral case we observe that we may choose sequences {fn } and {on } of step func tions such that w .p.l
85
Stochastic calculus r
J I fn (s) -f(s) I d s 0
-+
0
t
J I on (s) - o(s) 1 2 ds -+ 0 0
and the sequence of processes
xn (t) = x, (0) +
t
t
J fn (s) ds + J on (t) dz(t)
0
0
converges uniformly w.p.l to x (t). Then the sequence of processes Yn (t) = u(t, xn {t)) also converges uniformly w.p.l to y(t). Taking the limit as n -+ oo below we obtain Ito's lemma
Yn (t)- Yn (O) =
t
J [u,(s, xn (s)) + ux (s,xn (s))fn (s) 0
+ � uxx (s, xn (s)) a� (s)) ds +
I
J ux (s,xn (s)) on (s) dz(s).
0
Before we move on to some examples using Ito's lemma we state a straightfor ward generalization of this result.
k Lemma 4.2 (Generalized Ito). Let u(t, x): [0, T] x Rd -+ R denote a continu ous nonrandom function such that its partial derivatives u,, ux ux - x . are conI I I . tmuous, where ·'
u,
=
ux .
=
ux;xi
=
I
a ar u (t,x), a aX; u(t, x), a u(t, x), . a-. ax IxI
Suppose that x(t) = x(t, w): ferential
i = 1 , 2, . . , d, .
. · l,J � � d'
[0 , T] x n -+ Rd is a process with stochastic dif
dx(t) = f(t) d t + o(t) dz(t),
Stochastic methods in economics and finance
86
f(t) f(t,
= where w): both arguments, and
[0, T]
x
n -+ Rd
is measurable in
(t, w) i.e. measurable in
a(t) = a(t, ) [0, T] n -+ Rd Rm. Here a is a (d m)-matrix-valued function, nonanticipating in [0, T] and finally z{t) = z(t' ) [0, T] n -+ Rm is a m-dimensional Wiener process. Let y (t) = u(t, x(t)). Then the process y (t) also has a differential on [0, T] given by dy (t) = [u1(t,x(t}) + ux (t,x(t))f(t) + � f f ux x/t, x(t)} [a(t)a' (t)];;] dt ; (4. 1 1 ) + ux (t, x(t)) a(t) dz(t). w :
x
w :
X
X
X
This completes the statement of the generalized Ito's lemma. Note that the double summation can also be written as
uxx ux
= is a (d x d)-matrix whose elements are k-vectors. Thus, an alwhere o' � is ternative expres i n for ·X ·
dy (t)
We recapitulate the analysis of this section by emphasizing three points. First, Ito's lemma is a useful result because it allows us to compute stochastic differentials of arbitrary functions having as an argument a stochastic process which itself is assumed to possess a stochastic differential. In this respect Ito's formula is as useful as the chain rule of ordinary calculus. Secondly, given an Ito stochastic process with respect to a given Wiener process and letting y (t) = to be a new process, Ito's formula gives us the stochastic differential of is also given with respect to where the same Wiener process. Thirdly, an inspection of the proof of Ito's lemma reveals that it consists of an application of Taylor's theorem of advanced calculus and several probabilistic arguments to establish the convergence of certain quantities to appropriate inte grals. Therefore, the reader may obtain Ito's formula by applying Taylor's theorem instead of remembering the specific result. This is illustrated below in three cases.
z(t)
u(t,x(t)) y(t),
x(t) dy (t)
Stochastic calculus
87
Case 1: This corresponds to the generalized Ito's lemma. Given
/(t, w): [O, T) x n -. Rd , a(t, w): [0, T ) X n ... R d X Rm,
z (t, w): [0, T) X n -. Rm ,
that satisfy the assumptions of Ito's lemma and the process x(t) having stochastic differential
dx;(t) =f; (t) dt + a;,(t)dz,(t),
with i = 1 , 2, . .. , d, and r =
1 , 2, . .. , m , we define y(t) = u(t , x(t)): [0, T] x Rd --. R k .
Note that consistency among the ranges of the three processes holds
dx(t)
= f(t) dt
+
dz(t).
a(t)
Now apply Taylor's theorem, make the necessary substitutions, and rearrange terms. Note that t is dropped and summations are from 1 to d:
dy(t)
= =
u1dt + L uxl_dx; + � LLUx I. x - dx;dxi I u1dt + L Ux .(f;dt + a;,dz,) + i LL ux · x .(f;dt + a;,dz,) (f;dt + ai, dz,) l
1
I = u1dt + L Ux _f;d t + L Ux . a;,dz, + � LL Ux - x Ja;, aj,) dz,dz� t I 1 1
= u1 dt + uxfdt + i LL ux -x _ [a a' ];i dt + ux adz I '
= (u, + uxf+ i LLllx - x .fa a' ];j) dt + ux a dz. '
I
Note that the following multiplication rules have been used
d t X d t = (d t)2 = 0, dt x dz, = 0,
dz, x dzq = 0
for
r
dz, x dzQ = dt
for
r
=I= q ,
= q.
Stochastic methods in economics and finance
88
Case 2:
In this case we partially specialize the results of case
I . Suppose that
f(t, w): [0, T] X n -+ Rd ,
o(t, w): [0, T] X n -+ Rd,
z (t, w): [0, T] x !L -+ R 1 • Also, let the process x (t) have differential dx;(t) = f;(t) d t + o;(t) d z (t), with i = I , 2 , ... , d. Define y (t) = u (t,x(t)) = u (t, x1 (t), ... , x (t)): [0, T] x Rd -+ R. d Note symbolically that consistency of dimensionalities holds: dx (t)
=
f(t) d t
+
o (t)
dz (t).
As in case 1 , we apply Taylor's theorem, make the necessary substitutions, re arrange terms, and use the multiplication rules (d t)2
=
0;
(dz?
=
d t;
dt x dz = 0
to obtain (all summations are from dy(t)
= =
Case 3:
1
(4. 1 2)
to d; t is dropped):
ut dt + � ux . dx; + � �� ux · x · dx; dxi '
' I
ut d t + � ux .(f;dt + a1 dz) + l :r:rux ·x · (f;dt + O;dz)Ujdt + oidz) l
I I
=
utdt + :r ux Jidt + :r ux . oidz + � :r:r ux ·x · oioidt
=
(u t + :r ux .fi + � :r:r ux x . o1 oi) dt + (:r ux . oi) dz. 1
'
'
'
' I
I
I
In this case we further specialize case 2 by assuming that f and o are real-valued functions. This case is the one stated in Ito's lemma. We proceed im mediately with the calculations:
Stochastic calculus
dy(t)
=
= =
89
u,d t + ux dx + t uxx (dx)2 u,dt + ux (fd t + odz) + � uxx (fdt + odz)2
u,dt + ux fdt + ux odz + � uxx o2 dt = (u, + uxf + � uxx o2 ) d t + ux adz.
This is the result of Ito's lemma. Note that the multiplication rules of (4.12) are applied to yield
S.
Examples
Several examples from economics and fmance which illustrate the use of Ito's lemma are presented in Chapters 3 and 4. Here we give some examples of Ito's lemma which are of a mathematical nature. Example I . Let y (t) = u(x(O), nonzero real number. Then, dy(t)
t) = x(O)e0 t with dx(t) = ax(t) d t, where a is a
=
u,dt + ux dx + � uxx (dx)2
=
ax(O) e0, d t.
Thus,y(t) solves dx(t). Example 2. Suppose thaty(t) = u(x(t)) = ex ( t) with y (O) = eX ( o ) > 0. Let dx (t) = - �
o2 (t) dt + o(t)dz(t).
Ito's lemma yields
dy(t)
=
=
=
=
u,dt + ux dx(t) + ! uxx (dx (t))2 ex (t) [ - � o2 (t) dt + o(t) dz(t)) x + � e ( t ) [- � o2 (t) dt + o(t) dz(t)F - t e-X(t)o2 (t) dt + ex o(t) dz(t) + � ex (t) o2 (t) d t ex (t) o(t) dz{t).
90
Stochastic methods in economics and finance
Note that the process
y(t)
= y(O) ex ( t )
=
y(O) cxp
{- I i
o2
(s) d s +
I
o(s)dz(s)
}
satisfies the equation of dy(t) for t E [0, T]. Example 3. Suppose that y(t) = u(x (t)) = ex(t ) , where
dx(t) = - � dt + dz (t) and x (O)
= 0. Applying Ito's lemma we obtain:
dy(t)
= = =
u1dt + ux dx(t) + � uxx [dx(t)]2 xt x e ( ) [ - � d t + d z(t) ] + � e ( t ) [ - � dt + dz (t)]l ex (r) dz(t).
2
and it illustrates that, given the stochastic
t dy(t) = ex ( ) dz(t) = y(t) dz(t)
(5.1)
This is a special case of example differential equation
with initial condition y(O)
y(t) = 1
exp { - �
= ex ( O ) = 1 , its solution is (5.2)
t + z(t)}.
This result is useful because it illustrates the difference between ordinary dif· ferential equations and stochastic differential equations. Observe that if (5 .1) were an ordinary differential equation its solution would be y(t) = exp [z(t)], which is different from (5 .2). Example 4. Suppose that y(t) = u(x1 , x2 ) = tz(t) with
From Ito's lemma we compute for i =
I 2 ,
91
Stochastic calculus
dy(t)
= =
=
u, d t + � uxr. dx;(t) + � �� uxr·x·dx;(t) dxi(t) I z(t) d t + tdz(t) + � [dtdz{t) + dz{t)dt] z (t) d t + tdz(t).
Example S. Suppose that y(t) = u (t, x1 , x2 ) =
tx1 x2 with
dx 1 (t) [1 (t) d t + a1 (t) dz(t), dx2 (t) = a2 (t) dz(t). =
Applying Ito's lemma we compute,
dy(t)
u, dt + � ux1 dx;(t) + � �� ux dx;(t) dxi(t) r I = x1 x2 d t + tx2 dx 1 (t) + tx1 dx2 (t) + + � [tdx 1 (t) dx2 (t) + tdx2 (t) dx 1 (t)] = x 1 x2 d t + tx2 [[1 d t + a1 dz] + tx1 a2 dz + + � [ta1 a2 d t + ta1 a2 dt] (x1 x2 + fx2 f1 + IOt a2)dt + {tx2 a1 + tx1 a2) dz. =
·x .
=
Example 6. Suppose y(t) u (x (t)) = u(z (t)), where u is assumed to be twice continuously differentiable with respect to x,x(O) z {O) = 0 and dx{t) = dz(t). Ito's lemma gives =
dy(t)
= =
=
u,dt + ux dx(t) + � uxx [dx{t)F u'(z (t)) dz(t) + � u"(z(t)) dt.
(5.3)
Here a prime denotes ordinary derivative. If we write becomes
y (t) = u(z(t)) = u (O) +
t
{5.3) in integral form it
t
J u'(z(s))dz(s) + � J u"(z(s)) ds.
0
(5.4)
0
From (5.4), solving for the second term in the right-hand side, we obtain: t
t
J u'(z (s))dz(s) = u(z(t)) - u(z(O)) - � J u"(z (s)) ds.
0
0
(5.5)
Stochastic methods in economics and finance
92
Eq. (5 .5) is usually called the fundamental theorem of calculus for Ito's stochastic
integral and it is useful because it expresses the integral in the left-hand side of (5.5) only in terms of an ordinary integral.
The above examples illustrate the use of Ito's lemma in a mathematical con
text which may seem far removed from applications in economic analysis. How
ever, we hasten to add that Ito's lemma has found numerous applications in finance
and economics and here we mention three broad areas of application : first, the modern theory of contingent claim valuation, with its primary model being the
celebrated option pricing formula of Black and Scholes� second, the formulation
of appropriate budget constraints in a stochastic context with Merton's work on
optimum consumption and portfolio rules being a primary model, and third, the formulation and analysis of optimal stochastic control problems with section
i n this chapter providing a useful illustration.
6.
10
Stochastic differential equations
In this section we study conditions that guarantee the existence and uniqueness
of solutions of stochastic differential equations of the Ito type. Such equations were first introduced in section 2 above in (2.1 2). Recall that such equations can be transformed into an integral equation as in tivated the analysis on stochastic integration. be
Consider a probability space
[0, oo ) , and let us write an d
(3.4) and that it was (3 .4) that
mo
(n, !F, P) and the interval [0, T] which can also
Ito stochastic differential equation, such as
x(t) = f(t,x(t))d t + o(t, x(t)) dz(t),
(6. 1 )
with initial condition
x(O, w) = c(w) = c. Eqs.
(6.2)
(6. 1 ) and (6.2) define a stochastic initial value problem and can be expressed
as a stochastic integral equation t
t
x (t) = c + J f(s,x(s))ds + J o(s, x (s)} dz(s). 0
As before,
(6.3)
0
x(t) is a real-valued stochastic process, f and are real-valued func tions measurable on [0, T] x R and z(t) is a Wiener process. We assume that x (t) o
is ff,-measurable and note that � must be independent of the a-field generated
Stochastic calculus
93
by zu - zt , u � t, for all t � 0. For our analysis it is enough to choose for � the smallest a-field with respect to wltich the initial random variable condition c(w}, and the random variables zs, s � t, are measurable. We are now ready to give a defmition. A stochastic process x (t) is called a solution of (6.1) and (6.2), i.e. ofan Ito stochastic differential equation on [0, T], if it satisfies the following three prop erties: ( I ) x (t) is .¥,-measurable, i.e. nonanticipating for t E [0, TL (2) the functions f and a are such that w .p.l T
J
0
l f(t,x(t)} l d t < oo
and
T
J
l a(t,x(t)} l 2 d t < oo ; and
0
(3) eq. (6.3) holds for all t E [0, T] w .p.l. In the theory and applications of stochastic differential equations questions of existence and uniqueness are important for analytical reasons. Usually, re searchers are interested in establislting the existence and uniqueness of their model described by a stochastic differential equation before they proceed to study the various probabilistic properties of the solution. Below we state the existence and uniqueness theorem for stochastic differential equations. In the next section we state various fundamental properties of such solutions. Theorem 6.1 . Consider eq. {6.1) with initial condition given by (6.2) and sup pose that the following assumptions are satisfied. {I) The functions f(t, x) and a (t, x) are defmed for t E [0, T] and x E R and are measurable with respect to all their arguments. (2) There exists a constant K > 0 such that for t E [0, T] and x E R , y E R,
1 /{t, x) - f(t,y) I + I a(t,x) - a(t,y) I � K l x - y I ,
(6.4) (6.5)
(3) The initial condition x(O, w) in (6.2) does not depend on z (t) and Ex(O, 2 w) < oo . Then there exists a solution of (6.1) satisfying the initial condition in (6.2} which is unique w.p.l, has continuous sample paths x (t) w.p.l,and sup, Ex(t)2 < oo . For a proof of tltis theorem see Gihrr.an and Skorohod (1972, pp. 40-43). The theorem is established by using a Picard- Lindelof iteration argument similar to the one used in ordinary differential equations and the Borel-Cantelli lemma.
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Stochastic methods in economics and finance
Obviously , the theorem holds for vector-valued functions x (t), z (t) and f(t), and the matrix-valued function o(t) with appropriate dimensionalities. Equation (6.4) is called the Lipschitz condition and (6.5) is called the Re striction on growth condition. Uniqueness of solutions means that if x 1 (t) and x 2 (t) are two solutions then (6.6) Note that if the functions f(t, x) and a (t, x) are defined on [0, T] x R and the assumptions of theorem 3.1 hold on every finite subinterval [0, T] of [0, oo ), then we say that the stochastic differential equation has a global solution defined for t E [0, oo ). This is the case for autonomous stochastic differential equations, where f(t, x) = f(x) and o(t, x) = a(x). For eq. (6.4) to hold it is sufficient that the functions f(t,x) and o(t,x) have both continuous partial derivatives of first order with respect to x for every t E [0, T] which are bounded on [0, T] x R. As a special case of {6. 1 ) consider the autonomous stochastic differential equa tion, treated in Gihman and Skorohod (1972, p. 1 06), d x(t) = f(x(t)) d t + o(x(t)) d z (t), where f(x) and a(x) are functions defined for a(x) satisfy the following conditions:
(6.7)
x E R. Suppose that f(x) and
I f(x) I + I a (x) I � K (I + I x I )
(6.8)
holds for some constant K > 0. Also, for each C> 0 there is a positive real num ber Lc such that for I x I � C and I y I � C we have
l[(x) - f(y) I + I o(x) - o(y) I � Le i x -y I .
(6.9)
Then for each random initial condition c {w) not dependent on z (t) there exists a unique solution of (6.7) satisfying the initial condition. This result is useful in many applications which yield autonomous stochastic differential equations. In such cases, eqs. (6.8) and (6.9) are used to establish the existence and uniqueness of solutions. Finally, let us consider an important special case of the class of stochastic dif ferential equations, namely the linear equations. Suppose that x(t) is a d-dimen sional process, a (t) is a (d x d)-matrix-valued function, o(t) is a (d x d)-matrix valued function, and z (t) is a d-dimensional Wiener process. Consider
Stochastic calculus
dx(t) = a(t)x(t) d t + o(t) dz(t)
95
{6.10)
with initial condition x(O) = c. Assume that the conditions for existence and uniqueness of solutions are satisfied and let us ask the question: What docs the solution look like in this special case? We claim that the solution of (6.1 0) on [0, T] has the form
x(t) = (I)
� + I -1 (s) o (s) dz(s� ,
(6. 1 1 )
where ¢(!) is the fundamental matrix of
x (t) = a(t)x (t).
(6.12)
Let us verify the claim. Start with equation
y (t) = c +
t
J ¢ -1 (s) o (s)dz{s)
{6.13)
0
which can be written in differential form as
dy(t) = ¢-1 {t) a (t) dz (t).
(6.14)
Observe from (6 . 1 1 ) and (6.13) that X
(t) = f/> (t)y {!).
{6.15)
Now we are in a position to apply Ito's lemma to show that dx(t) has the ap propriate form as in (6.10). To do so we use the two equations {6. 1 5) and (6.14). We have
a a [¢ (t)y {t)l dy{t)) dx{t) = at [¢(t)y {t)] d t + ay a2 + ! 2 -2 [¢ (t)y (t)l [dy(t)F ay = � (t)y (t) d t + f/> (t)dy (t) + 0 + 0 =
a(t)¢ (t)y {t) d t + o(t) dz{t}
=
a(t)¢ (1)¢-1 (t)x(t) d t + o(t)dz(t)
=
a(t)x (t) d t + o (t)dz{t) .
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96
To illustrate {6.1 1 ) consider the linear case given by
dx{t) = ax(t)d t + a{t) d z {t),
(6.16)
with x(O) = c. In this case since the matrix a is autonomous we know from ordi nary differential equations that the system x (t) = ax(t) has as fundamental matrix
¢(t) = ea t , t E [0, T], t 1 where ¢- (t) = e - a . Applying (6.1 1) we get
� I if>-1 (s) o(s) dz(s� � I o(s z(s�
x (t) = if>(t) = �·
=
+
e-
+
t
cea +
t
"'
)d
J ea
0
For a more detailed discussion of the issues discussed above and for additional related topics see Arnold (1974) and Gihman and Skorohod (1972). 7. Properties of solutions
The solutions of ordinary differential equations satisfy, under certain condi tions, two well-known properties called the property ofdependence of solutions on parameters and initial data and the property ofdifferentiability ofsolutions. It is natural to ask whether stochastic differential equations satisfy similar prop erties. In this section we show that the property of dependence of solutions on para meters and initial data holds, but the property of differentiability of solutions does not hold. The latter is due to the fact that if x (t) is the unique solution of an Ito stochastic differential equation then x(t) depends on the Wiener process z (t) whose nondifferentiability anywhere implies the nondifferentiability of x (t). However, if we defrne differentiability in the mean square we can show that under certain assumptions x(t) is mean square-differentiable. Beyond these two properties, solutions of stochastic differential equations satisfy some additional properties that are probabilistic in nature, i.e. the solutions are Markov processes
97
Stochastic calculus
and under certain assumptions they are diffusion processes. We now proceed to establish precisely the various properties. Consider the stochastic differential equation which depends on a parameter p, written as
dx(p, t) = f(p, t, x) d t + a(p, t, x) dz (t),
(7.1)
with initial condition x(p, 0) = c(p) for t E [0, T] and p E �, where a-' is the para meter set. Denote by x (p, t) the solution of (7 .1) and let Po E g>_ The property of dependence ofsolutions on parameters is established by the next theorem. Theorem 7.1. Let x(p, t) be the solution of (7.1) and suppose that f(p, t, x) and a(p, t, x) satisfy for all p the conditions of existence and uniqueness, i.e. (6.4) and (6.5). Also, assume the following: (7.2) (2) for every N> 0,
I x I � N, t E [0, T], as p
lim sup ( lf(p, t, x) - f(p0, t,x) I + I X
-+
Po then
a(p, t,x) - a(p0 , t, x) I ) = 0;
(3) there is a constant K, independent of p such that for t € [0, T]
Then, as p -+ Po , sup t
p
l x(t,p) - x(t,p0 ) l -+ 0.
(7.3)
For a proof of this theorem see Gihman and Skorohod {1972, pp. 54-55). As a special case of theorem 7.1 suppose that the functions f and a are inde pendent of the parameter p, i.e. let
dx(p, t) = f(t, x) d t + a(t, x)dz(t),
(7.4)
with initial condition x(p, 0) = c(p) for t E [0, T]. Here we maintain the depen dence of initial condition on the parameter p. As a corollary of theorem 7 .l we now obtain the property of the stochastically continuous dependence of solu tions on initial data, i.e. for eq. (7 .4) assuming existence and uniqueness of its solution, (7 .2) implies (7 .3).
Stochastic methods in economics and finance
98
We now define mean-square-differentiability for a stochastic process. Let x (t) be a stochastic process with t E [0, T] and let u E [0, T] be a specific point. We say that x(t) is mean-square-differentiable at t = u, with random variable y(u) as its derivative, if the second moments of x(t) and y {u) exist and satisfy for h > 0 EI
[x(u + h) - x(u)]/h - y(u) 1 2
�
0
as h � 0.
We apply the above definition to a stochastic process depending upon a para meter and given by t
x(s, t, c) = c + J f(v,x (s, v, c)) dv +
t
s
J a(v, x(s, v, c)) dz (v), s
{7.5 )
where v E [s, T], 0 � s < t � Tand x(s, s, c) = c E R. A question naturally arises: Under what assumptions is x (s, t, c) of (7.5) mean-square-differentiable with re spect to initial data c? The answer is given by the next theorem. Theorem 7.2. Suppose that /and a of (7.5) are continuouswith respect to (t,x)
and they have bounded first and second partial derivatives with respect to the argument x. Then for given t = u, u E [s, T], the solution x (s, u , c) is twice mean square-differentiable with respect to c and
a
u
& x(s, u, c) = 1 + J fx (v, x(s, v, c)) s u
a x (s, v, c) d v ac
a + J ax (v, x(s, v, c)) - x(s, v, c) dz(v). ac s For a proof of this theorem and its generalization see Gihman and Skorohod (1972, pp. 59-62}. Next we recall the defmition of a Markov process and estab lish the fact that solutions of stochastic differential equations are Markov pro cesses. A real-valued process x(t) for t E [0, T] defined on the probability space (n, .tF, P) is called a Markov process if for 0 � s � t � T and for every set B e &I, with 9'1. denoting the Borel a-field of R, the following equation holds w .p.l:
P[x(t) E B 1 .� ] = P[x(t} E B I a(x(s))].
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99
Here .'f_; is the a-field generated by x (s) for s E [0, T] and a(x(s)) denotes the a-field generated by the single random variable x(s). In words, the definition says that for a Markov process the past and the future are statistically independent when the present is known. For a Markov process x(t) we obtain from the theory of conditional probability the existence of a transition probability function, de noted by P(s, x, t, B), which yields the probability that x (t) E B, given that x (s) = s, for 0 � s � t � T. Theorem 7.3. Consider the stochastic differential equation (6 . I ) with initial con dition as in (6.2) and suppose that a unique solution x(t), t E [0, T] exists. Then x (t) is a Markov process whose initial probability distribution at t = 0 is c and whose transition probability is given by
P(s, x, t, B) = P[x(t) E B I x (s) = x ]. For a proof see Arnold (1 974, p. 147). A special class of Markov processes is the class of diffusion processes. A real-valued Markov process x(t), t E [0, T], with almost certainly continuous sample paths, is called a diffusion process if its transition probability P(s, x, t, B) satisfies the following three conditions for every s E [0, T],x E R and E > 0: th
(1) lim
I
f
t-s ly-x I > «=
-
P(s, x, t , dy) = 0;
(7.6)
(2) there exists a real-valued function f(s, x) such that th
lim
I
f
t-s ly-x 1 .-
-
€
(y-x) P(s,x, t, dy) = f(s,x);
(7.7)
(3) there exists a real-valued function h (s, x) such that tls
lim
1t-s
f (y-x)2 P(s,x, t, dy) = h(s,x). ly-x l .- t:
(7.8)
Note that f is called the drift coefficient and h is called the diffusion coeffi cient and they are obtained from conditions (2) and (3). We remark that condi tion ( 1 ) says that large changes in x (t) over a short period of time are improbable. Theorem 7 .4. Consider the stochastic differential equation (6.1 ) with initial con dition as in (6.2) and suppose that a unique solution x (t), t E [0, T] exists. Sup-
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Stochastic methods in economics and finance
pose also that the functions f and a are continuous with respect to t. Then the solution x(t) is a diffusion process with drift coefficient f(t, x) and diffusion coefficient h(t,x) = a2 (t,x}. For a proof qf this result see Arnold (1 974, p. 1 5 3). The usefulness of this last theorem can be explained as follows. Note from the definition of a diffusion process that the transition probability function P(s, x, t, B) is crucial in defining the drift coefficient f(t, x) and the diffusion coeffi. cient h(t, x). A question arises. Suppose that a diffusion process is given with coefficients f and h. Can we obtain P(s, x, t, B) from f and h? The answer is yes. Actually the decisive property of a diffusion process is that the transition proba bility P(s, x, t, B) is uniquely determined under certain assumptions by the co efficients f and h. This fact is surprising because f and h involve only the first and second moments which, in general, are not sufficient to define a distribu tion. Therefore given a stochastic differential equation that satisfies the hypo theses of theorem 7 .4 we know that its solution is a diffusion process which under certain assumptions could yield a transition probability without actually having to find an explicit solution. How is this done? We answer the question by making certain definitions and by stating relevant theorems. To each diffusion process with coefficients f = (fi), i = 1 ; 2, , d and h = (h;i), i,j = 1 , 2 , . , d we assign the second-order differential operator .
� =
...
.
�fi(s,x) z
a
a
-
X·'
+ 4 � � h;j (s, x) z
1
a
a
2 a
X·I X · I
(7.9)
gg can formally be written for every twice partially differentiable function g(x). For real-valued f and h (1 9) becomes .
!'2
= f(s ' x)
dx
d
-
d2 . + 2! lz (s x) dx2 '
(7.10)
Denote by Es,x expectation conditioned upon x at time s. The following theorem is important in our analysis. Theorem 7.5. Suppose that x(t) for t E [0, T) is a d-dimensional diffusion pro cess with continuous coefficients f(s, x) and h(s, x) and that conditions (1 ), (2)
and (3) in (7 .6), (7 .7) and (7 .8) hold uniformly in s E [0, T]. Let g(x) denote a continuous, bounded, real-valued function such that for s < t, t fixed,x E R d , u(s,x) = Es ,x g (x (t)) =
f g(y)P(s,x, t, dy). Rd
(7 .1 1 )
Stochastic calculus
101
Suppose u has continuous bounded partials
au . 32 u . 3x; ' 3x;3xi '
1 <:: i,j <:: d.
Then u (s, x) is differentiable with respect to s and satisfies the partial differen tial equation
au as
-
+
!'}u = 0,
(7.12)
with boundary condition u (s, x)-+ g(x) as s
-+
t.
For a proof of this theorem see Gihman and Skorohod ( 1969, p. 373). Eq. (7 . 1 2) is called Kolmogorov 's backward equation because differentiation is with respect to the backward arguments s and x, where s < t and x < y. Kolmogorov's back ward equation (7.12) enables us to determine P(s, x , t,y) which is uniquely de fined if we know (7 .1 1 ). If (7 . 1 2) has a unique solution for g a real-valued con tinuous and bounded function, we can for given f and h calculate u (s, x) and then from it calculate P(s, x, t, y). More specifically, for stochastic differential equa tions a corollary of theorem 7.5 is the following result. Theorem 7.6. Suppose that eq. (6.1) with initial condition as in (6.2) satisfies the hypotheses of theorem 7.4 and therefore has as a unique solution x (t) which is a diffusion process. Also, assume that f and a have continuous and bounded first- and second-order partial derivatives with respect to x. Let g(x) denote a continuous bounded real-valued function with continuous and bounded first and second-order derivatives and let for 0 <:. s <:. t <:. T and x E R,
u (s, x) = Es,x g(x(s, t, x)). Then u and its first- and second-order partial derivatives with respect to x and its partial derivative with respect to s are continuous and bounded and satisfy the equation
a u (s, x) + fiu(s,x) = O as
as
S -+ t, u (s,x) -+g(x).
The proof follows from theorem 7.5. This result shows that the study of stochastic differential equations and their solutions is closely related to the study of second order partial differential equations of the form of (7 .1 2).
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For the special class of autonomous stochastic differential equations such as (6.7) we immediately obtain as a corollary of theorems 7.3 and 7.4 that their solutions are Markov and diffusion processes. The last result of this section refers to the moments of solutions. Theorem 7.7. Consider the stochastic differential equation (6.1) with initial con
dition as in (6.2) and suppose that it has a unique solution. AJso, assume that
where n is a positive integer. Then the solution x (t), t E [0, T], of(6.1) satisfies E l x (t) l2 n � ( 1 + E i c l 2 n ) ec1 , E I X (t) - c 1 2 n � D( 1 + E I c 1 2n ) t eCt ,
where C = 2n (2n + 1 ) K 2 and D are constant depending only on n, Tand K of (6.4). For a proof see Arnold ( 1974, pp. 1 1 6-1 1 8). 8. Point equilibrium and stability A particular solution of a stochastic differential equation that plays an important role in many applications is the equilibrium solution. Consider eq. (6.1) defined for t E [0, T]. If there exists a nonrandom constant c such that for all t E [0, T], f(t , c) = a (t, c) = 0,
(8.1)
with x(t, w) = c for all t E [0, T] and w .p.l, then we say that c is a point equilib rium solution, or simply an equilibrium solution. Another terminology for the same concept is stationary point or steady state point. Observe that the defini tion specifics c, if it exists, as a nonrandom constant, or equivalently, as a point distribution. Various other definitions of the concept "equilibrium solution" exist. For a survey of such definitions the reader may consult Majumdar (1975). A differ ent definition than the one presented above will be given in the next section. Suppose that a point equilibrium solution exists for an autonomous stochastic differential equation of the Ito type such as
Stochastic calculus
103
(8.2)
dx(t) = f(x(t)) d t + o(x(t)) dz(t)
for t E [0, T]. In this section we study the stability of the equilibrium solution of an autonomous stochastic differential equation such as (8.2). For convenience and without loss of generality we assume that 0 is an equilibrium solution. We use two approaches to stochastic stability that yield similar results to acquaint the reader with both methods. We call these two approaches the Gihman-Skoro hod approach and the Liapunov-Kushner approach. 8. 1.
The Gilzman-Skorohod approach
This method is presented in Gihman -Skorohod ( 1972, pp. 145 - 1 5 1 ). Consider eq. (8.2) and suppose that for t E [0, oo) it has a unique equilibrium solution which is 0 such that
f(O) = o(O) = 0.
(8.3)
We say that the 0-equilibrium is stable if for any € > 0 there exists an 1J > 0 such that if I x I < 1J then as t � oo P [lim x(t) = 0] �
1 - €.
(8.4)
Here x = x (0, w), i.e. x is the initial condition and in (8 .4) the probability is con ditioned on x. If for any € > 0 there exists an 1J > 0 such that (8.4) holds for x E (0, 1J) or x E (-7], 0), then we say that the 0-equilibrium is right-stable and left stable, respectively. Conditions for stability are given in the next theorem.
(8.2) the conditions in (8.3) hold and that there exists an 1J > 0 such that o(x) > 0 for 0 < I x I < 1]. For the stability of the 0-equilibrium ( l ) from the right; (2) from the left; and (3) both, it is necessary Theorem 8.1. Assume that for eq.
and sufficient that for some respectively, where /1
=
and /2
=
{
J exp J
0�
0
f
- 11
exp
u�
1J
> 0, ( 1 ) /1 < oo ; (2) /2 < oo ; and (3)/1 + 12 < oo ,
}
2f(y) dy du 2 (y) 0
{- f
u
- 11
2/(y) 0
2
(y)
(8.5)
}
dy du.
(8.6)
Stochastic methods in economics and finance
104
For a proof see Gihman and Skorohod (1972, pp. 1 46- 148). Thls theorem is useful because it reduces the study of stochastic stability to a computation of (8.5) and (8.6). Sometimes it is difficult to establish the stochastic stability property of an equilibrium solution, possibly because such a property may not hold in a given model. In such a case the researcher may ask whether the solutions are bounded. In many applications the property of boundedness of solutions is the next most desirable property to that of stochastic stability. It comes as no surprise that equa tions similar to (8.5) and (8.6) are used to establish boundedness from above, as the next theorem states. Theorem 8.2. Assume that eq. (8.2) with initial condition x (O) = c has a unique solution denoted by x (t) with a(x) > 0 for x E R. Let
/1 (x) =
X
_[
and
/2 (x) = If /1
00
J
x
exp
exp
{
{
[ U
-
u
-
J
o
2f(y)
y
}
du
(8.7)
2[(y) dy d u . 02 (y)
(8.8)
02 (y)
d
}
(x) < oo and /2 (x) = oo, then for t E [0, T]
For a proof of this theorem sec Gihman and Skorohod (1972, p. 1 19). We now present the elements of the Liapunov-Kushner approach. 8. 2
The Liapunov- Kushner approach
This method is developed in Kushner (1967, pp. 27-76) and is an extension of the deterministic indirect stability method of Liapunov. Before we state the main theorem we discuss some preliminary notions. First, we introduce the function V(x) which plays the role of a Liapunov function. Its properties are described in theorem 8.3. Secondly, we define the set Qm = {x: V(x) < m < oo }. It is assumed that Qm is a bounded set.
Stochastic calculus
105
Tltirdly, �m V(x) is the differential operator given by (7 .1 0) and it plays a role equivalent to the role of the trajectory derivative in deterministic stability. It denotes the average time rate of change of the process V(x(t)) at time t given x (t) = x. For an equation such as (8.2) we have (8.9) Theorem 8.3. Let x (t) be a unique solution of (8.2) and suppose that V(x) is a continuous non-negative function wltich is bounded, and has bounded, continu ous first- and second-order derivatives in Qm . If qfim V(x) � 0 in Qm the x (t) � :X, where x E {x: �m V(x) = 0} n Qm with probability at least 1 - V(x)/m. For a proof of this theorem see Kushner ( 1 967, p. 42). To illustrate theorem 8.3 consider the simple example given by Kushner (1967, pp. 55-56). Let x(t) be the solution of an Ito equation of the form dx = ax d t + axdz.
(8. 1 0)
Choose as V(x) = x2 and note that for each set Qm = {x: x2 < m < oo}, V(x) is non-negative, bounded and has bounded first- and second-order derivatives in Qm . Applying (8.9) we get
If 2 a + a2 < 0, then �m V(x) � 0, and by theorem 8.3 , x (t) � 0 w.p.l because m can be made arbitrarily large. Two remarks are in order. First, both approaches to stability yield a result that shows the solution of an Ito equation not just staying within a small neigh borhood of the 0-equilibrium solution but actually the solution approaching the 0-equilibrium. Technically we describe this stability as a local asymptotic stochas tic stability. If the 0-equilibrium is stable and x(t) � 0 w.p.l for any nonrandom initial condition x e R , then we say that the 0-equilibrium is globally asympto tically stable or that stochastic stability in the large holds. Secondly, as in the deterministic case, finding an appropriate Liapunov func tion to use in establishing stability is usually difficult. Kushner (I 967, pp. 606 1 ) describes a method presented below to guide the researcher in ltis efforts for Ito equations such as (8.2). A similar method is also presented by Feller (1 954). As this method is described note its similarity to the Gihman-Skorohod method, and particularly eq. (8.5).
Stochastic methods in economics and finance
106
Assume that a Liapunov function exists with V(O) = 0 and V(x) > 0 for x =I= 0. Suppose that V(x) is a continuous and bounded function and has bounded continuous first- and second-order derivatives so that g;m V(x) is defined in some open bounded set Qm with (8.9) holding. Suppose that �m
V(x) = f(x) Vx (x) + � a2 (x) Vxx (x) � 0.
{8.1 1 )
From {8.1 1 ) we obtain:
Vxx (x) 2/(x) � - _:_ __:.::.......: .. a2 (x) Vx (x)
(8.12)
The function that satisfies (8. 1 2) when strict equality holds is
V(x) =
fx o
exp
{
-
J
u
}
2/(y) dy d u , a (y) 2
(8.13)
provided that the integral in {8.13) exists. Thus, we see that eqs. (8.5) and (8.6) used in the Gihman-Skorohod method play the role of Liapunov functions. For further study of stability see Ladde and Lakshmikanthan (1 980, pp. 56-9 1 ). 9. Existence of stationary distribution
A broader concept than that of the equilibrium solution discussed in the previous section is the concept of stationary distribution, or steady state distribution, or equilibrium distribution. Such a stationary distribution is obtained from the so lution of a stochastic differential equation which is a time independent random variable, i.e. x (t, w) = x (w ). The problem of the existence of a stationary distribution for a stochastic dif ferential equation has been solved in the mathematical literature only for certain special cases. Some of the pioneering work in this area includes Feller {1 954), Tanaka {1 957) and Khas'minskii ( 1 962) among others. Their results are sum marized in Mandl (1968) and have been applied in economics in the work of Bourguignon (1 974) and Merton (1 975a). Below we present conditions for the existence of a stationary distribution of an autonomous Ito stochastic equation of the form of (8.2). In the next chapter we will apply the result to obtain the stationary distribution for the neoclassical economic growth model un 1er uncer tainty.
Stochastic calculus
107
Consider eq. (8.2) and suppose that f(x) and o(x) are continuously differ entiable on [0, oo ) with o (x) > 0 on (0, oo ] and (8.3) holding. From theorem 7.4 we know that the solution of (8.2), x(t), is a diffusion process taking on values in the interval [0, oo ]. Furthermore, the endpoints of the interval [0, oo ] are ab sorbing states, i.e. if x (t) 0, then x(u) = 0 for u > t and similarly if x (t) = oo , then x(u) = oo for u > t. In general, a stationary distribution will always exist in the sense that x will either ( l ) be absorbed at one of the boundaries or (2) it will have a finite density function on the interval (0, oo ) or (3) it will have a discrete probability mix of (1) and (2). Possibility (2) is the nontrivial case which is of interest for (8.2) under the stated assumptions. Note, however, that in case (2) the boundaries are in accessible in the sense that, as e -+ 0, =
(9. 1 )
P[x(t) � e] -+ 0, and also P [x(t) � 1/d -+ 0.
(9.2)
A necessary and sufficient condition for (9 .l) and (9 .2) to hold is that eqs. (9 .3) -(9.5) are satisfied, where X
f
/2 (u)du = oo ,
(9.3)
f
/2 (u) du = oo ,
(9.4)
f
/1 (u) du
(9.5)
0
00
X
00
0
where /1 (u) and /2 (u ) are defined as follows:
I
and
/1 (u) = 2 exp o (u)
{u } { Ju } 2[(y) f o2 (y) dy
exp
y
2f(v) dv du. o2 (v)
Stochastic methods in economics and finance
108
If conditions (9 .3)-(9 .5) arc satisfied a stationary distribution, denoted by 1r(x), exists and is given by 1T
(x) =
m
2
o (x)
exp
{
with m chosen so that J;
x
}
2f(y) dy , f o (y)
1T
2
(9.6)
(x) dx = 1 .
10. Stochastic control
In this section we establish various propositions from stochastic control which have been found useful in economic applications. The analysis is intuitive. For a rigorous analysis see Fleming and Rishel (1975). Consider the problem: V( · )
J (k(t), t, oo) = max E1
00
J e- ps u (k(s), v(s)) ds
(10. 1 )
t
subject to the conditions d k(t) =
T(k(t), v(t)) d t + o(k(t), v(t)) dz(t), k(t) given.
(1 0.2)
Here v = v(t) = v(t, w) is the control variable, k = k(t) = k(t, w) is the state vari able, p � 0 is the discount on future utility, u denotes a utility function, T is the drift component of technology, and a is the diffusion component. Note that Er denotes expectation conditioned on k(t) and v(t). The problem described in (10.1) and (10.2) is a stochastic analogue stated and studied by Arrow and Kurz (1970, pp. 27-51). A standard technique for our problem, as in the case of Ar row and Kurz, is Bellman 's Principle of Optimality according to which "an op timal policy has the property that� whatever the initial state and control are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision". See Bellman (I 957, p. 83). Problem (10. 1 ) and (10.2) is studied here for the undiscounted, finite horizon case, i.e. for p = 0 and N < oo . This case is further subdivided in certain subcases as the analysis below indicates.
109
Stochastic calculus
10.1.
Maximum principle in one dimension
Consider the special case of (10.1) and (10.2) where the problem becomes
J (k(t), t,N) = rna� E1
N
J
(10.3)
u (k, v)ds
t
subject to conditions
dk = T(k, v) d t + a(k, v) dz, k(t) given.
(10.4)
Using Bellman's technique of dynamic programming, problem (10.3) and (1 0.4) can be analyzed as follows:
J (k(t), t, N) = m� E1
N
J
u(k, v) ds
t
t+Lit
=
m�x E1
J t
= m�x E1
t+Lit
J
[f
u(k, v) ds + m� Et+Li t
N
J
u (k,v)ds
t+Lit
u(k, v) ds + J (k(t + �t), t + �t, N)
t
t+Lit
=
m� E1
�
u(k, v) ds + J(k(t + �t), t + �t, N�I
= max E1[u(k(t), v(t)) �t + J (k(t), t,N) v
(10.5) Observe that Taylor's theorem is used to obtain ( 1 0.5) and therefore it is assumed that J has continuous partial derivatives of all orders less than 3 in some open set containing the line segment connecting the two points (k(t), t) and (k(t + �t), t + �t). Let (10.4) be approximated and write
�k T(k, v) �t + a(k, v) �z + o(�t). =
(10.6)
Insert (10.6) into (10.5) and recall the multiplication rules of eq. (4.12) of Chap ter 2. The result is
Stochastic methods in economics and finance
1 10
(10.7) For notational convenience let
( 10.8) Using (10.8), eq. {10.7) becomes
0 = max E, [u(k(t), v(t)) At + D.J + o{At)]. v
(10.9)
Eq. (I 0.9) is a partial differential equation with boundary condition
aJ (k(N), N, N) = o. ok
(1 0.10)
Pass E, through the parenthesis of {I 0.9) and after dividing both sides by At, let At � 0 to conclude
0 = max [u (k(t), v(t)) + J, + Jk T(k(t), v{t)) + � Jkk a2 (k(t), v(t))]. v
(10. 1 1 )
Eq. ( I 0.1 1 ) is usually written as
- J1 = max [u (k (t), v(t)) + Jk T(k(t), v(t)) + v + t Jkk a2 (k(t), v(t))] and is known as the
theory.
(10.12)
Hamilton-Jacobi-Bellman equation of stochastic control
Let us proceed further with our analysis. We define the costate variable p(t) in the next equation as
p(t) = Jk (k(t), t, N).
(1 0.1 3)
From (I 0.13) it follows immediately that
{10.14)
Stochastic calculus
111
Using (10.13) and (10.14) we may rewrite ( I O.l2) as - J, � m�x ll
(k,v,p, :�).
(10. 1 5)
where H is the functional notation ofthe expression inside the brackets of (1 0.1 2). Assume next that a function exists that solves the maximization problem of (10.15) and denote such a function by
v
vo vo (k, p, aapk ) . (1 0.16) Note that v0 is a function of k(t) and t alone along the optimum path, because Jk is a function of k(t) and t alone. In the applied control literature, and more specifically in economic applications, v0 is called a policy function. Assuming then that a policy function v0 exists, (1 0.15) may be rewritten as - J1 m�x ll (k.v,p, :� ) ll (k,v0 ( k , p, ::) .P. :� ) ap ) . (1 0. 1 7) IIo ( k,p, ak =
=
=
=
This last equation, (10.17), is again a functional notation of the right-hand side expression of (I 0.1 2) under the assumption of the existence of an optimum controI 1.e.
v0 , .
(10.18)
Equipped with the above analysis, our final goal in this subsection is to derive a system of stochastic differential equations describing the behavior of the state and costate variables, i.e. find expressions for and dp. An expression for is almost readily available to us from (I 0.4), (1 0.16) and (1 0.18); in particular
dk
d
k T(k, v0) d t a(k , v0) dz = llg (k,p, :�)dt + u (k ,p, :� ) d z =
+
=
II� d t + adz.
dk
(10.19)
112
Stochastic methods in economics and finance
Note that oH0 fop = H� = T, used in the derivation of (10.19), is obtained from (I 0.18). Next we derive an expression for dp. Use the definition of p(t) given in (10.1 3), eq. (10.4), and Ito's lemma to get dp = Jk , d t + Jkkdk + � Jkkk (dk)2 = Jk 1 dt + Jkk ( Td t + adz) + k Jkkk ( Td t + adz)2
(10.20) To simplify eq. ( 10.20) we compute Jk t from (10.17) assuming that equality holds for mixed partial derivatives op 10 °2 P + hP k ok ok2 = II� + T Jk k + � a2 Jk k k
- J,k
Ilko + Hpo
=
·
(10.2 1 )
Note that to obtain ( 1 0.2 1 ) first we use the facts that H� = T and H�k = i a2 , both resulting from partial differentiation of ( 1 0.18), and secondly, the defini tion in (10.14). Substituting ( 10.2 1 ) into (1 0.20) we reach our desired result ,
- H� dt + a Jkk dz .
=
( 1 0.22)
We summarize the above analysis in a proposition.
Proposition I 0.1
(Pontryagin Stochastic Maximum Principle). Suppose that k(t) and v0 (t) solve for t E [0, N] max E v
0
N
J
0
u (k (t), v(t)) dt
subject to the conditions dk = T(k(t), v(t)) d t + a(k(t), v(t)) dz, k(t) given.
Then, there exists a costate variable p(t) such that for each t, t E [0, N]: (I) v0 maximizes H(k, v, p, opfok) where
(
H k, v, p,
ap ak
)
1 13
Stochastic calculus
ap = u(k , v) + pT(k , v) + 2. a2 · ak
(2) the costate function p(t) satisfies the stochastic differential equation dp = - llZ dt + o(k, v0)Jkk d z � and (3) the transversality condition holds p(k(N), N) =
aJ (k(N), N, N) ;p: 0, ak
p(N)k (N) = 0.
Next we proceed to make some generalizations. 1 0.2. Generalizations
First, mathematically speaking the optimal path obtained from Pontryagin's Stochastic Maximum Principle is the solution of two stochastic differential equa tions subject to certain conditions. More specifically, we rewrite these equations together d k = H� d t + a dz ,
(10. 19)
dp = - H� d t + aJkk dz,
(10.22)
k (0) given,
(10.23)
p(k(N). N) = 0.
(10.24)
Secondly, it is easy to generalize proposition 1 0.1 by introducing a bequest function B(k(t), t). If this were to be the case the maximization problem would become
E.[J
N
m�x
J
u (k(t), v(t)} dt + B(k (N), N))
0
subject to the same conditions (1 0.4) as before. Proposition 1 0.1 holds with a new transve�sality condition, i.e. p(k(N), N) =
aB (k(N), N). ak
Stochastic methods in economics and finance
1 14
Thirdly, generalizing (I 0.3) and ( I 0.4) in another direction by allowing dis counting, we need not repeat the previous analysis. Let J(k(t), t, N) = m�x E,
N
J
e-ps u (k, v) ds
t subject to (10.4) and k (t) given. Write Pt
W(k(t), t,N) = e J(k(t), t, N)
and also
Jt = - {e-P' W } = - pe -P t W. dt
d
Thus, the Hamilton-Jacobi-Bellman equation is now transformed into p W = max [u(k(t), v(t)) + Wk T(k(t), v(t)) + � a2 (k(t), v(t)) Wkk). v
The remaining analysis can then be patterned after what is done above. Finally ,consider thegeneral multidimensional time dependentcase described by J(k(t), t, N)) = m�x E,
subject to
[ tJ
N
J
u (k (s), v(s), s)ds + B(k(N), N)
dk;(t) = T; (k(t), v(t). t) dt + a;(k(t), v(t), t)dz;(t), i = 1 , 2, ... , n , k (t) given.
For this problem the Hamilton-Jacobi-Bellman equation is - J1(k(t), t, N)
=
max [u(k(t), v(t), t) + J� T(k(t), v(t), t) v
+ � tr(Jkk a(k(t), v(t), t) a' (k(t), v(t), t))] H(k(t), v(t), p (t), pk (t), t) = max v = H0 (k(t), p (t), pk (t), t).
As before, primes denote transpose. The multidimensional analogue of the costate stochastic differential equation becomes
Stochastic calc:u/us
dp . = Hk0 . d t + I
I
n
j= I
�
1 15
Jk .k · a-I dzI. I I
provided that dz; is not correlated with dzj for i =I= j. A detailed analysis may be found in Bismut (1973). 1 0.3.
Constraints on controls
We now study a more general problem in stochastic control theory which is formulated as follows: J (k(t), t, N) = m�x E,
[ J u (k(s), v(s), s) ds + B(k(N), N) ]
(10.25)
t
subject to the conditions
";
dk; {t) = T; (k(t), v(t), t) dt + - � ;=I 1
i = 1 , 2, . . ., n , k(t) given;
Q=
g2 (k(t), v(t), t) � 0,
aii · (k(t), v(t), t) d zii I· ' I
1 , 2, ... , L.
(10.26)
(10.27)
In other words, the control stochastic process v(t) satisfies a set of L inequality constraints. Note that it is assumed that dzii; are Wiener processes that satisfy :
covariance (dzrir , dzsi ) = pYJr. .c? d t , 1 S .�
where Prir sis is the correlation coefficient which is independent of k(t) and v(t). We proceed with the analysis of this problem and at the end we summarize the results in a proposition, as we did earlier. We write the recursive equation by Bellman's Principle of Optimality: t+At
J (k(t), t, N) = m�x E,
J t
u(k(s)� v(s), s)ds + l(k(t + �t), t + At, N).
Now, if the approximation t+At
E,
J r
u(k(s), v(s), s) ds = u (k(t), v(t), t) �t + o(At)
Stochastic methods in economics and finance
1 16
is valid, then we may write J (k(t), t , N) = max [u(k(t), v(t), t)6t v
+ E,J (k(t + D.t), t + 6t, N) + o(6t) ] .
Put, 6J (t) = J (k(t + 6t), t + !::& t , N) - J (k(t), t, N).
Thus we obtain 0 = max [u(k(t), v(t), t)6t + E, !::& J(t) + o(!::& t)]. v
( 1 0.28)
Use Taylor's Theorem to expand D.J(t) around (k(t), t), assuming J is twice dif ferentiable. Then 6J(t) = J1 l::& t + J� l::& k + ! 6k' Jkk D. k + o(!::& t).
(1 0.29)
By taking conditional expectation of !::&J(t) in (10.29) we have E1f::&J(t) = J, l::& t + E, [J� 6k] + E, [ 4 l::& k ' Jkk l::&k ] + o(l::& t).
(I 0.30)
So, we have to compute E, [J� !::& k ] and E, [ 4 6k' Jkk l::&k ]. We do so next. From (10.26) write n;
. . + o (l::& t), l::&kI. = T.I !::& t + • � a11. . • D.z1/1• =I 1
,, and by taking conditional expectation of ( I 0.3 1 ) we have E, l::&k; = E,
lr; 6t + . i= l L J;
(1 0.3 1 )
a;; . !::&z;; . + o (!::& t)l = T1 6t + o(6t). I
J
I
because E, l::&z;;I. = 0 for all i andi;· With this information available we compute:
{
"
}
n
E, {J� !::& k } = E, . � J� I. !::& k1 = .� J� . E, (l::& k1) 1= 1 1=1 "
= .� J� . T1 6 t + o(l::& t). I J
I=
l
1 17
Stochastic calculus
Furthermore,
E, {(Ak)'Jkk (Ak)}
=
=
{ , J, E, {,�, .#, E, }
because E, {�zri �zsi } = Prj si �� + o (At). r r Collecting the resulfs above ( f0.30) becomes
E,�J(t)
=
·
ll
J1 �t + _L Jk I T; !::& t t= 1 "
+
n
11r
"s
r
s
l L L L L r= s= i = 1 i = 1 l
I
and also
-J1 (k(t), t, N) ��
Jk k a j a j P j + r s r r s s r rsjs �� o(�t),
max {u(k(t), v(t),
=
v
t) !::& t
+ ¢(k(t), v(t), t, N) �� + o (At)}, where ¢ is defined by n
¢(k(t), v(t), t, N) = L Jk . T; + � � . Jk r ks arj asjs Prjrsjs . r i= l r,S.J 1
r •ls
(I 0.32)
But notice now that ¢ is a function of (k, v, t, N) since each T; and a;i; are func tions of (k, v, t , N). Therefore we can replace the maximization problem by the simpler one, i.e. max [u(k(t), v(t), t) �� + ¢(k(t), v(t), t, N) �� + -J(k(t), t. N) !::& t = vECg(k( t ), t )
+ o(�t)], where the constraint set is defined by
Cg (k(t), t) {v E R "' : g(k(t), v, t) � 0}. =
We summarize the results in
118
Stochastic methods in economics and finance
Proposition 10.2 (The stochastic maximum principle with constraints). Suppose that J(k, t, N) is twice continuously differentiable in (k, t) and that the optimal k (t) and v(t) are such that for all r
Er
r+ Ar
J r
u(k(s), v(s), s) d s u (k(r), v(r), r) Llr + o (Llr). =
Then, at each time t, the optimal control v(t) solves max
vECg(k(t),t)
[u (k(t), v, t) + <1> (k(t), v, t, N)] ,
where
Cg(k(t), t) = {v E R m : gQ(k(t), v, t) � 0,
�=
l , 2, ... , L }
and J(k, t, N) must solve the partial differential equation
-J,(k, t, N) =
max
vECg
[u (k, v, t) + <1> (k, v, t, N)],
with defined as in (I 0 3 2) with boundary condition .
J(k, N, N) = B(k, N, N) for all k. 1 1 . Bismut's approach Bismut ( 1 973) has applied the general methods of convex analysis developed by Rockafellar ( 1 970) to problems of optimal stochastic control. In this section we present an intuitive exposition of Bismut's ideas as found in Bismut ( 1 975). Al though a specific application of Bismut's method is postponed until the next chapter, the analysis below of optimal stochastic control attempts to interpret in terms of economic concepts the methods and results of mathematical theory. In particular, there arc two important concepts in optimal stochastic control that are useful in economic applications. These are the concept of risk-taking and information processing. As befor� the problem can be formulated as:
Stochastic calculus
max E0
T
J
u(k, v, t, w) d t
0
1 19
(1 1 . 1 )
subject to the conditions
dk =f(k, v, t, w) d t + a (k, v, t, w) dz, ( 1 1 .2)
k(O) k0 given. =
Here u can represent an instantaneous utility or profit function, k denotes capi tal stock, v is the investment decision, and w is the environmental factor. For simplicity we study the one-dimensional case. We assume that an increasing sys tem of information is available from the family of a-fields {.�, : t E [0, T]} and that this information includes the past values of k and z and also f and a. Here, as before, z is a Wiener process. Under this assumption the expected mean and variance of the capital increment consecutive to any decision v are thus known. Let p1 denote the marginal value of capital at time t which is given by 3E t T
p1 = a k
J u(k, v, s, w) ds,
( 1 1 .3)
t
i.e. p, is the partial derivative with respect to k of the conditional expectation of the utility function from time t with v being an optimal policy. Bismut (1 973, p. 387) and Bismut ( 1 975, p. 242) assume that p1 may be written as t
P, = Po + J 0
J 1/s dz t
Ps d s +
0
s
+ A11,
( 1 1 .4)
where Ps is the infinitesimal expected rate of growth of p, /Is is the infinitesimal conditional covariance of p with z , 1l1 is a predictive term, M0 = 0, which is the best estimate at time t of a given random variable and which is independent of z . Its infmitesimal increments have then a null conditional expected value at each time. This decomposition corresponds to the idea that p 1 can be decomposed in the sum of p0 , of the second term that gives the expected infinitesimal increment of p 1 at each time, of the third term which integrates uncertainties in the accu mulation process, and finally of the last term which integrates the information on environmental factors M,. Define now .;t' by .tf = u (k, v,
t, w) + pf(k, v, s, w) + Ha(k, v, s, w).
( 1 1 .5 )
120
Stochastic methods in economics and finance
Bismut ( 1 973, p. 40 1 ) proves that the following relations hold for optimal v: a.Yf
-
-
ov
0,
dp =
a.Yf
ok
( I 1 .6) dt + 1/dz + di\1,
Pr = 0.
( 1 1 .7) ( 1 1 .8)
As in the preceding section dk is given by ( 1 1 .2) with v being optimal. Note that eqs. ( 1 1 .6)- (1 1 .8) and ( 1 1 .2) are very similar to the equations stated in proposi� tion 1 0 . 1 . More specifically, eq. (I I .6) follows from the maximality ofv, whlle eq. ( 1 0 . 1 9) and ( 1 1 .2) with v maximal arc identical. Eqs. ( 1 0.24)and ( 1 1 .8) are the same and they denote the transversality condition. Finally, ( I 0.22) and (1 1 .7) are the only equations that differ. Note that Bismufs random variable II in ( 1 1 .7) corresponds to the random variable Jkk a in ( 1 0 .22), while the term diU in ( 1 1 .7) has no analogous term in ( 1 0.22). Roughly speaking, Bismut's correspondence of primal and dual variable is as follows: f -+ p,
( 1 1 .9)
a -+ H,
( 1 1 . 1 0) (1 1 . 1 1 )
Let us now interpret the various variables above to uncover their economic meaning. First note that ..>f' in ( 1 1 .5) is the sum of instantaneous utility or profit, plus the expected infinitesimal increment of capital valued at its marginal ex� pected value, minus the risk associated with a given investment policy valued at its cost. The instantaneous attitude towards risk is given by H and it is positive if the individual is risk-taking and negative if he is risk�averting. Next we interpret dp or -dp, i.e. the conditional expected rate of depreciation in the marginal value of capital. From (I 1 .7) we see that - dp is the sum of capital's contribu tion to utility or profits plus capital's contribution to enhancing the expected value of the increment of the capital stock, minus its contribution to increasing the conditional standard deviation of the increment of the capital stock valued at the cost of risk, minus two other terms: II dz and d1H. To interpret Hdz note from ( 1 1 .2 ) that dz = ( 1/a) (dk - fdt)
(1 1 .12)
Stochastic calculus
121
which, upon multiplication with H, yields
Hdz = (H/a) (dk - fdt).
( 1 1 . 1 3)
The term Hdz in ( 1 1 .7) is then a correction term in the evolution of the marginal value of capital which evaluates in terms of p the difference between dk and E(dk), where E(dk) = fdt. The last term, d.l\1, denotes changes in the predic tion of long-term uncertainties which can either increase or decrease the value of capital. Intuitively speaking, M incorporates information which is not con tained in past values of z and while z contains all short-term uncertainties that appear in the accumulation process, M is a prediction of the long-term uncertain ties. This concludes the discussion of Bismut's approach. 12. Jump processes In this section we develop the generalized Ito formula for Jump processes will be modeled by a Poisson process that describes the arrival of ran dom events. When a Poisson event arrives a jump in the state variable takes place and this jump will be distributed according to a preassigned density function. After the generalized Ito formula is developed, we will then develop the maxi mum principle by following much the same approach that we did for the case of diffusion processes. Our approach remains intuitive.
jump processes.
I 2. I.
Generalized Ito formula
We will develop the generalized formula for the case of mixed Poisson and Brown ian processes. Consider the following:
dx(t) = f(t,x) d t + a(t,x) dz(t) + g(t,x) dq(t).
(1 2.1)
Here, for R = (- oo , oo),
f(t, x): [ 0, oo) x R -+ R, a (t, x): [0, oo) x R R, -+
g(t,x): [O, oo) x R -+ R. Also, {z(t)}�==o is a standardized Wiener process and
{q(t)};:0
is a Poisson pro-
Stochastic methods in economics and finance
122
cess assumed to be distributed independently of {z(t)}�= in order to keep things o simple. Let >.. D. t + o (D.t) be the probability that q(t) jumps once in (t, t + D.t). Let the amplitude A of the jump be random with density function p(a), i.e. p(a)da is the probability of a jump amplitude contained in (a, a + da) up to higher order terms in da. Assume that the probability that q(t) jumps more than once in (t, t + D.t) is o(D.t). Thus, the probability that q(t) is constant on (t, t + D.t) is 1 >.. D. t + o(D.t). The generalized Ito formula for the one-dimensional case may be stated. See Kushner ( 1 967, p. 1 8) for a rigorous statement and proof. Gihman and Skorohod ( 1 972, p. 263) have a more complete treatment of generalized Ito formulae than Kushner.
-
Proposition 1 2.1 (Generalized Ito formula). Let F(t, x) be twice continuously differentiable in (t, x). Let there exist a closed and bounded interval such that {a I p(a) > 0} C /. Let D.F = F(t + D.t, x + D.x) and let E1 D.F denote conditional expectation conditioned on x (t) = x. Then
E1 1!.F =
{ F1(t,x) + Fx (t,x)f(t,x) + � Fxx (t,x) a2 (t,x) + X (..£ [F(t,x + g(t, x)a) - F(t,x)]p(a)da) }l!.t + o(At).
To establish the theorem note that
E1D.F (E;' D.F) >.. D.t + (1 - >.. D.t) Ei* D.F + o(D.t}, =
(1 2.2)
where Ei means expectation conditioned on the occurrence of the Poisson event and Ei* means expectation conditioned on the Poisson event not occurring. Eq. (1 2.2) may be written ( 1 2.3} The first term in eq. (1 2.3) is the conditional expectation of the change in the function F given that the Poisson event does not occur. The second term of ( 1 2.3) is where the action is. Notice first that the term Ei'* D.F � 0, as D.t � 0. Thus, we only have to look at the term Ei' D.F to calculate the second term on the right-hand side of eq. ( 1 2 .3) up to o(D.t). Performing these calculations we get
E** D.F= F D.t + F E** t D.x2 t D.x + 2! F E** t
t
+ >.. D. t
X
XX
( f [F(t,x + g(t, x)a) - F(t, x)]p(a) da ) + o(D.t). aE/
( 1 2.4)
123
Stochastic calculus
We are able to write A.D.t E:* D.F
=
o(D.t) because Et'' AF
�
0 as D.t � 0. Also,
A.AtEjAF= A.llt E: f [F(t + At,x + [At + aAz + ga + o(D.t)) aEI
aEl
- F(t,x)]p(a) da = A.D.t f [F(t,x + ga) - F(t,x))p(a)da + o (D.t), because fD.t + a D.z � 0 as D.t � 0, since and variance D.t. The formula (12.4) reduces to
E1 D.F
=
D.z is normally distributed with mean 0
F1 D.t + Fx fD.t + � Fxx a2 At + 'A
( f (F(t, x + ga) - F(t, x)) p(a) da) D.t + o(D.t), aEJ
(12.5)
which is the generalized Ito formula. 12.1.
The maximum principle for jump processes
Consider the problem
J(k(t), t, N) = max E1 v(.)
subject to the conditions,
[JN t
u(k(s), v(s), s) ds + B(k(N), N)l
J
( 1 2 .6)
dk(s) = T(k(s), v(s), s) ds + a(k(s), v(s), s) dz (s) + g(k(s), v(s), s) dq(s),
(12.7)
where all notation is as in section 1 0 except that the jump component
g(k(s), v(s), s)dq(s) has been added. Here the jump amplitude A is distributed with density p (a), i.e. independent of {z(t)}�= o as in the statement of the generalized Ito formula. Thls assumption is made to keep matters simple. Let us follow the procedure in section 1 0 together with the generalized Ito formula to discover the stochastic maximum principle. Calculate for (k, v, t) given at t
124
E1
[t+At !
Stochastic methods in economics and finance
�
u d s + J(k(t + At), t + At, N) - J(k(t), t, N
= u(k, v, t) At + J,(k, t, N) At + lk (k, t, N) T(k, v, t) At
+ t Jkk (k, t, N) a 2 (k, v, t)At + 'A At
f [J(k + g(k , v, t) a, t, N) - J(k, t, N)] p (a) da + o (At).
aE/
( 1 2.8)
Eq. ( 1 2.8) is obtained from the generalized Ito formula applied to E1AJ. Let ¢(k, v, t , N) denote the right-hand side of (1 2.8) divided by At. Then we write Proposition 12.2 (Maximum principle). Assume that J satisfies the hypotheses of the generalized Ito formula. Then the optimal control function v 0 (k, t , N) is found by choosing v to solve max ¢ (k, v , t, N) v
and J is determined by the partial differential equation
0
0 = max ¢ (k, v, t,N) = ¢ (k, v (k, t, N), t,N) v
with the boundary condition
J (k , N, N) = B (k, N). Obviously the maximum principle can be generalized in a straightforward way to multidimensions, constraint sets on the controls v, correlated Wiener and Poisson process, and so on, at this level of heuristic argument. Of course, to do these generalizations rigorously would take a lot of detailed mathematics. 13. Optimal stopping and free boundary problems In Chapter 1 , section 8, we introduced the reader to some basic notions on opti mal stopping. In that chapter we considered a sequence of random variables Y1 , By imposing certain assump Y2 , and their corresponding rewards X1 , X2 , tions on these sequences of random variables we were able to obtain the existence of an optimal stopping rule. In this section we rely on van Moerbeke ( 1 974) to .••
•••
•
Stochastic calculus
1 25
treat the continuous time case, where, instead of a sequence of random vari ables Y1 , Y2 , , we consider a Wiener process z 1 starting at z0 (w) = 0. Note that we write the function as g(z, and by this we mean that at time t when the state of affairs is z = z 1 (w), the value of the reward is given by g(z, The for a random period of time T = r (w) is written as •••
reward average reward Eg(z + z7, t + )
t)
t).
r ,
(13.1}
where the arguments o f g, i.e. (z + z7, + r), denote the space - time of the Wiener process starting at (z, In other words, suppose that we start playing a game at time with a state of affairs z and play for a random period T, where T � The state of affairs corresponding to this random period is determined by the underlying Wiener process and we denote it by z7 Having started at with z and played for T periods giving us z7, eq. ( 1 3 . 1 ) estimates the expected reward of such a game. Our primary interest is with a finite time interval [0, where We assume that g and all its partials are continuous for < oo and E [0, < < oo and have limits as A discontinuity is permitted at = but in (z, is infinitely differentiable this case we require that (z) = g(z, except for a few isolated jumps. Here g(z, denotes the left limit of g(z, at The function h is called the and with no loss of generality we assume that it is non-negative. The assumption that h (z) � 0 is not restrictive because if h (z) < 0 in a given interval it will make sense to stop earlier, before hitting the final = Having defined the functions g and and stated the assumptions made about them, we now impose upon a condition called the Tychonov condition. If the functions
t
t).
t
T- t.
•
T t T
t
T].
t-. T.
lz
t T. =
T} - g T-) T-) final gain
t T.
g growth
ag a2g ag g, at ' , az ' az 2 h,
a 2g azat
,
,
t
T], t T,
t)
lz,
a3g , az3
a1z a3h a2 1z , , , az az 2 az3
uniformly in any finite strip are bounded by eo(z z ) when z I tends to then g is said to satisfy the Suppose we start at time with a state of affairs z and we play a game up to time < oo. Under these circumstances the g(z, is obtained by maximizing (I 3.1) over all stopping times T, with T � The stopping timeT achieving this maximum is called the Obviously, the optimal reward function g is important for our analysis and we would like to characterize
I
T
oo ,
Tychonov condition. t optimal reward t) T-t. optimal strategy .
[t, T],
126
Stochastic methods in economics and finance
it in some way. This is done by means of the concept of an excessive function. A function [, bounded below, is called excessive in an open domain D of R 2 if ( I ) Ef(z + zT , t + ;) � f(z, t) for every stopping time ; not exceeding the first exit time ;0 from D, and (2) Ef(z + zTn, t + ;,1 ) f(z, t) for every sequence of stopping times T11 < ;0 such that P[;n � 0) = 1 . Note that if f is sufficiently differentiable, excessivity in the domain D is the same thing as �
for all (z, t) in D. Equipped with the definition of excessivity we characterize g as the smallest excessive function exceeding g. The Tychonov condition implies that g is finite and continuous. Next we distinguish between two regions: a continuation region Cwhere g > g, which means that it pays to play the game, and a stopping region S where g = g, which means that quitting is best. Since g is continuous the continuation region C is open. We assume that the continuation region C has a continuously differ entiable boundary z ; s(t), except possibly for a few isolated points where I d s/d t I may blow up. The boundary separating these two regions is the optimal stopping boundary. The Tychonov condition imposed on g helps us to conclude that the optimal strategy is to play as long as you remain in C and stop when you hit the optimal boundary. Let ;0 denote this hitting time. Then
g(z, t) = Eg(z +
ZT0 ,
t + To).
(13.2}
Therefore, our purpose is to find the optimal stopping boundary. At this point it is appropriate to remark that there is a beautiful interplay be tween our problem and the theory of partial differential equations. Although we proceed rapidly to reach our present goal of finding the optimal stopping boundary, we would like to encourage the interested reader to consult van Moer beke (1 974) and some of the many references he cites. l laving said this, note that the problem of finding g and the optimal strategy can be solved by convert ing it into a free boundary problem for the heat equation. From (1 3.2} we con clude that g is parabolic in the continuation region C which means that first g is excessive, and secondly Eg(z + zT U' t + iu) = g(z, t), where 'u is the first exit time from any open set U with compact closure. Parabolic functions satisfy the backwards heat equation and conversely solutions of the backwards heat equation which are bounded below are all parabolic functions. Therefore,
1 27
Stochastic calculus
( 1 3.3) g=g
(13.4)
at the boundary of C
and
(1 3.5)
g(z, T) = g(z, T).
Furthermore, because of our earlier assumption about the continuously differ· entiable boundary of the continuation region C, the optimality of g implies that
ag
ag
3z (s(t), t) = � (s(t), t),
(13.6)
and also for (y , u) E C, as (y, u) � (s(t), t) we have that
a
ag
t (s(t), t), 3z (y, u) � az
(13.7)
(13.6) and (13.7} holding at points (s(t}, t), where I dsfd t I < oo Eqs. (13.5}, (13.6) and ( 1 3 .7) are called the smooth fit equations. Let us reflect for a moment. What eqs. ( 1 3 .3) - ( 1 3 .7 ) describe is an initial
with both
.
and boundary value problem with two boundary conditions, which means that our problem is overdetermined unless we choose to keep the boundary free. Sup· pose that we keep the boundary free. Then it seems plausible that eqs. (13.3) (13.7} can determine both the boundary s(t) and the optimal reward g. We close this section by stating Theorem 13.1. Let C be an open set in t � T < oo with a continuously differ entiable boundary curve z = s(t), except possibly for a finite number of isolated points where d s/d t blows up. Let a Tychonov·type function u satisfy
au ) a 2 u = 0 in C, a t + 2 az2 u = g at (z, t) = (s(t), t), u (z, T) g(z, T), =
au
ag
az = 3";
at
(z, t) = (s(t), t),
if i ds/dt I < oo ,
Stochasri<' methods in economics and finam:e
1 28
u >g
in C and
vg
II = - +
ot
1
2
-
u
=g
o 2g �0 oz2
--
elsewhere,
in the complement of C.
Then u is actually g and s(t) is the optimal stopping boundary. For a proof of this theorem sec van Moerbeke ( 1 974). The above brief analysis can serve as an introduction to the topic of optimal stopping in continuous time. 14. Miscellaneous applications and excercises and { Y(t), t e T} are real-valued stochastic processes defined on the same probability space and such that E(X(t)) < oo and E( Y(t)) < oo for all t E T. The covariance function of these two processes de noted by rx y (s, t), is defined for s < t by
( 1 ) Suppose that
{X(t), t E T}
rx y(s, t) = cov(X(s), Y(t)) =
E ([X(s) - E (X(s))] [ Y(t) - E ( Y(t))]).
This definition extends (4 . 1 2) of Chapter 1 . When the two processes are the same then the covariance function is called the autocovariance function and is denoted by rx (s, t). Show the following two simple facts: var X(t)
=
rx (t. t) for t E T,
rx (s,t) = rx (t, s)
for s, t e T.
{X(t), t E T} such that E(X(t)) < oo for all t E T, is continuous in the mean square at time t if as lz � 0. (2) A stochastic process
lim E(X(t + h) - X(t))2
�
0.
Let llx (t) = E(X(t)) and suppose that llx (t) is continuous in t, t E T, and that rx (s, t), s E T and t E T, is jointly continuous in s and t. Show that under these hypotheses {X(t), t E T} is continuous in the mean square. (3) Consider eq. (2. 1 1 ) of tltis chapter which we rewrite for convenience below
x (t + {).f) - x(t) =f(t , x(t)) D.t + a(t, x(t)) [z(t + D.t) - z(t)] + otD.t).
Stochastic calculus
129
Show that:
E (x(t + !1t) - x(t})
=
f(t, x(t)) l1t + o(t1t),
var (x(t + l1t) - x (t))
=
a2 (t, x(t)) E (z (t + At) - z (t))2 + o(At}
=
o2 (t, x(t)) At + o(l1t).
(4)
Consider eq . (2. 1 I ) with backward differences which we write as follows:
x(t) - x(t - !1t) = f(t, x (t)} l1t + o(t, x (t)) (z (t) - z (t - !1t)) + o (l1t}. Assume that the functionsfand a are continuous. Show that:
E (x(t) - x(t - l1t))
=
f(t - !1t, x(t - !:l t)) !:lt + ox (t - At, x(t - At)) a(t - At, x(t
- f1t)) f1t + O (f1t), var (x(t) - x(t - !1t))
=
o2 (t, x(t}) �t + o(l1t).
Compare the results of this exercise with the results of the preceding exercise to conclude that the mean of the increment of the process depends on the type of the difference that is used while the variance of the increment of the process is not so affected. (5} Consider the partition of [s, t] C [0, T]
s0 = to < t 1 < ... < tn t, =
max
o
I t;+ 1 - t; I � €,
[t; , t;+ 1 ) C [s, t),
where € > 0 and arbitrarily small. Suppose that z (t, w) is a Wiener process with unit variance and suppose that as € � 0, A and B satisfy:
E I A - � z(t;) [z(t;+ 1 ) - z (t;)1 1 2 �o, I
E I B - � z {t;+ 1 ) [z (t;+ 1 } - z{t; )l l 2 �o. '
Show that B-A = t-s. (6} Suppose that z (!) is a Wiener process with unit variance and consider the
Stochastic methods in economics and finance
1 30
stochastic integral fs1 Use the definitions of the Ito integral 1 5 of this chapter to obtain: first, Stratonovich integral in sections 3
z(u}dz(u).
and
and the
t
J z(il)dz(u) = � [z2 (!) - z2 {s)] - � (t - s), s
when the integral is interpreted as an Ito integral; secondly, t
J z(u)dz(u) = � [z2 (t) - z2 (s)], s
when the integral is interpreted as a Stratonovich integral. In particular, note that the Stratonovich integral satisfies the integration by parts formula of ordinary calculus, while the Ito integral does not. Note that integration by parts yields t
t
J z(u)dz(u) = z(t)z (t) - z(s)z (s) - J z(u)dz(u), s
s
or equivalently t
J z(u)dz(u) � [z2 (t) - z2 (s)]. =
s
See Stratonovich ( 1 966, p. 365). (7) Suppose that
dx 1 (t) = x�(t)dt dz(t), dx2 (t) = x2 dz (t). Use Ito's lemma to compute dy(t) for each of the following: (a ) y(t) = u(t,x1 ,xJ) = x 1 (t)x2 (t), (b) y(t) = u(t,x 1 ,x2 ) = t[x1 (t)x2 (t)], (c) y(t) = u(t,x 1 , x2 ) = z(t) [x 1 (t)x2 (t)]. +
(8) Consider the system of stochastic differential equations with initial con ditions
Stochastic calculus
131
x1 (0) = x2 (0) = z (0) = 0, dx1 (t) = d z (t), dx2 (t) = x1 dz (t), where z(t) is a Wiener process with unit variance . Use Ito's lemma to verify that the solution of this system of equations is given by x 1 (t} = z(t), t
x2 (t) =
(9)
J z (t}dz (t).
0
Solve the following two stochastic differential equations of first order:
(a) dx (t) = a0x (t) d t + a0 d z (t), where a0 and a0 are nonzero constants. {b) dx(t) = a(t)x (t} d t + a(t) dz (t), where a (t) and a (t) are arbitrary nonrandom functions of time. ( 1 0) Consider the stochastic equation dx(t} = - x (t) dt + a(t)x (t} d z (t), where a(t} is an arbitrary nonrandom function of time. Discover sufficient con ditions for the stability of the 0-equilibrium. (1 1) Consider the second-order stochastic system dx1 {t) = x2(t)dt, dx2 (t) = -x1 (t) d t - ax1 (t) dz (t). Kozin and Prodromou ( 1971) study the sample stability of this system and the interested reader is encouraged to consult their paper. Unlike the straightforward stability analysis of the nonstochastic system where a = 0, the stochastic stability analysis of the above system is quite technical . ( 1 2} Consider a stochastic control problem having a linear quadratic objec tive function:
132
Stochastic methods in economics and finance
- W(x(t)) = m� n E, subject to
00
J
s=t
e-P(s-t) {a(x(s))2 + b(v(s))2 }ds
dx(t) = v(t)dt + ox(t) dz(t).
where, a > 0, a > 0, b > 0 and p > 0. Discover an optimum solution. Note that this problem and one of its generaliizations are discussed in the next chapter. IS.
Further remarks and references
Most economists are familiar with the analysis of discrete time stochastic models from studying econometrics. In this chapter we usc briefly the discrete time case in order to motivate the analysis of the continuous time case. Problems related to discrete time stochastic models, beyond the introductory econometrics level such as in lntriligator ( 1978), arc presented in some detail in Chow (1 975), Aoki ( 1 976), and Bertsekas and Shreve ( 1978). Also, note that Sargent ( 1 979) has a chapter on linear stochastic difference equations. The modeling of uncertainty in continuous time is presented in Astrom (1970), Balakrishnan (1973), Fried man (1975), Soong (1973), Tsokos and Padgett (1 974), and Gihman and Skoro hod (1969, 1 972). Ito's stochastic differential equation appeared in Ito (1946, 1 950) and was later studied in some detail in Ito (195 1 b). In his 1946 paper Ito discovered that certain mathematical questions, raised by Kolmogorov and later by Feller about partial differential equations related to diffusion processes, could be studied by solving stochastic differential equations. Later in his 1 95 1 Memoir, Ito expanded the Picard iteration method of ordinary differential equations to establish theo rems of existence and uniqueness of Ito stochastic differential equations. It is worth remarking that Ito's stochastic differential equation is one among many stochastic equations. Syski ( 1967) considers a basic system of random dif ferential equations of the form
dx
dt = f(x(t), y(t), t) for t E T, with initial condition x(t0) = x0 , where {, x and y can be vectors of appropriate dimensions, and he classifies such equations into three basic types: random differential equations with (1) random initial conditions, (2) random inhomogeneous parts, and (3) random coefficients. The Ito stochastic differen-
Stochastic calculus
1 33
tial equation is a special class of random differential equations which, however, is important for several reasons. First, the conditional mean and the conditional variance as functions are sufficient statistics for Ito equations. Thus, for Ito equa tions the calculations of the conditional mean and the conditional variance func tions completely determine the whole process. This is analogous to the statistical fact that the mean and variance are sufficient statistics in normal distribution theory. Secondly, the Ito equation exhibits a nonanticipating property which is use ful in modeling uncertainty. In other words, if the true source of uncertainty in a system is dz and we want the differential equations not to be clairvoyant, then the evolution of the state variable x in the next instant should depend only on uncertainty evolving in that instant. An Ito equation is consistent with this lack of clairvoyance. Thirdly, the Ito equation has solutions which when they exist have nice prop erties. Section 7 describes these properties and it suffices to say here that the Mar kov property and the diffusion property are useful properties with a well de veloped theory available about them. Finally, although economists are just dis covering the usefulness of the Ito equation, it is worth noting that this equation has found important applications in the engineering literature, and particularly in the areas of control, filtering and communication theory. The discussion of section 3 concentrates on Ito's integral . Basic references are Ito ( 1 944, 1 95 l b}. Doob (1 953) presents in some detail Ito's original ideas with some extensions. See also Doob ( 1 966) for an explanation of the connec tion between Wiener's integral and Ito's generalization. Recently, Astrom (1 970) and Arnold ( 1 974) have presented simplified discussions of Ito's integral. A de tailed account about stochastic integrals is given in McKean ( 1 969). A differ ent approach to stochastic integration and in general to stochastic calculus may be found in McShane (1 974). Wong and Zakai ( 1965} discuss the convergence of ordinary integrals to stochastic integrals. We note that section 3 is a brief intro duction to stochastic integration having as its purpose to motivate and supply a definition for the classical Ito stochastic integral. For the reader who is interested in pursuing his study on stochastic integrals we suggest Metivier and Pellaumail ( 1 980), McKean { 1 969) and Kussmaul (1977). These books develop the theory according to the Ito prototype. The idea of defining stochastic integrals with re spect to square integrable martingales, suggested in Doob ( 1 953), was extended in Kunita and Watanabe (1 967). Further extensions of the stochastic integral with respect to a special class of Banach-valued processes is presented in Meyer (1 976) and with respect to Hilbert-valued processes in Kunita ( 1 970). Naturally, the various extensions of the concept of a stochastic integral have implications on Ito's lemma and the study of stochastic differential equations. Metivier and
Stochastic methods in economics and finance
1 34
Pcllaumail ( 1980) provide detailed references of such extensions. In this book we usc Ito's integral because we chose to analyze Ito's stochastic differential equations. The reader, however, should be informed that an alterna tive approach to stochastic integration has been proposed by Stratonovich (1966). To illustrate the difference between the Ito and the Stratonovich integral we con sider the simple special case t
J z(u)dz(u),
(15.1)
s
where z(t) is a Wiener process with unit variance. In this special case, for a par tition of the form of (3.8) the analysis of section 3 in particular lemma 3.4 in this chapter showed that Ito's integral denoted by A is defined such that as e � 0, then
E I A - � z(t;) [z(t;+ 1 ) - z (t;)] l
1 2 � o.
( 1 5 .2)
Recall that (15 .2) comes from the two equations in (3.1 1 ) of this chapter. Stra tonovich (1966, p. 363) defines his stochastic integral, in our special case, as
E
S-
]
[z(t;) + z(t;+ 1 ) 2 [z(t;+ 1 ) - z(t;)l fL
2 � o,
( 1 5.3)
where S denotes the Stratonovich stochastic integral. Note that the Stratonovich stochastic integral is a particular linear combination of the integrals A and B de fined in (3 .1 1 ). More specifically,
S - -I A + -I B
-2 2 ,
where A is Ito's stochastic integral, and A and JJ arc as in (3.1 1 ). The Ito and Stra tonovich integrals are not greatly different. Actually, Stratonovich has developed formulae representing one integral in items of the other, and these formulae are not complicated. See Stratonovich (I 966, p. 365). However, because of the na ture of stochastic calculus these two integrals have different properties. The Ito integral and the Ito differential equation maintain the intuitive idea of a state model. Also, the Ito integral has the useful properties that it is a martingale and it preserves the interpretations that the expectation of dx in (2. 1 2) is [dt and the conditional variance of dx is d t. The main disadvantage of the Ito integral is that it does not preserve the differentiation rules of ordinary calculus as Ito's lemma demonstrates. The Stratonovich integral preserves many computational
a2
Stochastic calculus
135
rules of ordinary calculus but it does not have the just stated advantages of the Ito integral. For further details about the relation between these two integrals see Meyer (1976). So far, all economics and finance applications of stochastic calculus have used the Ito integral because the associated Ito differential equations provide a mean ingful modeling of uncertainty , as explained earlier. Consequently, Ito's lemma is important for computing stochastic differentials of composite random functions. Put differently, Ito's lemma is a formula of a change of a variable for processes that arc stochastic integrals with respect to a Wiener process. Note that Ito's lemma first appeared in Ito ( 195 I a) and later in Ito (1961 ). We repeat here what we said earlier, namely that Ito's lemma and Ito's integral were discovered by Ito as he was working on partial differential equations related to diffusion pro cesses. This original research led also to the development of stochastic differen tial equations. It was Ito, and later Doob, Gihman and Skorohod among others, who established the field of stochastic differential equations. At this point we will discuss briefly one method for solving stochastic differ ential equations of first order as presented in Gihman and Skorohod (1972, pp. 33-39). Consider the equation on [0, T], dx(t) = f( t , x(t)) d t + a(t,x(t)) dz(t)
( 1 5 .4)
and note that it can be written in integral form as
x (t) = x(O) +
I
t
J f(s,x(s)) d s + J a(s, x (s)) dz (s),
0
( I 5.5)
0
where x(O) is the initial condition. The idea of the technique is to discover ap propriate transformations of ( 1 5 .4) so that the right-hand side of ( 1 5 .5) has a more convenient form, i.e. the unknown function does not appear on the right hand side of ( I S .5). We illustrate this technique with an example and refer the reader to Gihman and Skorohod ( 1 972, pp. 33 -39) for the detailed mathematics and for more examples. Consider the equation dx( t) = a0x(t) d t + a0x(t) dz,
(1 5.6)
with a0 and a0 constants. To solve this equation consider the substitution y = log x and usc Ito's lemma to obtain:
Stochastic methods in economics and finance
136
dy
=
1 1 dx + X 2
-
( - -I ) (dx) = - (a0xdt + a0xdz) X2
I
2
X
1 1 2 2 - - a0 x d t 2 X2
Integrating this last equation we get:
y(t) - y(O) =
t
t
J (a0 - � a� ) d t + J a0 dz
0
,
0
which can be written as:
y(t) = y(O) + (a0 - � a�) t + a0z(t). Recalling the substitution y = log x, i.e. x that the solution of ( l 5 .6) is:
x(t)
�
eY( t )
�
x (O) exp
( 1 5 .7)
= e Y , for x(O) = eY (O ) , we conclude
{ (a0 - a; ) t + a0z(t)} .
Thus, finding the appropriate substitution and using integration can lead to find ing the solution of stochastic differential equations. In sections 8 and 9 we discussed the problem of stochastic stability and we distinguished between stability of a point equilibrium and stability in the sense of convergence in distribution to a steady state distribution. Mathematicians have studied stability for a point equilibrium where noise has disappeared. while the area of stability in the sense of convergence in distribution to a steady state dis tribution independent of initial conditions has not received much attention. Even so, both areas remain quite open for future research. One of the first papers in the stochastic stability of point equilibrium is the work of the Russian mathema ticians Kats and Krasovskii ( 1 960) where they extend the deterministic Liapunov ( 1 949) method. Note that Antosiewicz ( 1 958), Borg ( 1 949), Hahn ( 1 963, 1967), Cesari (1 963), J lartman ( 1 964), Krasovskii ( 1 965), La Salle and Lcfschetz (1961), La Salle ( 1 964), Massera (1 949, 1956), and Yoshizawa ( 1 966} are among the basic references for Liapunov deterministic method to stability. In the United States, Bucy (I 965) and Wonham (I 966a. 1 966b ), among others, contributed to the solution of problems of stochastic stability in the spirit of the Liapunov method. Kushner ( 1 967a) gives a complete account of the major stochastic sta bility results following the Liapunov method. The reader may find Kushner's ( 1 972) stochastic stability survey useful for an introduction to the subject. AI-
Stochastic calculus
137
though general results about stochastic stability are not abundant, some progress has been achieved in the stability of linear stochastic systems. Some of these results, with many references, appear in Kozin (1972). Our distinction between stochastic stability of a point equilibrium and sta bility in the sense of convergence in distribution independent of initial condi tions should not imply that these are the only two concepts of stochastic stability. Within the area of stochastic stability of point equilibrium several definitions and theorems exist. Our section 8 gives only one notion of stochastic stability of point equilibrium. Now we mention two more definitions of stochastic stability of a point equilibrium to illustrate the scope of this important area of mathe matical research. For example, if we let x(t) denote the solution of a stochastic differential equation such as (8 . 1 ) on [0, ), with x = 0 being the equilibrium solution, we then say that tlze 0-equilibrium is stable in the mean if the expecta tion exists, and given € > 0 there exists an 11 > 0 such that I x I = I x(O, w) I < 11 implies oo
(
E s� p I x(t) I
)<
t:.
Another definition is this: the 0-equilibrium is exponentially stable in the mean if the expectation exists, and if there exist constants a, (3 and 11, all greater than zero, such that I x I = I x(O, w) I < 11 implies
E ( I x(t) I ) < (3 1 x(O, w) I e -a t for all t > 0. Several other definitions are available in the mathematical literature. See Kozin ( 1972) and Kushner ( 1 97 1 ). A simple illustration reported in Kozin ( 1 972, pp. 1 42 - 1 92) may be ap propriate at this point to give the reader an indication of the mathematical curi osities that arise in stochastic stability. Consider the Ito equation dx(t) = ax(t) d t + ax (t) d z , where a and a are constants and z is a Wiener process with unit variance; suppose that the initial condition is x(O) x0• The solution of this equation, w .p.l, is =
x(t) ; x(O) exp
[(a - a; ) t + uz (t)}
and the nth moments of this solution are given by
138
�(�a - 2 ) nt
Stochastic methods in economics and finance
E(xn (t)) = xn (0) exp
02
02112
+ -2
��-j "
From this last eq�ation Kozin ( 1 97 2 , p. 1 92) concludes that there is exponen tial stability of the nth moment provided
a
,a
Thus, for n = I , < 0 implies that the first moment is exponentially stable, but note that higher moments are unstable. For n = 2 < - a 2 guarantees the expo nential stability of the first and second moments, but higher moments are un stable. Recall that the Liapunov- Kushner method of section 8 applied to the same equation showed stability of x(t), i.e. of the sample paths w.p .l provided < - a2 /2 but nothing was said there about the stability of the moments. It is hard to give an economic interpretation to an economic model described by an Ito equation whose point equilibrium is stable but whose higher moments are unstable. Therefore we need to distinguish between the sample path behavior and the moment behavior and actually choose to give priority regarding stability characteristics to one or the other behavior. The area of stochastic stability in the sense of convergence in distribution to a steady state distribution remains an open field for research. It is difficult to establish the existence of an equilibrium distribution in a general setting, let alone prove its stability. For special cases we have existence theorems for an equilibrium distribution. The basic results in this area arc reported in the book by Mandl ( 1968) and some of the original work first appeared in Feller ( 1 954), Tanaka (1957) and Khasminskii ( 1 962). Merton (I 975a) has a stochastic stability result in a special case of a continuous growth model which he obtains by assuming constant savings functions being maximized over the set of constant savings func tions. This is a very special stability result. The problem of stochastic stability in a continuous stochastic optimal growth, i.e. the problem of proving that opti mal stochastic processes converge in distribution to a steady state distribution, remains open. Brock and Mirman (I 972) proved this result for the discrete sto chastic growth model. Concerning the topic of stochastic control, the problem stated in eqs. (I 0 . 1 ) and (I 0.2) is a stochastic version o f the deterministic control problem studied by Arrow and Kurz ( 1 970, pp. 27 -5 1 ) We chose to study (1 0 . 1 ) and ( 1 0.2) be cause most economists are familiar with the deterministic control problem analy zed by Arrow and Kurz ( 1 970) and the present stochastic version will enable the reader to compare the familiar with the new mathematical issues and results of the stochastic extension.
a
,
.
Stochastic calculus
139
The analysis of section I 0 uses Bellman's principle of optimality as it first ap peared in Bellman (I 957), and subsequently popularized in books such as Drey fus (1 965), Hadley (1 964) and Mangasarian ( 1 969), along with stochastic analy sis to derive stochastic conditions of optimality. Aoki (1 967), Astrom (1 970), Kushner ( 1 97 1 ) and Bertsekas (1 976) have presentations of stochastic control at the introductory level. However, unlike the area of deterministic optimal control where many books are available, such as Anderson and Moore ( 1 97 1 ), Athans and Falb (1 966), Berkovitz (1 974), Bryson and Ho ( I 979), Kwakernaak and Sivan (1 972), Strauss ( 1 968), Pontryagin et al. (1 962), Hestenes (1 966), and Lee and Markus (1 967), the literature on stochastic optimal control is not yet very large. At an advanced level the reader is encouraged to consult the book by Fleming and Rishel ( 1 975) and the papers of Benes ( 1 97 1 ), Bismut (1973, 1 976), Davis ( 1 973), Fleming ( 1969, 1 97 1 }, Kushner (1965, 1967b, 1975), Rishel (1 970) and Won ham (I 970). We note that Fleming and Rishel ( 1 975, ch. 5) provide a rigorous analysis for the stochastic optimal control problem which supplements our heuristic approach. In what they call the verification theorem, Fleming and Rishell ( 1 97 5 , p. 1 59) give suf ficient conditions for an optimum, supposing that a well-behaved solution exists for the Hamilton -Jacobi- Bellman nonlinear partial differential equation with the appropriate boundary conditions. In section 1 2 we present a generalized Ito formula and a maximum principle for jump processes. Our approach is intuitive and its aim is to familiarize the reader with some of the mathematics used in Merton (1971 ). A rigorous analysis would require an analysis of the exact mathematical properties of the term g(c, x)dq (t) of eq. ( 1 2 . 1 ) and the meaning of the jump process integrals. Kushner (1967, p. 1 8) and Doob ( 1 953, p. 284) discuss some of these issues. For a more detailed treatment see Dellacherie ( 1 974). It is appropriate to note that as economists apply the techniques of stochastic control theory to economic models a need will soon develop for a stability analysis of such stochastic models. Recent economic research in the deterministic case by Araujo and Scheinkman (1 977), Benveniste and Scheinkman (1977, 1979), Scheinkman ( 1 976, 1 978), Brock and Scheinkman ( 1 977, 1 976), Brock ( 1 976, 1 977), Cass and Shell (I 976), McKenzie ( 1 976 ), Magill ( I 977a), Samuel son (1 972), and Levhari and Liviatan ( 1 972), among others, has demonstrated how a wide class of economic problems arise from deterministic optimal control whose stability properties are crucial for correctly specifying such models. The availability of mathematical results concerning the stability of deterministic con trol systems such as Gal'perin and Krasovskii (1 963}, Hale ( I 969), Hartman (196 1 ), Hartman and Olech ( 1962), Lefschetz ( 1 965), Mangasarian ( 1 963, 1 966}, Markus and Yamabe (1 960), Rockafellar (1 973, 1976), and Roxin (1965, 1 966), have helped economic researchers. On the other hand, the nonavailability of
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Stochastic methods in economics and finance
enough mathematical results on the stability of systems of stochastic differential equations arising from a stochastic control context may delay economic research in this area. For a sample of such economic problems see Brock and Magill ( 1979) and Magill (I 977b). Many applications of stochastic methods in economics and finance deal with stochastic control in discrete time as opposed to the continuous time case dis cussed in this chapter. We chose to include in the next two chapters discrete time stochastic applications to enrich the reader's education, although we have not discussed explicitly discrete stochastic methods in this chapter. There are many similarities, however, and the various applications will illustrate them in what follows. It suffices to mention here that Kushner ( 1 9 7 1 ) establishes various similarities and relationships between discrete and continuous time stochastic models. Discrete time stochastic problems are usually less complicated than the corresponding continuous time problems. The latter may be approximated by the discrete time procedure using time intervals of length Since h may be made arbitrarily small, relationships between discrete and continuous time models can be established. In finance, Merton ( 1 978) approximates the continuous time model by using only elementary probability methods to derive the continuous time theorems. In so doing he uncovers the economic assumptions imbedded in the continuous time mathematical theory.
lz.
CHAPTER 3
APPLICATIONS IN ECONOMICS
For the person who thinks in mathemat ics, and does not simply translate his ver bal thoughts or his images into mathemat ics, mathematics is a language of discovery as well as a language of verification. II.A. Simon (1977, p. xv)
I . Introduction In this chapter we present several examples to illustrate the use of stochastic methods in economic analysis. Some applications use specific results from the previous chapters, while some other applications introduce new techniques.
2. Neoclassical economic growth under uncertainty The research of Bourguignon (1 974) and Merton (1 975a) has extended the neo classical model of growth developed by Solow ( 1 956) to incorporate uncertainty. Such an extension uses Ito's lemma as a tool for introducing uncertainty into the deterministic model. Consider a homogeneous production function F(K, L ) of degree 1 , where K denotes units of capital input and L denotes units of labor input. From homo geneity we obtain that F(K/L, 1 ) = f(K/L) = f(k ), where k = KfL. For equilib rium to obtain, investment must equal saving, i.e . .
dK dt
K = - = sF(K , L), O < s < I ,
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Stochastic methods in economics and finance
s is the marginal propensity to save. The Solow neoclassical differential equation ofgrowth for the certainty case is obtained as follows (a dot above a where
variable denotes its time derivative): k=
dk
dt
=
( ) = dr L d
K
KL - LK L2
=
k. L
-
L
L.
K = sf(k) - nk, L
(2. 1 )
where LjL = n, i.e. it is assumed that L (t) = L (O)e" 1 ,
(2.2)
L (O) > O, O < n < I .
The existence, uniqueness and global asymptotic stability properties of the steady state solution of the neoclassical differential equation in (2.1) are presented in Burmeister and Dobell ( 1 970, pp. 23- 30). Suppose now that instead of LjL = n, labor growth is described by the sto chastic differential equation, d L = nL dt + oLdz.
(2.3)
The stochastic part is dz, where z = z (t, w) = z (t ) is a Wiener process defined on some probability space (n, !F, P). In the engineering literature dz is usually cal led white noise. The drift of the process, n , is the expected rate of labor growth per unit of time and the variance of the process per unit o f time is o2 Note that dL/L = n d + adz says that over a short period of time the proportionate rate of change of the labor force is normally distributed with mean n d t and variance o2 dt. The new specification of the growth of labor n i (2.3) alters the neoclassical differential equation of growth. To compute the stochastic neoclassical differen tial equation ofgrowth we make use of Ito's lemma. We are given that •
t
,
dK = sF(K L)dt, dL = nL d t + oLdz. Letting k = KJL , Ito's lemma yields ak ak ak dk = - d t + - dL + - d K + at aL aK +
�[:��
(dK)2 + 2
a�:L
(dK} (dL) +
:;_� J (dL)2
Applications in economics
= -
=
� (nL d t L
+
(�
Ka2 L 2 d t aL dz) + _!_ sF(K, L)dt + ]_ L 2 L
[sf(k) - (n-a2 )k ]dt - kadz.
)
143
(2.4)
Note that if a = 0 for all k E [0, oo), then eq. (2.4) yields as a special case the certainty differential equation of neoclassical growth in (2.1}. Comparing eqs. (2. 1 ) and (2.4) we see that because of the new specification o f the labor growth in (2.3}, Ito's lemma has enabled us to obtain random fluctuations in the changes of the output per labor unit ratio dk. These random fluctuations are due, of course, to the random fluctuations of the labor growth. Thus, uncertainty with respect to labor growth via Ito's lemma is translated into uncertainty with re spect to output per labor unit, which is consistent with our intuition that fluc tuations in an input are expected to cause fluctuations in output.
3. Growth in an open economy under uncertainty Let F(K, L) be a homogeneous production function of degree such an economy be saving = sF(K, L )dt + pK dz,
1
and let saving in
(3.1)
where , as before, s is the marginal propensity to save and pKdz indicates a ran dom inflow of capital from the rest of the world. Observe that we do not explain the causes of p K dz because our purpose in this application is to illustrate ItO 's lemma and how uncertainty in the rest of the world causes uncertainty in the domestic economy. Here we assume that there are ad hoc random inflows of capital but we hasten to add that one possible explanation of pK dz may be the random fluctuations in the differential between domestic interest rates and the average interest rate prevailing in the rest of the world. Next, as in the previous application, we assume that the behavior of labor growth is given by dL = nL d t + aL dz. The model consists of the labor growth equation and the equilibrium condi tion equation dK = sF(K, L)d t + pKdz.
(3.2)
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144
Note that (3.2) says that over a short period of time the proportionate rate of change of the capital stock is normally distributed with mean sF(K, L)dt and variance p2 d t. Thus, in this application we have two sources of uncertainty, i.e. uncertainty due to fluctuations in the labor force and uncertainty due to fluctu . ations in the inflow of foreign capital. Both such fluctuations are expected to influence the changes in the domestic output per labor unit and the precise for mulation is obtained by computing dk. To obtain dk, k = K/L, we use Ito's lem ma which yields
dk = +
K 1 [nLdt + aLdz) + [sF(K, L)dt + pKdz] L L2
�[
-2
C2) pKoL d t + ::; o2 L2 dr]
= - .£. nL dt - .!_ aLdz + sF(K, L) dt + p � d z L L L2
1 L
/}
2 pKaL dt +
-LK a2 L 2 dt 3
= (sf(k) - (n-a2 + pa)k ]d t - (a-p)kdz.
(3.3)
Obviously, if p = 0 and there is no inflow of foreign capital, the last equation reduces to eq. (2.4).
4.
Growth under uncertainty: Properties of solutions
Consider the stochastic differential �equation of economic growth derived in sec tion 2 of this chapter:
dk = [sf(k) - (n-a2 )k]dt - akdz,
(4.1 )
with initial random condition k(O, w) = k(O) = k0 > 0. Suppose that (4.1) has a unique solution k (t, w) = k (t) for t e (0, oo) What are the properties of such a solution? The answer is provided by the following theorems. .
Theorem 4.1 (The Markov property). Suppose that the stochastic differential eq. (4. 1 ), with initial random condition k0 > 0, has a unique solution. Then its
Applications in economics
145
unique solution k(t), t E [0, ) is a Markov process whose initial probability distribution at t = 0 is k0 and whose transition probability is P(s, k, t, B) = P[k(t) E B I k(s) = k]. The proof follows immediately from theorem 7.3 of the previous chapter. This theorem is a useful result, particularly for economic policy considera tions. Suppose that for an economy the process of capital per worker is described by the Markov process k (t). Given that the economy has capital per worker k at time s, the economic policy makers may be interested in knowing the probability that at some future time t the capital per worker will fall within the interval oo
We now establish that k(t) is a diffusion process. Theorem 4.2 (The diffusion property). Suppose that the stochastic differential eq. (4.1), with initial condition k0 > 0, has a unique solution. Then its unique solution k(t), t E (0, oo), is a diffusion process with drift coefficient [sf(k(t)) (n -a2 )k(t)] and diffusion coefficient a2 k2 (t). The proof of this theorem follows from theorem 7.4 of the previous chapter if we note that eq. (4.1) is autonomous stochastic differential equation and therefore the continuity with respect to t is vacuously satisfied. The economic significance of this result may be made clear by a comparison of various systems. For example, in the deterministic model of economic growth we obtain a differential equation whose present state determines its future evolu tion. The Markov property of the stochastic differential equation of growth is richer in content because the current state completely determines the probability of occupying various states at all future times. The diffusion property goes even further: it describes changes in the process of capital accumulation per worker during a small unit of time, say D.t, as the sum of two factors. The first factor, i.e. the drift coefficient, is the macroeconomic average velocity of the random motion of capital accumulation when k (s) = k. The second factor, i.e. the dif fusion coefficient, measures the local magnitude of the fluctuation of k(t) k (s) about the average value, which is caused by collisions of the process of capital accumulation with economic and noneconomic variables undergoing a random movement. A straightforward generalization of eq. (4. 1) is when s and a are functions of k, written as s(k) and a(k), instead of being nonrandom and non-negative con stants. Assuming this to be the case we may rewrite (4.1) as an
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Stochastic methods in economics and finance
dk [s(k)[(k)- (n-a2 (k))k)dt - a(k)kdz. =
(4.2)
Similar analysis as the one just completed can show that if a unique solution of (4.2) exists then it will satisfy the Markov and the diffusion properties. S.
Growth under uncertainty: Stationary distribution
The tools of stochastic calculus can be used to further study neoclassical growth under uncertainty. In this section we are interested in the existence of a station which is assumed to be the solu ary distribution of the stochastic process tion of eq. (2.4). As discussed in section 9 of the previous chapter, the stochastic process for is completely characterized by the drift and diffusion coefficients. Using the results of section 9 of Chapter 2, the stationary distribution for k, de noted by is given by
k(t)
k 1r(k), (n- a2 )y dy 7T(k) = a2mk2 exp �2 Jk s[(y)a2y2 J 2 k sf(y) mk 2 exp a2 f y2 dy =
-
nI
°
2
[
.
-
--
J
(5 . 1 )
.
Note that (5. 1 ) for the special case when the production function is given by a Cobb - Douglas function, 0 < a < 1 , becomes
i.e.[(k) k0, 2s 7T (k) mk- 2nfo 2 exp IL(l-a)a 2 k-(1 - o:)lJ . In eqs. (5 .1) and (5 2 ) m is determined so that J;' 7T{y)dy =
=
(5 2 )
-
.
.
=
I . For further
analysis of (5 . 1 ) and (5 .2) see Merton ( I975a). The remainder of this section follows Merton (1 975a) in comparing the ex pected stationary value of per capita output with the steady state certainty value. To achieve this goal we need a brief analysis and a technical lemma. Consider (2.4) and suppose that it has a stationary distribution denoted by 7T. Let be a twice continuously differentiable function and use Ito's lemma to compute i.e.
g(k)
dg(k), dg(k) g'(k)dk + � g" (k)(dk)2 = (g'(k)[s[(k) - (n- a2 )k] + � g"(k)a2 k2 ) d t - g'(k)a2 k2 dz. =
Applications in economics
147
The following lemma is useful.
g(k}
Lemma 5.1 . Suppose is twice continuously differentiable and that (2.4) has a stationary distribution rr. Also, assume that
(g'(k}a2 k2 rr(k}) =
lim
k-o
lim
k-�
(g'(k)o2k2 rr(k)] = 0.
Then (5.3} For a proof see Merton (1975a, p. 392). as a special case to compute the expected We use lemma 5 . I with stationary value of per capita output, i.e. We use (5 .3) as follows:
g(k) = k
E(f(k)}.
E(l [sf(k} - (n-a2 }k] + 0} = 0.
(5.4)
•
From (5.4) we obtain
E(sf(k)) = (n- a2 )E(k) and thus
E(f(k)) = IZ-02 s E(k).
(5.5)
The result in (5 .5) is of particular interest because it allows for a comparison = with the certainty case. For example, let < a < I . From growth theory we know that the certainty estimate of the steady state per capita output
f(k} ko:, 0
IS
(5.6) How do (5.5) and (5.6) compare? Obviously
1 - o:) 2 n-a s o:/( E(f(k}) = E(k) > ( n ) s -
·
(5.7)
Eq. (5.7) illustrates that the certainty estimate is biased since (5.5) is larger than
Stochastic methods in economics and finance
148
(5 .6) and that therefore care must be taken in using the certainty analysis, even as an approximation of stochastic analysis.
6. The stochastic Ramsey problem
(I
optimal saving
In this section we follow Merton 975a) in determining the policy function under uncertainty. The problem is to a saving policy T - such that we
t)
s*(k,
fmd
T
J u(c)dt
maximize E0
(6. 1 )
0
subject to
dk = (sf(k) - (n-a2 )k)dt - akdz and k(t) � 0 for each t w .p. l , and in particular k(1) � 0. Here, u is a strictly concave, von Neumann-Morgenstern utility function of per capita consumption c for the representative consumer. Note that (6.2) c (l -s)[(k). =
To solve this stochastic maximization problem we use Bellman's Optimality Principle, as in section 1 0 of Chapter 2 . Let
J(k (t), t, 1)
=
m�x Et
T
J [(1-s)[(k)]dt. t
u
(6.3)
The Hamilton-Jacobi-Bellman equation for (6.3) is given by
{ u [(I-s)f(k)] + 2 + _!_ a2 k 2 } . s
0 = max
aJ ar
-
aJ sf(k) - (n-a2 )k ] + +ak [
a J 2 a k2
(6.4)
The first-order condition to be satisfied by the optimal policy
0 = u' [(1-s*)f(k)](-f(k)) +
�� [(k),
s* from (6.4) is
Applications in economics
149
which becomes
aJ . u , [(l-s*)f(k)] = ak u'
(6.5)
s*
s*,
Note that means du/dc. To solve for in principle, one solves (6.5) for and and then substitutes this solution into (6.4) as a function of which becomes a partial differential equation for Once (6.4) is solved then its solution is substituted back into (6.5) to determine s* as a function of and The nonlinearity of the Hamilton-Jacobi-Bellman equation causes difficul ties in finding a closed form solution. One way of overcoming this difficulty is by letting oo, in which case the partial differential equation is reduced to an ordinary differential equation. This is done next. Observe that is a time-homogeneous process and that is not a function of time; thus from (6.3) we deduce that
k, T-t
aJjak,
J.
k
T-t.
T-+
u
k
aJ = - E, {u[(l-s*(k, T- t))f(k(T-t)))}. (6.6) at Suppose that an optimal policy exists, f is a well-behaved production function, and n -a2 > 0, then as T-+ lim s*(k, T-t) = s*(k, ) = s*(k), there will exist a stationary distribution for k associated with the optimal policy s*(k) and denoted by n*. Let T in (6.6) to obtain . a' (6.7) hm 3i = -E*(u[(l-s*)f(k)]) = --B, oo
oo
{ }
-+
oo
E*
where is the expectation operator over the stationary distribution n* and B is the level of expected utility of per capita consumption in the Ramsey optimal stationary distribution. Use (6.7) in (6.4) to write as oo,
T-+ aJ 1 a2J 0 = u [(l-s*)f(k)] -B + - [s*[(k) - (n-a2)k] + - -- a2k2 • ak 2 ak2 Next we differentiate (6.5) with respect to k:
::� = u"[(l-s")f(k)) ( (1-s")((k) - �s: f(k) ) .
(6.8)
(6.9)
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Stochastic methods in economics and finance
Finally, substitute (6.5) and (6.9) into (6.8) and rearrange terms to conclude d S* 0 = (--2I a2 k 2ju· ) -- + (fiu ' "
dk
- -2 a2 k 2 u "( )s * + I
(6.10) Note that (6.1 0) is a first-order differential equation for s* with boundary con dition as t � oo,
lim E0 {J(k (t), t)} = 0. In (6.10), if namely
a=
0, the classical Ramsey rule of the certainty case is yielded,
u's*f- u 'nk + u - B = 0, which is usually rewritten as
B-u , s*f-- nk = -u, where B is the bliss level of utility associated with maximum steady state con sumption and k s*f- nk along the optimal certainty path. =
7. Bismut on optimal growth In this section we give an application of Bismut's approach to optimal stochastic control presented in section 1 1 of Chapter 2. Here we follow Bismut ( 1 975). Consider a one-sector optimal growth model with the usual notation i.e. k is capital per worker, f(k) is a well-behaved production function, s is the marginal propensity to save, u is a concave utility function and p is the discount rate. The problem is to maximize the expected discounted intertemporal utility assuming u ' (0) = oo,
m!x
E0
00
J e-Pt u ((l -s)f(k))d t
0
subject to the constraints
(7 . 1 )
App/icatiorrs itt economics
151
(7.2)
dk = sf(k)dt + a(k, sf(k))dz, k(O) = k0 > 0. The transformed Hamiltonian function, i.e. :II
( 1 1 .5)
of Chapter
2,
is written as
(7.3)
= u((l -s)f(k)) + psf(k) + H a(k, s[(k)).
Maximize .f(' in terms of s, where 0 < s < 1 , to get
- u ' (c)f + pf + Ha1[ = 0, which becomes, after dividing by [,
(7.4)
u'(c) = p + Ha1 .
Note that c is per capita consumption and a1 is the partial derivative of a with respect to investment where investment equals sf(k ). Next, we write eq. (1 1 .7) of Chapter 2 as it applies to our case . We have dp = - [( 1-s)f'(k)u'(c) + psf'(k) - pp + H(ak
+ sf'(k)a1 )] dt + (7 .5)
where ak denotes the partial derivative of a with respect to k. Eq. (7 .5) may be rewritten as dp
=
{- (p + Ha1 )['(k) - Hak + pp} dt + /ldz + d1l1.
(7.6)
Eqs. (7 .4) and (7 .6) uncover important economic reasoning. Eq. (7 .4) indicates that the consumer will consume up to the point where the marginal utility of his consumption is equal to the expected marginal value of capital in terms of utili ty, minus the marginal risk of investment valued at its cost. If, for example, the consumer is a risk-averter with - H > 0, this will tend to make the consumer con sume more than he would with the same p when no risk is involved. Let R denote the cost of capital. The consumer pays p + Ha1 to the producer and the cost of the producer is Rk - /Ia. The equation of profits is (p
+ Ha1 )f(k) + Ha(k, J) - Rk,
(7.7)
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152
which once maximized in
k gives (7.8)
Then the profit rate in terms of the marginal value of capital
(
)
H . H a f, (k) +-a R r== 1 +p p p k I
p, denoted by r, is (7.9)
Now look for a moment at (7 .6) and rewrite it as
dp
,
dz dt
-= {-(p +Ha1)[ (k) -Hak + pp } +H dt
Multiply both sides of (7 .1 0) by
+ dM dt
·
(7.10)
(1/p) and take its expected value to conclude (7 . 1 1 )
Use (7
.1 1) to rewrite (7 .9) as 1 ' r + E dp dt P
( )
P=
which is the neoclassical relation between interest rate the expected inflation rate. Finally, compare and from (7 .9) to obtain
(7. 1 2)
p, rate of return r, and
f' (k) r r - (Hfp)ak f'(k) 1 + (11/p)a1 ' where the instantaneous risk premium f'(k) - r is equal to (Hfp) (ak + ra1) 1 + (Hfp)a1 =
Having presented several applications of stochastic calculus to economic growth we next discuss the concept of rational expectations which recently has received attention from several economists. Rational expectations use the tech niques of stochastic analysis and they are incorporated in several applications in this chapter and the next.
Applications in economics 8.
153
The rational expectations hypothesis
In this section we present the
rational expectations hypothesis postulated by
Muth ( 1 9 6 1 ) which has found many applications in stochastic economic and financial models. Theorists realize that to make stochastic models complete, an expectations hypothesis is needed. What kind of infonnation is used by agents and how it is put together to frame an estimate of future conditions is impor tant because the character of dynamic processes is sensitive to the way expecta tions are influenced by the actual course of events. Muth ( 1 9 6 1 ) suggests that expectations, since they are informed predictions of future events, are essentially the same as the predictions of the relevant economic theory. This hypothesis may be restated: the subjective probability distributions of outcomes tend to be distributed, for the same information set, about the objective probability distri butions of outcomes. This hypothesis asserts that: ( 1 ) infonnation is scarce and the economic system generally does not waste it; (2) the way expectations are formed depends specifically on the structure of the relevant system describing the economy; and (3) a prediction based on general information will have no substantial effect on the operation of the economic system. The hypothesis does not assert that predictions of economic agents are perfect or that their expecta tions are all the same . From a theoretical standpoint there are good reasons for assuming rational expectations because: ( 1 ) it is a hypothesis applicable to all dynamic problems, and expectations in different markets would not have to be treated in different ways; (2) if expectations were not moderately rational there would be opportu nities to make profits; and (3) rational expectations is a hypothesis that can be modified with its analytical methods remaining applicable in systems with in complete or incorrect information . As an illustration of the ideas above, we present Muth 's
(I 96 1 )
model of
price fluctuations in an isolated market, with a flXed production lag, of a com modity which cannot be stored. The model is given by: {8. 1 ) demand, C(t) -{3p(t) (8.2) P(t) 'YPe(t) + (t) supply , (8.3) market equilibrium, P(t) C(t) where P(t) represents the number of units produced in a period lasting as long as the production lag, C(t) is the amount consumed , p (t) is the market price in the tth period, pe(t) is the market price expected to prevail during the tth period on the basis of infonnation available through the (t- 1 )st period, and finally (t) is = = =
u
u
an error term. All the variables used here are deviations from equilibrium values.
1 54
Stochastic methods in economics and finance
Put (8. 1 ) and (8.2) in (8.3) to get p(t) = -('Y/�)pe(t) - (1/(3)u (t).
(8.4)
Note that u (t) is unknown at the time the production decisions are made but it is known and relevant at the time the commodity is purchased in the market. Suppose that the errors have no serial correlation and that Eu (t) = 0. Then the prediction of the model in (8 .4) is (8.5) In (8.5) E p(t) denotes the prediction of the model or the theory and it is objec tive, while pe(t) denotes the subjective prediction of the firms. If the prediction of the model were different from the expectations of the firms, there would be opportunities for profit. Note that we do not use parentheses with the expecta tion operation to simplify the notation. The rationality assumption given by (8.6) states that such profit opportunities could no longer exist. If ('Y/�) * - 1 in (8.5), then the rationality assumption implies that (8.7) i.e. the expected price equals the equilibrium price. Let us now introduce more realism into our illustration by allowing for ef fects in demand and alternative costs in supply. We assume that part of the shock variable may be predicted on the basis of prior information. From (8.4), taking conditional expectation, we write Ep(t) = - ("t/f3)pe(t) - ( l /(3)Eu (t),
(8.8)
and using the rationality assumption in (8.6) we obtain (8.9) which yields
pe(t) = - ( 1 /((3 + 'Y )] Eu (t).
(8. 1 0)
Applications in economics
155
If the shock is observable, then the conditional expected value may be found directly. If the shock is not observable, it must be estimated from the past histo ry of variables that can be measured. In this latter case we shall write the u's as a linear combination of the past history of normally and independently distributed random variables x (t) with zero mean and variance o2 , i.e. 00
i=O
u (t) = 1: w(i)x(t-i),
(8. 1 1 )
Ex (i) = 0,
(8.12)
Ex (i)x (j) =
{ 02 0
if i = j, if i =Fi.
{8.13)
The price will be a linear function of the same independent disturbances and will be written as
00
i=O
(8.14)
p (t) = 1: W(i)x(t-i). Similarly, from (8.12) 00
i= 1
pe(t) = W{O)Ex (t) + 1: W(i)x(t-i) =
:E i= 1 00
W(i)x(t-i).
(8.15)
Putting (8.14) and (8.15) in (8 . 1 ) and (8.2), respectively, and solving using (8.3) we have
W(O)x(t) +
(
)
oo 1 oo "/ 1 + - 1: W(z)x(t-i) = --1: w(z)x(t-i). (3
I
(3
0
(8.16)
Eq. (8.16) is an identity in the x's and therefore the relation between W(i) and w(i) is as follows: W{O) = W(i) = -
1 w(O), (3 1
f3 + 'Y
w (i) for i = 1 , 2, 3 ... .
(8. 1 7) (8.18)
Stochastic methods in economics and finance
156
Note that (8 . 1 7) and (8 . 1 8) give the parameters of the relation between the price function and the expected price function in terms of the past history of indepen dent shocks. The next step is that o f writing the expected price in terms of the history of observable variables, i.e. 00
pe(t) = � V(j)p(t-j). i
(8.19)
=I
Use (8.14), (8. 1 5) and (8.19) to obtain
pe(t)
= =
=
00
00
� W(i)x(t-z) = � V(j)p(t-j)
i= 1
[
j= I
]
-� V(j) � W(i-j)x (t-i- j)
1=
1
[ _f
-�
I= 1
J= 1
t=O -
J
V(j) W(i-j) x (t-i).
(8.20)
As before, we again conclude from (8.20) that the coefficients must satisfy
W(i) =
t
i= 1
V(j) W(i-j)
(8.21)
since the equality in (8 .20) must hold for all shocks. In (8 .2 1 ) we have a system of equations with a triangular structure which may be solved successively for V1 ,
v2 , . . .
As a particular illustration suppose that in (8.1 1 ) w(i) = 1 for all i = O, 1 , 2, ... , which means that an exogenous shock, say a technological change, affects all future conditions of supply. Then using (8.17) and (8.18) eq. (8.2 1 ) yields
pe(t)
=
- 00 ( {3 )j {3
'Y
-�
J=
1
'Y
+ ')'
p(t-j),
(8.22)
which expresses the expected price as a geometrically weighted moving average of past prices.
Applications in economics
157
9 . Investment under uncertainty In this application we follow Lucas and Prescott ( 197 1 ) who introduced an un certain future into an adjustment-cost type model of the firm to study the time series behavior of investment, output and prices. Their paper is rich in methodol ogical ideas and techniques and in this section we plan just to describe the model and state an existence theorem. Consider an industry consisting of many small firms each producing a single output, q, by using one single input capital, k , , under constant returns to scale. With an appropriate choice of units we may use k, to also denote production at full capacity and denote the production function as (9.1) Gross investment, denoted by x, , is related to capacity in a nonlinear way: (9.2) where h is assumed to be bounded, increasing, h (0) > 0, continuously differen tiable, strictly concave, and that there exists a l) , 0 < l) < 1 , such that l) h -r { 1 ). The last assumption means that 5k, is the investment rate that is needed to maintain the capital stock k,. The assumption of strict concavity is made be cause it gives rise to adjustment costs of investment and the model thus reflects gradual changes in capital stock as opposed to immediate passage to a long-run equilibrium level. Let p1 denote the product price and r the cost of capital with r > 0. Using the standard discount factor, /3, where f3 1 /( l + r), then ex post the present value of the firm, V, is given by =
=
(9.3) We use the notation kt , xt and pt interchangeably for industry and firm variabies. The objective of the firm is the maximization of the mean value of (9 .3) with the stochastic behavior of P, somehow specified. However, because we have omitted variable factors of production in (9 . I ) the firm will choose to produce at full capacity and the only nontrivial decision of the firm is the choice of an investment level. This investment level is decided, as usual, by comparing the known cost of a unit of investment to an expected marginal return. To simplify the firm's investment decision we place the burden of evaluating the income
Stochastic methods in economics and finance
158
stream changes due to a given investment to the traders in the firm's securities. We denote by w: the undiscounted value per unit of capital expected to prevail next period . The firm's problem can now be stated as x>o
max [ - x + (3k1h (xfkt )w*], r
(9.4)
w1k1 P,k, - x + (3k1h(x/k1)w:
(9.5)
- 1 + (3h ' (xfk1)w: :s;;, 0 with equality, if x > 0.
(9.6)
where x is the cost of investment and (3k t+ 1 w: is the next period value resulting from x. Using (9.2) we obtain (9.4). The maximization problem of (9.4) is sub ject to the two constraints =
and
Since (9.5) and (9.6) are solved jointly for x, and w: as functions of k1,p1 and w1, we can write the investment function as
x, = k1g(w,
�
p1),g' > 0.
(9.7)
The industry demand function is assumed to be subject to random shifts and is written as
P, = D(q1, u1) ,
(9.8)
where {u 1 } is a Markov process with a transition function p ( , · ) defined on R 2 • For given u 1 , D is a continuous, strictly decreasing function of q 1, p 1 = D(O, u 1) < oo, and with •
q
J D(z, u1)dz
(9.9)
0
bounded uniformly in u 1 and q. We also assume that D is continuous and in creasing in u 1 so that an increase in u 1 causes a shift to the right of the demand function. For given (k0 , u0) an anticipated price process is defined to 1be a sequence {pt } of functions of (u 1 , u 2 , .•• , u t ) , or functions with domain R • Similarly, an investment-output plan is defmed as a sequence {q1 , x1 } of functions on R 1 • We restrict the sequences {pt }, {q t } and {xt } to belong to L i.e. to be elements of +,
159
Applications itr economics
the class L with non-negative terms for all (t, u 1 , u2 , ••• , u1). Here L denotes the set of all sequences x {x1}, t 0, 1 , 2 , . . . , where x0 is a number and for t � 1 , x1 is a bounded measurable function on R1, bounded i n the sense that norm is finite, i.e. =
II x II = sup t
(u 1
1
sup •••
, 11
=
1)E E
1 I x (u 1 , '
.••
, u1 ) I
< oo.
Therefore , for any sequences {p1}, {q1} and {x1}, elements of L •, the present value V in (9 .3) is a well-defmed random variable with a finite mean. The objec tive of the firm then becomes the maximization of the mean value of V with respect to the investment-output policy, given an anticipated price sequence. To link the anticipated price sequence to the actual price sequence we assume that expectations of the firms are rational or that the anticipated price at time t is the same function of (u 1 , , u 1 ) as the actual price. We are now ready to de fmc an industry equilibrium for fiXed initial state (k, u) as an element {q� , x� , p� } of L + x L + x L + such that (9 .8) is satisfied for all (t, u 1 , , u1) and such that •••
•••
(9.10) for all {q,, x 1} E L + x L + satisfying (9 . 1 ) and (9 .2). Note that the expectation of the tth term in (9 .1 0) is taken with respect to the joint distribution of (u 1 , ,u 1). Having defmed industry equilibrium the question naturally arises of whether a unique equilibrium exists. Lucas and Prescott (1971) first show that a competi tive equilibrium leads the industry to maximize a certain "consumer surplus" ex pression, and then they show that the latter maxinlUm problem can be solved using the techniques of dynamic programming. Define the function s(q, u), q � O , u E R by •••
s (q, u) =
q
J D(z. u)dz, 0
(9. 1 1 )
so that for given u , s (q, u) is a continuously differentiable, increasing, strictly concave, positive and bounded function of q , and for given q, s is increasing in u . Note that s(q t , u t ) is the area under the industry's demand curve at an output of q1 and with the state of demand u1• Let the discounted consumer surplus, S, for the industry be 1
S = E ( 1�0 J1 [s (q1, u1) - x1 ) ) .
(9. 1 2)
160
Stochastic methods in economics and finance
We are interested in using the connection between the maximization of S and competitive equilibrium in order to determine the properties of the latter. Asso ciated with the maximization of S is the functional equation
v(k, u) = x;;o.o sup (s(k, u) -x + (3fv [kh(x/k),z )p(dz, u) ) . We now state a basic result of Lucas and Prescott (I 971 ).
(9.13)
v v0,
Theorem 9.1 . The functional eq. (9 .1 3) has a unique, bounded solution on the right-hand side of (9.13) is attained by a unique (0, oo) x R ; and for all and is the unique industry equilibrium , given In terms of given by
(k, u) x(k, u),
x(k, u).
k0
x1 x(k1, u1), kt+1 k1h[x(k1, u1)/k1), q, = k, , P1 = D(q1 , u1)
(9.14)
=
{9.15)
=
for t = 0,
I , 2 , ... , and all realizations of the process {u,}.
Proof. See Lucas and Prescott ( 1 97 1 , pp. 666-67 1 ). With the question of existence and uniqueness being settled Lucas and Prescott study the long-run equilibrium assuming ( 1 ) independent errors and (2) serially dependent errors. Below we state the results for the frrst case and refer the reader to the Lucas and Prescott paper for the results of the second case. Suppose that the shifts and are independent for s =1= t, i.e. the transition will function It follows in this case that does not depend on not depend on and from (9 .14) and (9 . 1 5) it follows that the time path of the capital stock will be deterministic, given by
p(z . u) u
u1
us
u.
x(k, u)
(9.16)
x(k1) x(k) ok,
v(k1, u).
where is the unique investment rate obtaining We defme a capital stock /cC > 0 to be a stationary solution of (9 .16) if and only if it is a solution to = since = l.
h (o)
{u1} there k0 > 0 and
Theorem 9.2. Under the hypothesis of independence of the process are two possibilities for the behavior of the optimal stock First, if if
k1 •
Applications in economics
f D(O, u)p (du) > o + [rfh ' (o)] holds, then plicitly by
161
(9.17)
k , will converge monotonically to the stationary value kc , given im·
f D(k, u)p(du) = o
+ [rfh' (o)] .
Or, secondly, if (9. 1 7) fails to hold, or if k0 = 0, then k1 will converge monoto· nically to zero.
Proof. See Lucas and Prescott (197 1 , pp. 671 -673). 10. Competitive processes, the transversality condition and convergence
The methods of Bismut (1973, 1975), briefly presented earlier, have been used by Brock and Magill (1 979) in an attempt to develop a general approach to the continuous time stochastic processes that arise in dynamic economics. In this section we follow Brock and Magill {1979) who show that under a concavity assumption, to be specified, a competitive process which satisfies a transversality condition is optimal under a discounted catching.up criterion. Let (!2, F, P) denote a complete probability space, � a a·field on n, and P a probability measure on .� Let I = [0, oo) denote the non·negative time interval and (/, Jt, p) the complete measure space of Lebesgue measurable sets ..II, with Lebesgue measure p. Let (n x /, :1{', P x p) denote the associated complete pro duct measure space with complete measure P x p and o·field .Yl' ::> y; x dt. Let (Rn , vfln ), with n � 1 , denote the measurable space formed from the n�imen· sional real Euclidean space Rn with a-field of Lebesgue measurable sets. tt " . Let
be an .tf·measurable function (random process) induced by the following sto· chastic control problem. Find an .)f-measurable control v(w, t) E U c Rs, s � I , such that for o > 0 sup J Je- 6 vEU n I
t u(w, t, k(w, t), v(w, t))dtdP(w), t
k(w, t) = k0 + Jt(w, k(w, ) v(w, T))dT + 0 T,
T ,
{10. 1 )
Stochastic methods in economics and finance
162 T
+
J o(w, 0
T,
k(w, T), v(w, T)) d z ( w, T),
(1 0.2)
where u
E Rl ;
f = (/1 , , f" ) E R" ; 0 = •.•
E R"m and k0 E K c R"
is a nonrandom initial condition; u(·, k, v),f(·, k, v), o(·, k, z,) are .W'�measur able random processes for all (k, v) in k x U c R" x Rs and u (w, ),/(w, . ), o(w, ), are continuous on I x K x U for almost all w, while z (w, t) E Rm , m � 1 , is a Brownian motion process. Let •
•
.� = .�(z (w, t)), T E [0, t]) denote the smallest complete o-ficld on n relative to which the random varia abies {z (w, T ), T E [0, t)} are measurable. We require that k(w, t) be .'F,-mea sureable for all t E /, so that /(·) and o( · ) are nonanticipating with respect to the family of a-fields {.�t' t E I}. To ensure the existence of a unique random process k(w, t) as a solution of {10.2) we assume that the Lipschitz and growth conditions of Chapter 2 are satisfied, i.e. we make the following assumption. Assumption 1 0.1 . Lipschitz and growth conditions: there exist positive con stants, et and /3, such that (i) II /(w, t, k, v) - !(w, t, k , v) I I + II o (w , t, k, v) - o (w, t, k, v) I I � a I I k for all (k, v), (k, v) E K x
-
k II
U, for almost all (w, t) E n x I, and
(ii) 11/{w, t,k, v) W + II o(w, t,k, v)l l 2 � {3(1 + II k 11 2 ) for all (k, v) E K x U, for almost all (w, t) E n x I. Here, as in previous sections, double bars denote vector norms. Recall from Chapter 2 that assumption 1 0.1 is sufficient for the existence and uniqueness of a solution of a stochastic differential equation. We will exhibit a sufficient condition for a random process to be a solution of the problem (1 0.1) and (1 0 .2) in tenns of a certain price support property,
163
Applications in economics
the nature of which is most clearly revealed by restr�cting this stochastic control in the manner of Bismut ( 1 973, p. 393) and Rockafellar (1970, p. 1 88) as fol lows. Consider the new integrand L (w, t , k , k , a) =
sup u(w, t, k, v) 1/(w, t, k, v) = k, a(w, t, k, v) = a vEV oo if there is no v E U such that -
f(w, t, k, v) = k, a(w, t, k, v) = a.
Note that L (w, t, · ) is upper semicontinuous for all (w, t) E n x I and L (w, t, k(w, t), k(w, t), a(w, t)) is Jf-measurable whenever k(w, t), k (w, t) and a(w, t) are Jf-measurable. We impose indirect concavity and boundedness conditions on the functions u (w, t, ), f(w, t, · ) and a(w, t,• ), and a convexity condition on the domain K x U by the following assumption. •
Assumption 10.2. Concavity-boundedness: L (w, t, · ) is concave in {k, ic, a) for all (k, k, a) E R n R n X R nm for all (w, t) E n X I and there exists 'Y E R, I 'Y I such that L ( ) < 'Y for all (w, t, k, ic, a) E n X I X R n X R n X R n m . <
.
00'
X
Let (k, ic, a) (k(w, t), k(w, t), a(w, t)) denote the Jf'-measurable random pro cess defined by the equation =
k(w, t) k0 + =
t
t
J k(w, r)d r + J a(w, r)dz{w, ;),
0
(10.3)
0
where k0 E K R n is a nonrandom initial condition, and where there exists an Jf-measurable control v(w, t) E U such that c
k(w, ;) = f(w, T , k (w, ;), v(w, ;)) ;
a(w, ;) = a(w, T, k (w, r), v(w, ;))
for almost all (w, t) E n x I. In view of assumption 10.1 f
n
(j o
I I k(w, r) 112
dr +
j o
II a(w, r) 112
)
d r dP (w) <
�
for all t E I.
( 1 0.4) denote the class of random processes satisfying (10.3) and (1 0.4),
We let 9 where k (w, ;), a(w, r) are f-measurable and nonanticipating with respect to
Stochastic methods in economics and finance
164
the family of a-fields {.¥,, t E /}. The control problem (10.1) and (1 0.2) then re duces to the following. Stochastic variational problem: Let L satisfy assumption 10.2, let L(w, t, · ) be upper semicontinuous for all (w, t)mE n x /, and let L(·.x. v,s) be .Jr'-mca surable for all (x, v, s) E R n x R " x R " . Find anfi'-measurable random process (k, k, a) E Y'such that sup nI II e-b t L(w, t, k(w, t), k(w, t), a(w, t))dtdP(w) . (10.5) (k ,k ,o)E5>
In order to give (1 0.5) a broad interpretation we introduce the following defini tion. Let % {IJ del)ote a class of .*"-measurable random processes (k, k, a). A random process (k, k,a) E %is optimal (in X) if c
T
lim inf I Je-6T (L(w, T, k, k , a) - L(w, r,k,k, a))dTdP (w) � O T-+00
n
0
for all random processes (k, k, a) E Jf: Next, let p(w, t): (n J,.f() (R n ,v ft'·1 ) denote an .Y't-measurable random price process dual to k (w, t). We let (p-op,p, 1r) = (p(w, t) - o p(w . t), p(w, t), 1r(w , t)) denote the .tt"'-measurable random price process defined by the equation
(1 0.6)
�
x
t
t
p(w, t) = Po + J p(w, T)dT + f 1T(W, T)dz (w, T), 0
(1 0.7)
0
where Po E R n is nonrandom and where p(w, r) and 1r(w, r) are$-measurable random processes, nonanticipating with respect to the family of a-fields {.�, , t E nm n n /}, with values in (R , j l ) and (R"m , . ll ), respectively, and which satisfy f
n
(]I I P(w, T) o
11 2
dT +
j I 1r(w, T) dT ) dP(w) o
112
< oo
(1 0.8)
for all t E /. Let fl'* denote the class of random processes defined in this way. The following concept is fundamental to all the analysis that follows. We de-
Applications in economics
165
fine a random process (k, k, a) E .t? to be competitive if there exists a dual ran dom price process (p - bp,p, 1r) E !'?* such that
- ...:... -
.
.. --, (p-bp),-k + -,...:. P k + tr(1r a ) + L (w, t, k , k, a)
;>
(-p- bp)' k + p 'k + tr( rr a' ) + L (w, t , k, k, a)
{10.9)
for all (k, k, a) E R n X R " X R n m , for almost all (w, t) E n X /. The economic interpretation .of this concept is thls: a competitive random process is a random process (k, k, a) E fiJ that has associated with it a dual ran dom price process Cp-bp, p , 1r) E &>* under which it maximizes profit almost surely, at almost every instant. For -(p- op) denotes the vector of unit rental costs, -rr denotes the matrix of unit risk costs induced by the disturbance matrix a, while ( 1 , p) is the vector of unit output prices, so that . ,, k + (p-op) k + tr(1ro ) L + p ,..:.. is the (imputed) profit which is maximized almost surely, at almost every instant, by a competitive random process. We also give a geometric interpretation which is this: the random process (p- bp, p , rr) e fP * genera�es supporting hyperplanes to the epigraph of -L(w, t, k, k, a) at the point (k, k, a) for almost all (w, t) E n X I. The hyperplanes parallel to a given supporting hyperplane indicate hyper planes of constant profit, so that the supporting hyperplanes are precisely the hyperplanes of maximum profit at each instant. Note that under assumption 1 0.2 a random process ( k, k, a) E f? is competi tive if and only if
(p {w, t) - Op (w, !), p(w, !), 7r (W, !)) E - oL(w, t, k(w, t), k(w, t)a(w, t))
(10.10)
for almost all (w, t) E n X /, where oL denotes the subdiff('rential of L (w, t, . ) . Eq. (10.10) is a generalization of the standard Euler-Lagrange equation. The Fenchel conjugate of -L (w, t, k, k, a) with respect to (k, a) will be cal led the generalized Hamiltonian
G(w, t, k,p, rr) =
sup rzm n (k ,o)E R x R
+ L (w, t, k, k, a)}.
{p' k + tr(rra') + (10. 1 1 )
Stochastic methods in economics and finance
166
G(w, t, k,
(w, t)
Observe that E p, 1r) is concave in k and convex in (p. 1r) for aJI " m n n n X I and is defined for all p , 7T) E R X R X R . Under assumption 1 0 .2 if ) is differentiable, a random process (k, k, a) E f1' is competitive if and only if
(k,
G(w, t,
·
I
t
k(w, t) = k0 + J GP (w, T) d T + J Grr(w, T)dz (w, T), 0 0
( 1 0 . 1 2)
and also
I
p(w, t) = Po + f [ - Gk (w, T) + op(w, T)]dT + 0 t
+
J 1r(w, T)dz {w, T).
( I 0 . 1 3)
0
stochastic Hamiltonian equa
Eqs. ( 1 0 . 1 2) and ( 1 0 . 1 3), which will be called the are a generalization of the standard Hamiltonian canonical equations for a discounted stochastic variational problem. Assume that for all (x, u, E R" x R " x R n m that
tions,
s)
L(w, t,x, s) = L (x, s) for all (w, t) E n X /, so that L is nonrandom and time independent. When ( 1 0.5) is fmite we deftn e the current value function W(k}: u,
u,
R"
W(k(t)) =
�up
(k ,k,o) E .?
E,
�
R
00
J e-o (r-r) L (k(w, T), k(w, T), a(w, T)} dT, t
( 1 0 . 1 4)
k
t,
k(t) K.
where E, denotes the conditional expectation given at time and where replaces as the initial condition in (10.3). is a concave function for all k E Note that under assumption 10.2, In establishing convergence properties, the following class of Mc!(enzie competi tive processes is of special importance. A random process (k, k, a ) e q> is Mc if it is competitive and if the dual random price process t) supports the value function
k0
W (k}
Kenzie competitive p(w, W(k(w, t)) - p(w, t)'k(w , t) � W(k) - p(w, t)' k for all k for almost all (w, t) E Q X /. " ER ,
( 1 0 . 1 5)
Applications in economics
167
If (k, k, a) fi' is McKenzie competitive then in (10.8) is determined by the condition p0 E a W{k0). We are now ready to state The9rem 10.1 . (Transversality condition). A competitive random process (k, k, o) E Jf"with dual price process CP - f>p, p, 1r) E #*, which satisfies the transversality condition lim sup E0e- 6 T p(w, 1)'k(w, T) � O E
Po
T-+oo
is optimal in the class %of random processes for which lim inf E0e- 6 p(w, T)' k (w, T) � 0. T-oo
T
(10.16)
Sec Brock and Magill ( 1979). The sample paths of a McKenzie competitive process starting from nonrandom initial conditions have a remarkable convergence property. Consider a point k0 E K and a McKenzie competitive process emanating from this point. Under as sumptions, which include a strict concavity assumption on the basic integrand L , a McKenzie competitive process emanating from any other point k0 E K con verges almost surely to the first process. This result, which has its origin in the dual relationship between the prices and quantities of a McKenzie competitive process, may be stated as follows. Theorem 10.2 (Almost sure convergence). Let assumption 10.2 be satisfied and let the function L be time independent and nonrandom as in (10.14). If two McKenzie competitive random processes (1 0.17) (k, k, a) E :� and (k, k, a ) E f¥>, with associated dual price processes (10.18) (p- fJp, p , 1r) E .�* and ("p-fJp, p, 1f) E .� *, starting from the nonrandom initial conditions
Proof.
satisfy the following conditions:
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Stochastic methods in economics and finance
{i) there exists a compact convex subset M
c
R" x Rn
such that for all t
E
I
(k(w, t), p(w, t)) = (k(w, t; k0), p(w, t; Po)) E M, (k(w, t), p(w, t) = (k(w, t; k0), p(w, t; Po) E ft1
for almost all W E n; (ii) there exists > 0 such that the function 11
V(k-k, p-p) = - (p-p) ' (k- k)
(10.19)
satisfies
fi2
V(k-k, p-p)� - 11 11 (k-k , p-p) 11 2
(10.20)
for all (k-k,p-p) e Y = {(k-k, p-p) l (k,p), (k,p) e ft1}; {iii) the value function is (a) strictly concave, {b) differentiable, and {c) strictly concave and differentiable, for all k in the interior of K , K = { k I (k,p) E M }; then (i), (ii) and (iii) (a) imply k(w, t) - k(w. t) -+ 0 w.p.l. as t -+ (i), (ii) and (iii) (b) imply p(w, t) - p(w, t) -+ 0 w .p.l. as t -+ (i), (ii) and (iii) (c) imply (k(w, t) - k(w, t),p(w, t) - p (w, t)) -+ 0 w.p.l. as t -+ oo Proof Sec Brock and Magill (1979, pp. 852-855). oo;
oo;
.
II.
Rational expectations equilibrium
In this section we follow Brock and Magill (1979) to show how the concept of a competitive process, in conjunction with the stochastic Hamiltonian equations (10.12) and (10.13), provides a useful framework for the analysis of rational expectations equilibrium. We examine in particular a rational expectations equi-
169
Applications in economics
librium for a competitive industry in which a fixed finite number of firms behave according to a stochastic adjustment cost theory by creating an extended inte grand problem analogous to that of Lucas and Prescott (1971 ) . Consider therefore an industry composed of N � 1 firms, each producing the same industry good with the aid of n � 1 capital goods. All firms have identical expectations regarding the industry product's price process, which is an �Yf-mea surable , nonanticipating process
r(w, t): (n x I,.Yt ) -+ (R+ , J t).
(1 1 .1 )
The instantaneous flow of profit of the i th fmn is the difference between its reve where and and its costs C; nue in = v ) denote the capital stocks and investment rates of the ith firm, and where f; arc the standard strictly concave production and and -C; adjustment cost functions. If l) > 0 denotes the nonrandom interest rate, then each firm seeks to maximize its expected discounted profit by selecting an $" measurable, nonanticipating investment processes
r(w, t)/ (k; (w, t)) v; (v;1, , (v1) (k;)
(vi (w, t)),
k ; = (kit, ... , ki")
•••
i v (w, t): (n x 1, .1(') -+ (R" , JI"),
such that sup Eo
00
f e-
O
T
i=
1,
..., N,
[r(w, T)[i (k ; (w, T)) - c; (vi (w, T))]dT'
v1(w ,t) 0 t t 1 k; (w, t) = k� + J vi (w, T)d T + J a; (k1 (w, T))dz (w, T), 0
0
a; (k ; )dz; = H1i , ag
(H1ik1 + ag )dz1i, l
�
i=
(1 1 .2)
(I 1 .3)
are n x n and n x I matrices with constant coefficients and T ) is an m-dimensionaJ Brownian motion process. This model is a simple stochastic version of the basic Lucas 1967) and Mortensen ( 1973) adjustment cost model, with the standard additional neoclassical assumption that the invest ment and output process of the ith firm have no direct external effects on the investment and output processes of the kth firm, for i if= On the product market the total market supply, given by where _
z' (w,
(
k.
1 (w, t) = Q5 :E ! (k1(w, t)), N
i= 1
170
Stochastic methods in economics and finance
depends in a complex way through the maximizing behavior of firms on the price process {1 1 . 1 ). On the demand side of the product market we make the simplifying assumption that the total market demand depends only on the cur rent market price Q0 (w, t) = 1/1- 1 (r(w, t)), r � 0,
where 1/I(Q) > 0, 1/1' (Q) < 0 and Q � 0. A rational expectations equilibrium for the product market of the industry is an .1t'-measurable, nonanticipating random process {1 1 .1) such that (1 1 .4) QD (w, t) = Qs (w , t) for almost all (w, t) E n X I. The firms' expectations are rational in that the anticipated price process coin cides almost surely with the actual price process generated on the market by their maximizing behavior. Consider the integra] of the demand function 'lt(Q) =
Q
f 1/J(y)dy,
Q � O,
0
so that 'It ' (Q) = 1/I(Q), 'It " (Q) = 1/J '(Q) < 0 and Q � 0. We call the problem of fmding N £-measurable, nonanticipating investment processes (u1 (w, t), . . , UN (w, t)): (Q X /,.tf) -+ (R n N ,.Jt"N) .
such that
(1 1 .5)
where (k 1 (w, t), . . , kN (w , t) satisfy (1 1 .2) and ( 1 1 .3), almost surely, the ex tended integrand problem. .
171
Applications in economics
Theorem I I . I .
(Rational expectations equilibrium). If the generalized Hamil tonian of the extended integrand problem (1 1 .5) is differentiable, if (k (w, t),p (w, t)) = (k1 (w, t), .. . , kN (w, t),p1 (w , t), . . , j5N(w, t) is a competitive process for (1 1 .5) which satisfies the transversality condition (1 1 .6) limT-+oo sup E0 e- 5 T p(w, T)' k(w, T) � 0, and if for any alternative random process k(w� t) with k0 = k0 (1 1 .7) lim inf E0 e-5 Tj5(w , T)' k(w, T) � 0, T-+oo then the Jf-measurable, nonanticipating random process
.
(1 1 .8) is a rational expectations equilibrium for the product market of the industry. Proof Since the generalized Hamiltonian for the extended integrand problem is differentiable, (k(w, t), p(w, t)) is competitive if and only if, writing (I 0.12) and (10.13) in shorthand form, d k; = h ;(p; ) d t + ai(k,.)dz; dp ; = (Bp i 'It' � f; (k i) f; - � 1Tiia ii. ] d t + 7T ;d z ; '
- (
i= 1
) . k
1
i= 1
where Jz'. = (C'. ;) -1 . Eqs. {1 1.8) and (1 1 .9) imply v
k
1
.
i = 1 , . . ,N ,
,i =
(I 1 .9)
.
1, .. , N. {1 1.10)
Eqs. ( 1 1 .6), {11.7) and ( 1 1 .8) are sufficient conditions for each firm to maxi mize expected discounted profit by theorem 10.1 . Eq. (1 1 .8) implies that (1 1 .4) is satisfied and the proof is complete.
Stochastic methods in economics and /iiUlnce
172
1 2. Linear quadratic objective function
In this section and the following one we present the analysis of particular sto chastic control problems which have found applicability in economics and finance. Consider the one-dimensional state and control problem - W(x(t)) = min Et v
J e-P (s- r ) {a(x(s))2 + b (v(s))2 }ds
s=t
subject to x (t) known and dx (t) = v(t)d t + ox(t)dz (t). Note that o > 0, a > 0, b > 0 and we try as a solution
p
> 0.
Since the objective function is convex
W(x) = -Px2 ,
( 12.1)
which yields W = -2Px
and W = -2P. (12.2) We now use the Hamilton-Jacobi-Bellman equation for the case with discount ing, i.e. X
XX
to write in our specific case -pPx2 = max { -ax2 -bv2 2 Pxv v
-
+
� ( -2Po2 x2 )}.
(12.3)
From (12.3) we obtain that -2bv-2Px = 0, or that V0
=
-Pxjb.
Substitute (12.4) into (12.3); then -pPx2
=
{ -ax2 - b ( - Pxb }2 - 2Px (- bPx } - Po2 x2 } ,
(12.4)
Applications in economics
173
which after simple algebraic manipulations is reduced to
1 - P2 + b
(p -o2 )P - a = 0.
(1 2.5)
Choose the largest root because of the convexity of the objective function and denote it by P+. A candidate solution that is optimum is given by
dx(t)= [- !
J
the solution of which is x (t) = x0 exp
13.
(1 2.6)
P. x (r) d t + ox(t)dz (r),
{ (- ! P. - 0; )
t}
t + oz ( )
.
(12.7)
State valuation functions of exponential form
Jn the previous section we saw that the state valuation function of the linear quadratic example turned out to be quadratic in the state variable . This is a gen eral principle in linear quadratic problems. We tum now to a type of problem where the state valuation function turns out to be of exponential form. In exam ining this type of problem we hope that the reader will pick up the common thread of technique that is used in searching for closed form solutions for the state valuation functions when they exist. We also hope that this will illustrate a technique of showing that a state valuation function of a particular closed form cannot satisfy the Hamilton-Jacobi-Bellman equation as well. In this way the search for closed form solutions to the Hamilton-Jacobi-Bellman partial differential equations is systematic instead of random guesswork. It is well to learn how to prove that solutions of a particular form do not exist as well as formulating hypotheses upon the objective and the constraints of the problem so that a closed form solution of a particular type does exist. Consider the problem J(x(t), t,N) = max subject to
E,
{1
D(s)u (c(s), s)ds + B(x(N), N) }
dx (s) = (b(s)x(sf<s> - c(s))d s + o(x (s), s)dz(s).
(13.1) (1 3.2)
Stochastic methods in economics and finance
174
Here D(s), c(s), u (c(s), s), x (s), B(x(N), N), b (s), (j(s), a(x (s), s), and d z (s) de note discount rate at time s, consumption at time s , utility of consumption at time s, state variable at time s which in economic applications is usually capital, bequest function of the state variable at time N, efficiency multiplier of produc tion function at time s , output elasticity at time s , standard deviation function at time s , and Wiener process that is standardized at each moment of time, respec tively. Notice that the problem in ( 13.1) and ( 1 3 2) is quite general. We shall see how general we can make it, and still get a closed form solution for the state valuation function J. The stochastic maximum principle applied to {13.1) and ( 1 3 2 ) gives .
.
0
where
= maximum {D(t)u (c, t) +�(J)}, c>O
!f (J) =
(13.3)
1
lim E { [J(x (t + tl t), t + tl t, N) - J(x (t), t, N)]/ tl t }.
�t-+0
(13.4)
The operator .!fJ( ) is just the conditional expectation of the instantaneous rate of change of J and is a general case of the operator defined in eq. (7.9) of Chapter 2. See also eq. (4.15) of Chapter 4. Let us calculate the partial differential equation given in eq. (13.3) for the constant relative risk-aversion class of utility functions given by ·
u (c(s), s) = (k (s) (c(s))1 - a(s))/(1 -a(s)).
(13.5)
c.
Here k (s) is just a constant independent of Assume that the optimal consump tion given by eq. (1 3.3) is positive. Then we can write (1 3.6)
In eq. (13.6) it is understood that everything is a function of time, and sub scripts denote the obvious partial differentiation. Insert the solution for c from eq. (13.6) into eq. (13.3) to get the partial differential equation (13.7)
Eq. (1 3.7) is obtained by the following steps. First, evaluate the operator in eq. (13.4) to get
Applications in economics
175
{1 3.8) Eq. (13.8) is obtained either by a direct application of Ito's lemma, or by ex panding J into a formal Taylor series discarding all terms of higher order than � t, and taking conditional expectation of the remaining conditional on informa tion received at time t. Secondly, insert the optimal value for c from eq. ( 1 3 .6) into eq. (13.8). Insert all of this into eq. (13.3) and rearrange terms to get eq. (13.7). Notice that everything in eq. (13.7) is a function of time, but we have suppressed notation of this dependence in order to simplify the expression. Now look carefully at eq. (1 3.7), especially the last three terms. If we tried as a candidate for J a function of the form
J(x, t, N) g(t, N)xe ( t) + [(t, N) =
(I 3.9)
then we should attempt to determine the unknown exponent e(t) by equating exponents and testing for consistency by examining the possibility of identical exponents on the x variable of all four terms of eq. (1 3.7). The exponents of x in eq. (1 3.7) for each of the four terms reading from left to right are:
(e- 1 ) (a-1 )Ia = e + exponent on (xe log x) = (e- 1) + (3 =
(e-2) + exponent on x from a2 •
(13.10)
Eq. (13.10) follows immediately by inspecting the partial derivatives of J, which we calculate by using eq. (13 .9), and list below in equations
(13. 1 1 ) J
X
=
egxe - l . J '
XX
=
e(e-1)gxe- 2
'
(13.12)
We see immediately from eqs. (I 3.1 1 ) and (13.12) that in our search for the most general problem that will give us a solution for the state valuation function of the form (1 3.9), we will have to assume the conditions listed below in order to get the same exponent for the last three equalities of eq. (13.10). First assume that
(13.13) Assumption ( 1 3 . 1 3) is needed to get rid of the term xe log x that obstructs the validity of eq. (13 . 1 1 ). Secondly, assumption
Stochastic rnetlrods in economics and finance
1 76
3.14) on the variance function is needed so that the last equality of eq. (13.1 0) is equal to e. Finally, assumption (13.1 5) {3 (s) 1 is needed for identity with the other terms of eq. (13.10). Thus, assumptions (13.13)- (13.15) allow us to assert that it is possible to find a function of the form in eq. (13.9) that will satisfy cq. (13.7). To solve for the unknown exponent e, which by eq. (13.13) must be con stant in time, we solve (e-1} (a- 1 )/a = e. (13.16) Thus, (13.17) e = 1 -a. We see now that a must be independent of time in order to obtain a solution of the form of (13.9). We have now solved for the exponent, e, in eq. (13.9). What remains is to solve for the functions g and f. In order to solve for the functions and/, write eq. (13.7) using eq. (13.10) as follows: (I
=
g
+ � e(e-1 )ghxe .
(13.18) Now, eq. (13.18) must hold for all x and therefore/, = 0 must hold. Hence,[ is independent of time. Now cancel xe off both sides of (13.18) to get 0 = (eg)(a- 1 )fa ka- 1 a(a-1 )- 1 + g, + beg + � e(e-l )glz = 0. Notice that (13.19) can be written in the form
(13.19) (13.20)
where g =.gr
Applications in economics
177
We see that (1 3.20) is a differential equation that looks formidable to solve. However, such a differential equation can be transformed. Experiment with transformations of the form (13.21) y =g'Y to discover that if-y 1/cx, then eq. (13.20) can be transformed into the form =
(13.22)
Cancel y a - 1 off of both sides of ( 1 3 .22) to get (13.23) 0 = a0 + a 1 y + cxy, which is a differential equation linear in y, y. Tltis is a standard fonn which can be solved regardless of whether the coefficients a0 and a 1 are dependent upon time or not. Here the coefficients a0 and a1 are defmed in the obvious manner by eq. (13.19). A boundary condition is needed before eq. (13.23) can be solved. This is obtained from the restriction (1 3.24) J(x,N,N) = B(x,N). Of course, we will not be able to solve eqs. ( 1 3 .23) and (13.24) for arbitrary be quest functions. To see how things go put (1 3.25) B(x,N) := O. Look now at eq. (1 3.9). From eqs. (13.9) and (13.25) we infer immediately that f(t,N) = 0. This is so because f,(t,N) = 0 for all t and f(N,N) = 0. Fur thermore, eqs. (13.25) and (1 3.19) imply g(N, N) = 0 and y(N, N) 0. (13.26) Thus, we have our boundary condition on eq. (1 3.23) and it can be solved for an explicit solutiony(t,N), from whichg can be calculated from eq. (13.21). What about the case of more general bequest functions than that given in eq. (13.25)? We see immediately by inspection of eq. (1 3.24) that if there is any hope of a solution for J of the form given by eq. (1 3.9), then the exponent on x in the function B(x, N) must be the same as in eq. (1 3.9). Thus, a more gener al class of bequest functions where x enters with the same exponent as that of =
e
1 78
Stochastic methods in economics and finance
the utility function may be treated without much extra effort. Thus, closed form solutions exist in this case as well. Now, retrace through our derivation using the Hamilton-Jacobi-Bellman equation, eq. ( 1 3 .7), and recall how much had to be assumed on the structure of the utility function, the production function, and the bequest function in order to get a closed form solution for J. We saw from eqs. ( 1 3 . 1 3) and (13.15) that the exponent on the utility function had to be independent of time and the ex ponent on the production function had to be unity at all points in time. Further more, the variance had to be proportional to the square of the state variable, but the constant of proportionality could vary in time . Also, all other coefficients could vary in time. It is worthwhile to note that other sources of randomness may be introduced into problem ( 1 3 . 1 ) and ( 1 3 .2) above and beyond the randomness in the change of the state variable, provided that these sources of randomness are independent of the Wiener process dz; that is to say the coefficients k , b and h may be gener ated by Ito equations as well. We may summarize this application by noticing that it is fairly illustrative of the method of searching for closed form solutions when they exist and deter mining assumptions that are necessary to put on the problem in order to get the existence of a closed form solution. Furthermore, we have approached the prob lem of searching for a closed form solution in such a way that illustrates the maximum amount of generality that one can have and still get the existence of a closed form solution. Once the solution has been found, then the optimal con trols may be solved for explicitly, and the optimal law of motion that describes the system may be written down in closed form as well. Quadratic objectives with linear dynamics or exponential objectives with linear dynamics are by far the most common examples where closed form solutions to the HJB equation are available.
14. Money, prices and inflation In this section we follow Gertler (1 979) to present a rational expectations macro economic model to illustrate the convergence of the state variables to a stable distribution over time. We begin by stating the deterministic system of structural relationships:
(14.1) In (M/P) = c In Y - mi,
c , m > 0,
(1 4.2)
Applications in economics
rrd = i\ (In Y - a In K ) + 1T *, i\. a > 0 , 0�¢� 1,
1 79
(14.3) (14.4)
1T = 1T*,
(14.5)
i = r + 1T*.
(14.6)
Equation (14.1) is a reduced form IS function relating the logarithm of real output positively to the logarithm of the capital stock and negatively to the real interest rate. It is assumed that the capital stock is constant. Eq. (14.2) is the LM function relating the logarithm of real money balances positively to the logarithm of real income and negatively to the nominal interest rate. Eq. (14.3) relates the desired rate of inflation, 1T d , to the aggregate excess effective demand and the anticipated inflation rate 1T *. Note that aggregate excess effective demand is the difference between output and the natural full employment level of output. Eq. (14.4) describes the price velocity constraint where actual current inflation rr is the sum of the ad hoc inertia inflation if and a convex combination of 1T d and if. Eq. (14.5) denotes myopic perfect foresight and it is the equivalent of rational expectations in the deterministic context. Finally, eq. ( 1 4.6) is a definitional identity where the nominal interest rate i is the sum of the real interest rate r and the anticipated inflation rate 1T *. Next we introduce uncertainty into the detem1inistic system by postulating In Y d t = b 1 ln Kd t - b2 rd t + dz.
(14.7)
Note that this last equation generalizes eq. (14.1) since the random tenn dz is being added. In (14.7), dz equals sy(t)dt, where s is a parameter and y(t) is a normally distributed random variable with mean zero, unit variance, and the y (t)'s are serially uncorrelated. The introduction of uncertainty also affects eq. ( 14.5). Assuming rational ex pectations instead of (14.5) we now postulate 1r *(t) = lim E[1r(t) 1 .�U ] , U-+t
(14.8)
where .OF is the a-field incorporating all information available at time u . Such information includes the structure of the model and both the initial values and the past behavior of the state variables. Using eqs. (14.2), (14.3) and (14.7) we obtain: u
Stochastic methods in economics and finance
180
ttdt = ¢Xw In (M/P)dt + ¢(Xwm + l)tt*dt + ( 1 -¢)7Tdt + ¢'A [(wmb.Jb2 ) - a] I n Kdt + [(¢Xmw)/b2 ) dz,
(14.9)
where for convenience we have w = b 2 /(m + cb 2 ). Eq. (14.9) describes the dy namic random behavior of actual inflation as a function of various parameters, of the logarithm of real money supply, the anticipated inflation rate, the inertia inflation of the system, the logarithm of the capital stock and uncertainty. From the assumption of rational expectations in (14.8) and (14.9) we conclude that
tt* = [mw/(1 - mcS w) In (M/P) + [ 1/(1-mcS w)] ii - [ cS/(1 -mcS w)]a In K + [ b1mcS wfb 2 (1-mcS w)] In K.
(14.10)
In this last equation we let cS = ¢'A (I -¢). Note that cS is the coefficient on excess demand in the price adjustment equation. Substituting (14.10) into (14.9) we have
ttdt = [ cS w/(1-mcS w)] In (MJP)dt + [1/(1-mcSw)) irdt - [cS/(1 -mcSw)]a In Kdt + [b1mcSwfb2 (1-mcSw)] In Kdt (14. 1 1 ) + (¢"A.mwfb2 )dz. At this point we assume that the money growth rate is deterministic and fixed at p.. Using Ito's lemma we obtain d In
(M/P) = pdt - ttdt + (¢Xmwjb2)2 dt,
(14.12)
where ttd t is as in (14.1 1 ). Finally, we write dir = {3 * (ttdt - 1i'dt),
(14.13)
dh = Alzdt + f' dt + dv,
(14.14)
which is an assumption about the evolution of the system's inertia inflation ii. We are now ready to study the problem of convergence. Substituting ttdt from (14. 1 1 ) into (14.12) and (14.13) we obtain the system of linear stochastic differential equations
where
181
Applications in economics
( (:P)) ' ( (= + n
h=
-l>wf(l -ml>w) -l/(1 - ml>w)
A -
f
{3*l> wf(l-ml>w) {3 *ml>w/(1 -m�w)
JJ.
•
[l>/(1 - ml>w)]a In K - [b1ml>wfb 2 ( 1 - ml>w)] In K
- [{3*l>/(l - ml>w)]a In K + [{3*b1 ml>wfb2(I - ml>w)] In K
( dv=
-(¢"Amw/b 2 )dz {3*(¢"Am wfb2)dz
1
,
= eA h (t) + J e A 0
)'
).
The solution of (14.14) is given by h (t)
)
{1 -s)
1
fds J e A (r- s) dv(s) ds. +
0
( 14.15)
Take the expected value of h (t) in this last equation to yield Eh(t) = e A h(O) + J e A (t- s> fds. t
(14.16)
0
Assume that A is negative-definite. Then from (14.16) we conclude that the mean, Eh (t), converges to a stable path. The necessary and sufficient conditions for stability are {3*m < 1
(14.17)
l> < l/mw.
(14.18)
and
Stochastic methods in economics and finance
1 82
Eq. {14.17) restricts the adjustment speed of the price inertia and requires that the demand for real balances cannot be too sensitive to the nominal interest rate. Eq. (14.18) restricts the size of 8 , the coefficient on excess demand in the price adjustment mechanism. Gertler (1979, p. 232) also shows that if A is negative-definite the variance covariance matrix of h (t) will converge to a stable value. The above discussion concludes our illustration and we refer the reader to Gertler (1 979) for further analysis of specific aspects. 1 5 . An N-sector discrete growth model In this application we follow Brock ( 1 919) to present an n-process discrete opti mal growth model which generalizes the Brock and Mirman ( I 972, 1 973) model. This section and sections 1 1 - 1 6 of Chapter 4 attempt to put together ideas from the modern theory of finance and the literature on stochastic growth mod els. Here we develop the growth theoretic part of an intertemporal general equilibrium theory of capital asset pricing. Basically , what is done is to modify the stochastic growth model of Brock and Mirman (1972) in order to put a non trivial investment decision into the asset pricing model of Lucas ( 1 978). The fmance side of the theory is presented in sections 1 1 - 1 6 of Chapter 4 and they derive their inspiration from Merton (1 973b). However, Merton's ( 1 973b) inter temporal capital asset pricing model (ICAPM) is not a general equilibrium theory in the sense of Arrow-Debreu, that is to say, the technological sources of un certainty are not related to the equilibrium prices of the risky assets in Merton (1973b). To make Merton's ICAPM a general equilibrium model, first the Brock and Mirman (1972) stochastic growth model is modified, and secondly, Lucas' ( I 978) asset pricing model is extended to include a nontrivial investment deci sion. This is done in such a way as to preserve the empirical tractibility of the Merton formulation and at the same time determine endogenously the risk prices derived by Ross (1 976) in his arbitrage theory of capital asset pricing. The model is given by
1 (3'� 1 t 00
maximize E 1
=
such that ct+ 1 + x t+ 1 -x, = x, =
N
.�
1= 1
xit , xit
u
(c,)
(15.1)
N
�1 [g;(xit ' r, ) - 8 f;, l,
� 0,
i
=
1 , 2, ... , N,
t=
l , 2, . . . ,
( 1 5 .2) ( 1 5.3)
Applications in economics
c, � 0,
t
=
1 , 2, ... ,
i 1 , 2, ... ,N,r1 historically given, =
1 83
(15.4) (15.5)
where E 1 , (3, u , c, . x,, g; , x;,. r, and f>; denote mathematical expectation condi tioned at time 1 , discount factor on future utility, utility function of consump tion, consumption at date t, capital stock at date t, production function of pro cess i, capital allocated to process i at date t, random shock which is common to all processes i, and depreciation rate for capital installed in process i, respectively . The space of {c,};: 1 , {x,};: 1 over which the maximum is being taken in (15.1) needs to be specified. Obviously, decisions at date t should be based only upon information at date t. In order to make the choice space precise some for malism is needed which is developed in what follows. The environment will be represented by a sequence {r,};: 1 of real vector valued random variables which will be assumed to be independently and identi cally distributed. The common distribution of r, is given by a measure p: Yi(Rm) [0, 1 ], where .'-Jf(R'" ) is the Borel a-field of R'" . In view of a well-known one to-one correspondence (sec, for example, Loeve, 1 977), we can adequately rep resent the environment as a measure space (Q, .o.F, v) , where n is the set of all sequences of real m vectors, .dF is the a-field generated by cylinder sets of the form n;: 1 A ,, where A, E �J(Rm), t I , 2, ... , and A1 = Rm for all but a finite number of values of t. Also v , the stochastic law of the environment, is simply the product probability induced by J..L , given the assumption of independence. The random variables r, may be viewed as the tth coordinate function on n , i.e. for any W {w, };: I E Q , r1(w) is defined b y r1(w) = W1• We shall refer to w as a possible state of the environment, or an environment sequence, and shall refer to w1 as the environment at date t. In what follows, .tF, is the a-field guaranteed by partial histories up to period t (i.e. the smallest a-field generated by cylinder sets of the form n;=l A T , where AT is in .� (R'") for all t, and A7 Rm for all T > t). The a-field -�t contains all of the informa tion about the environment which is available at date t . I n order to express precisely the fact that decisions c1,x, only depend upon information that is available at the time the decisions are made, we simply re quire that c1, x, be measurable with respect to .o.Ft . Formally the maximization in (15 .1) is taken over all stochastic processes {c,}�= 1 , {x, };: 1 that satisfy {15.2)-{15.5) and such that for each t = I , 2, . .. , c, x 1 are measurable .
==
=
==
184
Stochastic methods in economics and finance
cesses is .r-compact. While it is beyond the scope of this application to discuss existence, presumably a proof can be constructed along the lines of Bewley
(1977).
The notation almost makes the working of the model self-explanatory . There are N different processes. At date t it is decided how much to consume and how much to hold in the form of capital. It is assumed that capital goods can be cost lessly transformed into consumption goods on a one-for-one basis. After it is decided how much capital to hold then it is decided how to allocate the capital across the N processes. After the allocation is decided, nature reveals that r, and K;(x;, r1) units of new production are available from process i at the end of period t. But f>;x;, units of capital have evaporated at the end of t. Thus, net new output is K;(x;, , r1 ) - f>;xit from process i. The total output available to be divided into consumption and capital stock at date t + 1 is given by N
.�1 [g;(x;, , r1) - f>;x;,J + x1
t=
N
=
=
t= I
.� [g; (xi , r,) + (1- f> ;)X;r ] t N
(15.6)
-� f;(x;, , r,) = y '+ 1 '
l= 1
where
(15.7) denotes the total amount of output emerging from process i at the end of period t. The output y'+ 1 is divided into consumption and capital stock at the begin ning of date t + 1 and so on it goes. Note that we assume that it is costless to install capital into each process i and it is costless to allocate capital across processes at the beginning of each date t. The objective is to maximize the expected value of the discounted sum of utilities over all consumption paths and capital allocations that satisfy (15 .2)
(1 5.4).
In order to obtain sharp results we place restrictive assumptions on this prob lem. We collect the basic working assumptions below. A.l . The functions u ( ), /;( ) are all concave, increasing, and are twice contin uously differentiable. •
·
A.2. The stochastic process { r1} �= 1 is independently and identically distributed. Each r1 : (il, .t;f, p) � R m , where (f2, .�, JJ) is a probability space. Here n is the space of elementary events, 9f is the sigma field of measurable sets with respect
Applications in economics
185
to and is a probability measure defined on subsets B n, B E Yf. Further more, the range of r1, r, (fl), is compact. A.3. For each {x;1 }� r the problem in (15.1) has a unique optimal solution (unique up to a set of realizations of {r1} of measure zero). Notice that A.3 is implied by A.l and strict concavity of and {!;}� Rather than try to find the weakest possible assumptions sufficient for uniqueness of solutions to (15 .l ) it seemed simpler to assume it. Furthermore, since we are not interested in the study of existence of optimal solutions in this application we have simply assumed that also. Since the case N = 1 has been dealt with by Brock and Mirman (1972, 1973) and Mirman and Zilcha (1975, 1976, 1977), we shall be brief where possible. By A.3 we see that to each output level y1 the optimum c,,x,,x;1, giveny, may be written (15.8) c, = g(y,); x, = h(y,); x;, = h;(y,). J.l,
c
J.1
1 ,
1
u
,
1 .
The optimum policy functions g(· ), h(·) and h;(·) do not depend upon t be cause the problem is time stationary. Another useful optimum policy function may be obtained. Given x1 and r1 , A.3 implies that the optimal allocation {x;,}� and next period's optimal capi tal stock x 1 are unique. Furthermore, these may be written in the form (I 5.9) X;r = a;(x1, r,_ ) and (15.10) x = H(x,, r,). Eqs. (15 .9) and (15 .1 0) contain r, _ and r1, respectively, because the allocation decision is made after r,_ 1 is known but before r, is revealed, whlle the capital consumption decision is made after y is revealed, i.e. after r, is known. Equation (15.10) looks very much like the optimal stochastic process studied by Brock-Mirman and Mirman-Zilcha. It was shown in Brock and Mirman (1972, 1 973) for the case N = 1 that the stochastic difference equation (15.10) converges in distribution to a unique limit distribution independent of initial conditions. We shall show below that the same result may be obtained for our N process model by following the argument of Mirman and Zilcha (1975). Some lemmas are needed. 1
t+
1
t+ 1
1
t+ 1
Stochastic methods in economics and finance
1 86
Assume A.l . Let U(y . ) denote the maximum value of the objec tive in (15.1) given initial resource stocky1 • Then U(y1 ) is concave, nondecreas ing in y1 and, for each y1 > 0, the derivative U' (y . ) exists and is nonincreasing iny Proof. Mirman and Zilcha (1975) prove that Lemma 1 5 . 1 . • .
for the case N = 1 . The same argument may be used here. The details are left to the reader. Note that g(y ) in the last equation is nondecreasing since u " (c) < 0 and U' (y) is nonincreasing in y owing to the concavity of U( ). Lemma 15.2. Suppose that A.2 holds and u (c) � 0 for all c. Furthermore, as sume that along optima 1 E 1 (3 - 1 U(y1) � 0 as t � oo. 1
•
t ;:
if {c,};: 1 , {x1 };: 1 , {xu}� 1 , 1 , 2 ; ... , is optimal then the following conditions must be satisfied. For each i, t (15.1 1) u ' (c,) � (3E1 { u' (c,+ 1 ) t; (x;, , r1 ) } , (15.12) u' (c,)x;, = (3E, { u ' (ct+ 1 ) J;(x;p r,)x; r } and (15.13) lim E1 { (3, _ 1 u' (c,) x,) = 0. Proof. The proof of (15.1 1) and (15 .12) is an obvious application of calculus to {15.1) with due respect to the constraints c, � 0 and x, � 0. An argument anal ogous to that of Benveniste and Scheinkman ( I 977) establishes ( I 5 .13). By con cavity off; , i = 1, 2, ... N, U( ·) and by lemma 15.1 we have for any constant 'Y, Then
,_00
O < -y < l
187
Applications in economics
(15.14) But since U is nondecreasing1 -iny1 and each!; is increasing in X;, the l.h.s. of(I 5.14) 1 is bounded above by E1 /3 U(y1) which goes to zero as t � oo Since u ' � 0 and y1 � 0, the r.h.s. of (15.14) must go to zero as well. But by (15.12) as t � oo .
I
El il'- u'(c,l
[f
!i(x;, ' -
" '• - I
)(x;, t-
1
)] EliJ'- (f =
'
u'(c, - I ) xi, t- 1
)
= E1 /3 1 - 2 u '(c, _ 1 )x, _ 1 � o,
as
was to be shown. Lemma 15.3. Assume that u'(c) > 0, u"(c) < 0 and u'(O) = oo Furthermore, assume that fj(O, r) = O,fj'(x, r)> 0 and fj'(x, r) < 0 for all values ofr. Also sup pose that there is a set of r-values with positive probability such that fj is strictly concave in x. Then the function h(y) is continuous in y, increasing in y, and h(O) = 0. Proof. See Brock (I 979). Now by A.3 and (15.8)-(15.10) it follows thaty,+ 1 may be written ( 15.15) Y r + I = F(x,,r,). Following Mirman and Zilcha (1975) define (15.16) F(x) = min F(x, r), F(x) = max F(x, r), where R is the range of the random variable (U, PA, J.l) Rm which is compact by A.2. The following lemma shows that F and F are well defined. Lemma I 5.4. Assume the hypotheses of lemma 1 5.3 and suppose that each f;(x, r) is continuous in r for each x. Then F(x, r) is continuous in r. .
rER
rER
r:
�
Stochastic methods in economics and finance
188
Proof This is straightforward because
Yt+ 1 = � /j(xit• r, ) = � /j (fli (x, )x,, r, ) F(x,, r,). I I Since fli (x 1) is continuous in x1 > 0 and each f; (x, r) is continuous in r we con clude that F(x, r) is continuous in x and r. This concludes the proof. =
Let x, x be any two ftxed points of the functions
H(x) = h(F(x)); H(x) = h(F(x)),
(15. 17)
respectively. Then Lemma 1 5.5. Any two fixed points of the pair of functions deftned in ( 1 5. 1 7) must satisfy x < x . Proof See Brock ( 1 979). Finally, applying arguments similar to Brock and Mirman following:
( 1972) we obtain the
Theorem I 5.1 . There is a distribution F(x) of the optimum aggregate capital stock x such that F,(x) � F(x) uniformly for all x. Furthermore, F(x)does not depend on the initial conditions (x 1 , r 1 ) . Here F1 ( x) = P[x, < x].
Proof See Brock ( 1979). Theorem 1 5.1 shows that the distribution of optimum aggregate capital stock at date t, F1 (x), converges pointwise to a limit distribution F(x). Theorem 15.1 is important because we will usc the optimal growth model to construct equilibrium asset prices and risk prices. Since these prices will be time stationary functions of x,, and since x1 converges in distribution to F, we will be able to use the mean ergodic theorem and stationary time series methods to make statistical inferences about these prices on the basis of time series observations. More will be said about this in chapter 4. 1 6.
Competitive frrm under price uncertainty
In this application we follow Sandmo ( 1 97 1) to illustrate the use of stochastic techniques in the theory of the competitive ftrm under price uncertainty. These
Applications in economics
189
techniques arc rather elementary and make usc of concepts introduced in Chap ter 1 . Consider a competitive firm in the short run whose output decisions arc dom inated by a concern to maximize the expected utility of profits. The sales price, p, is a non-negative random variable whose distribution is subjectively deter mined by the firm's beliefs . The density function of the sales price is f(p) and we denote the expected value of the sales price by i.e. E(p) = We assume that the firm is a price taker in the sense that it is unable to influence the sales price distribution. Let u denote the von Neumann- Morgenstern utility function of the firm and 7t(x) the profits function, where x is output. We assume that u is a bounded, concave, continuous and differentiable function such that u'(1r) > 0 and u "(1r) < 0. ( 16.1) Thus, the firm is assumed to be risk averse. The total cost function of the firm, F(x), consists of total variable cost, C(x), and fixed cost B. We write ( 16.2) F(x) = C(x) + B, where C(O) = 0 and C'(x) > 0. ( 16.3) In the usual way we define the firm's profit function by (16.4) 1r(x) px - C(x) - B, and the firm's objective to maximize the expected utility of profits can be written as (16.5) E(u(px - C(x) - B)). The necessary and sufficient conditions for a maximum of (16.5) are obtained by differentiating (16.5) with respect to x; they are (16.6) E(u '(1r) (p - C'(x))) :::: 0 and (16.7) E(u "(1r)( p - C'(x))2 - u'(1r)C"(x)) < O. p,
=
J.l..
190
Stochastic methods in economics and finance
Suppose that eqs. ( 1 6.6) and ( 16.7) determine a positive, finite and unique solu tion to the maximization problem ( 1 6.5). For our analysis the basic question is: How does the optimal output under uncertainty compare with the well-known competitive solution under certainty, where price is equated with marginal cost? To provide an answer we proceed as follows. Rewrite ( 16.6) as
( 1 6.8)
E(u'(1r)p) = E(u'(1r)C'(x)) and subtract E(u'(7T)J.L) on each side of ( 1 6 8) to get .
E(u '(1r) (p - p.)) = E (u'(1r) [C'(x) - p.]).
(16.9)
Note that taking the expectation of ( 1 6.4) we obtain
E(1r) = E(p)x - C(x) - B = p.x - C(x) - B. Therefore, 1r(x) - E(1r) = px - p.x = (p - p.)x, or equivalently 1r(x) = E(1r) + (p - p.)x. If p � p., then from the last sentence we obtain that 1r(x) � E(1r) and therefore u' (1r) � u'(E{1r)). In general,. for all p we have that
u'(1r)(p - p.) � u '(E(1r)) (p - Jl).
(16.10)
Take expectations on both sides of ( 16.1 0) to get
E(u'(1r) (p - p.)) � u'(E(1r)) E(p - p.) = 0.
( 1 6. 1 1)
Observe that the zero in the right-hand side o f ( 16. 1 1) comes from the fact that u'(E(1r)) is a constant and E(p) = p.. Combine the result in ( 16.1 1), namely that E(u'(1r) (p - p.)) � 0 with eq. (16.9) to conclude that
E(u'(1r) [C'(x) - p.]) � 0,
(16. 1 2)
which finally implies
C'(x) � p.
(16.13)
because u'(1r) > 0 by (16.1). Our result in (16.13) says that optimal output for a competitive firm under price uncertainty is characterized by marginal cost being less than the expected price. If we characterize the certainty output as that quan tity where C'(x) = p., then we may conclude that under price uncertainty, out-
191
Applications in economics
put is smaller than the certainty output. This result is a generalization of McCall's ( 1967) theorem for the special case of a constant absolute risk aversion utility function. Next, suppose that x* denotes the positive, finite and unique optimum out put which is the solution to {16.6) and satisfies (16.7). Then x* will give a global utility maximum provided (16.14) E(u(px* - C(x*) - B)) � u(-B), where -B is the level of profit when x 0. Consider the left-hand side of{l6.14) and approximate it by a Taylor series around the point p p. to rewrite ( 16.14) as =
=
E(u(p.x* - C(x) - B) + u'(p.x* - C(x*) - B)x*(p - p.) + & u"(p.x* - C(x*) - B)x*2 (p - p.)2) � u (-B).
(16.15) Note that higher-order terms in the Taylor series have been neglected and also note that by definition the second term on the left-hand side of {16 .15) is zero. Rearranging the remaining terms in ( 16.15) and dividing through by u '(px* C(x*) - B) so as to make the expressions invariant under linear transformations of the utility function, we then obtain u(px* - C(x*) - B) - u( -B) I u"(p.x* - C(x*) - B) X*2 E(p - IJ.)2 �- u '(p.x* - C(x*) - B) 2 u' (p.x* - C(x*) -- B)
..:... � .._ _ ..:... _ .._ _ ....:. .._ ....:. .._ _ __:. ;._ _
·
(16.16) Observe that the factor -u "/u' on the right-hand side of (16.16) is the risk aver sion function evaluated at the expected level of profit for optimum output x*. The factor x*2 E(p - p.)2 denotes the variance of sales. Each of these two factors is positive and therefore from ( I 6.16) we conclude that p.x* - C(x*) - B > - B, (16.17) given that the utility function is strictly increasing. Finally from ( 16.1 7) we ob tain C(x*) < p., x*
(16.18)
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Stochastic methods in economics and finance
which says that at the optimum level of output of a competitive firm under un certainty, the expected price is greater than average cost. This fact further im plies that the firm requires strictly positive expected profit in order to operate in a competitive environment under price uncertainty. Therefore, price uncer tainty leads to a modification of the standard results of the microeconomic theo ry of the competitive finn in an environment of certainty. 17. Stabilization in the presence of stochastic disturbances In this section we follow Brainard ( 1 967) and Turnovsky (1977) to illustrate the effects of uncertainty in stabilization policy. To fully demonstrate the role of uncertainty we usc the simple case of one target, denoted by y, and one instru ment, denoted by x, related linearly as follows:
(17.1)
y = aX + U .
Here we assume that y and x are scalars while a and u are random variables having expectations and variances denoted by E(a) = a , E(u) = u , var(a) = a� , and var(u) = a�, respectively. Suppose that the policy-maker has chosen a target value y*. To fix the ideas involved, suppose that ( 1 7 . I ) is a reduced form equation between GNP, y, and the money supply, x, subject to additive, u, and multiplicative, a, disturbances. Given y*, the stabilization problem is to choose x so that the policy-maker will maximize his expected utility. Brainard (1967) uses a quadratic utility function, U, of the form u=
-
(y - y*)2 .
(1 7.2)
The problem then becomes max E(U) = max E(-(y - y*)2 ) X
X
(17.3)
subject to ( 1 7. I ) Substituting ( 1 7. 1 ) to ( 1 7.3) and taking expectations we obtain
(17.4)
where p denotes the correlation coefficient between a and u. Put (1 7.4) into ( 1 7.3), differentiate with respect to x, and set this derivative equal to zero to obtain the optimal value of x; it is given by
193
Applications in economics
( 17.5) a0 = au =
0, In ( 1 7 .5) xu denotes the optimal value under uncertainty. Note that if which means that certainty prevails, then the certainty optimal value of the in strument variable, denoted xc, is given by y* - u (17 .6) X = -c
a
Comparing (17 .5) and ( 1 7.6) we observe that the difference between the values of xu and xc depends on If 0 then xu = xc . This shows that for an addi tive random disturbance the values of xu and .xc are the same; this is called the certainty equivalence result. Therefore, to understand the role of uncertainty we must study the role of the multiplicative disturbances arising from the random variable a. To do so, suppose that 0; then ( 1 7.5) becomes a0 •
aa =
au =
( I 7.7)
from which we obtain that ( 17.8)
This last equation means that the policy-maker is more conservative when uncer tainty prevails. What are the implications of such a conservative policy? Inserting xu from ( 17.7) into ( 17.1) we solve for the target value achieved from xu in the special case under discussion, i.e. when � 0 and u u the result is a
y
=
aa(y* - u) +U. a2 + a 2 a
=
=
;
(1 7.9)
Without loss ofgenerality we assume in ( 1 7.9) thaty* > ii; this assumption simply says that the target value y* is greater than the expectation of the additive dis turbance term. From ( 1 7.9) we are now able to discover the implications of un certainty in economic policy. Observe that
Stochastic methods in economics and finance
194
u) + u) - y*
a (y* E(y) - y* = E l 7i 2 a 2 + aa
\
=
=
a 2 (y*
_
u)
a�(u - y*) < a2
+
(a2
+
2
a
�) (u - y*)
+a
o.
a
( 1 7. 1 0)
From ( 17.1 0), under the assumption that y* > u, we conclude that E (y) < y*, which means that the target variable will on the average undershoot its desired value. If we assume that a� = 0 and a� =I= 0, it can be shown in a similar way that E(y) = y*, so that y will fluctuate randomly about its target value. Thus even in the simplest possible case with one target on one instrument variable, the results of stochastic analysis are more general and richer than those of the certainty analysis.
18. Stochastic capital theory in continuous time In this section we conduct an analysis similar to that of section 8 of Chapter 1 for discrete time except that now we work with the case of continuous time. Consider the Markov process {X1 } �= 0 with t e [0, oo ). Most of the time we sup pose that {X1 } �=o is given by the Ito stochastic differential equation
a
dX = [(X, t)d t + (X, t)dz,
(18.1)
-y (t, X, T) =
(18.2)
where dz is normal with mean zero and variance dt. Consider the problem sup
t< T < T
E [e- X7 I X(t) = X]. n
Here the supremum is taken over the set of measurable stopping times, i.e. the events { T � s } depend only upon {Xr } for r < s . The existence of optimal stopping times and of critical boundaries becomes a very technical matter in continuous time as the analysis of section 1 3 of Chapter 2 has shown. Hence, we shall proceed heuristically making use of some ideas from section 1 3 of Chapter 2. Actually the basic ideas are simple, intuitive and quite pretty when unencumbered by technicalities. We define the continuation region C(t, T) for problem (1 8.2) by
Applications in economics
C(t, T) = { (s, X) I -y(s, X, T) > e-nX, t � s � T}
195
(1 8.3)
Note that solutions X(s, T) to (18.4) -y(s, X, T) = X(s, T)e- rs play the role that the critical numbers {X,} played in the discrete time case Just as in the discrete time case we would like the critical numbers to exist and to be unique so that X(s, T) is a function and not a point to set mapping. We want to describe the boundary of C(t, T) by a function. We will be particularly interested in C(O, T) C(T) and C(O, ) = C. Fortunately a theorem by Miroshnichenko (1975, p. 388) gives us what we need under mild assumptions. In order to motivate Miroshnichenko's theorem we proceed heuristically as follows. Put =
oo
R (s, X) = e- rsx.
(18.5)
E
Since (t, X) C(T) therefore -y(t, X, T) > R (t, X). Now sample paths are con tinuous. Therefore if llt is small enough, it will always be worthwhile to con tinue on from (t + 11t, X(t + At)). Since the value at (t, X) is the maximum of the value of stopping before t + At and continuing on after t + llt, it follows that (t, X) C(T) implies e
-y(t, X, T) = E[-y(t + llt, X(t + llt), T) I X(t) = X].
(18.6)
In order to shorten notation, put (1 8.7)
for any function H(t, X). Let A(t, T) = {{s, X) I LR(s, X) > 0, t � s � T}.
Then we claim that A (t, D C C(t, n. To prove our claim let
(s, X) e -y(t, T) = { (s, X) I -y(s, X, T) = R(s, X)}.
Hence,
Stochastic methods in economcs i and finance
196
R (s, X) =
sup
s <. r <:. T
E [R(r, X(r)) I X(s) = X].
{18.8)
In particular, for ClOY t.s � 0, R (s, X) � E[R(s + As, X(s + .6s)) I X(s) = X].
(1 8.9)
Expand the r.h.s. of (1 8.9) and let .6s --. 0, to obtain (18.10)
R(s, X) � R(s, X) + LR(s, X).
Thus, LR (s, X) � 0 and "t(t, T) c {(s, X) I LR(s, X) � 0},
(18.1 1)
C
from which we conclude that A(t, T) C(t, T). This ends the proof of the claim. Clearly for (s, X) C(t, T) we have R(s, X) = "f(S, X, T). Furthermore, for (s, X) C(t, T), ( 18.6) holds. Hence, for (s, X) C(t, T), for At small, we must have by expanding the r.h.s. of (1 8.6) in a formal Taylor series about (t, X) and taking expected values E
E
E
"t(t, X, T) = "t(t, X, T) + "f,At + 'YxfAt + � 'Yxx a 2 At + o (At).
(18.12)
Cancel "t(t, X, T) off of both sides of (18.12), divide by At, and take At � 0 to obtain (18.13)
Equation (18.13) is the fUndamental partial differential equation ofoptimal stopping theory. It only holds on the continuation set. It is so important that it ls worthwhile to repeat how it was obtained. For (t, X) in the continuation re gion, by continuity of sample paths, E[R(t + At, X(t + At)) I X] < E[ "t(t + At, X(t + At), T) I X]
for At small. Hence, the value "t(t,X, T)must equal E ["t(t + At,X(t+ At), T) I X] for At small. Thus, "t(t, X, T) E[ "t(t + At, X(t + At), T) I X]. =
Expanding the r.h.s. of this last equation in a formal Taylor series we get ( 1 8.1 3).
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197
We are now at the point where Miroshnichenko's theorem (1975, lemma 1, p. 388) should have meaning for the reader. Theorem 18.1 (Miroshnichenko). Under mild regularity conditions on f and the following obtains: (1) A(t, T) C(t, T), and (2) C(t, T) is open and each connected component of C(t, T) contains at least one connected component of the set A (t, T). In particular, if the set A is connected, then it consists of a single connected component and the set C(t, T) also consists of a single connected component, i.e. it is connected. We gave an heuristic proof of part (1) when we established the claim. The reader is referred to Miroshnichenko ( 1975, pp. 388--389) for the proof of part (2). Theorem 18.1 is useful in obtaining a sufficient condition for uniqueness of critical numbers in the time independent case. To find such a condition, work with current values. Put (18.14) W(t, X, T) = er1-y(t, X, T). Note that Wand increase as T increases. Suppose that finite limits exist as T� oo and that these limits are values for the infinite horizon problems. From ( 18.14) we obtain a
c
'Y
The fundamental equation (18.13) becomes (18.15) In the time independent case the term W1 = 0, so that the fundamental partial differential equation becomes an ordinary differential equation. This is pleasant and will be exploited heavily in what follows. What does the connectedness of A { (t, X) I Lg(t, X) > 0} boil down to in special cases? For example, if g(t, X) = e rrx, then t (18.16) Lg = -re- r X + e- r r!> 0 if and only iff/X> r. Hence, iff is time independent, we may state. Theorem 18.2. Iff and are time independent, R (t, X) = e- rrX, A(X) =fiX =
-
a
Stochastic methods in economics and finance
1 98
is decreasing in X with X.
A(O) > r, A(oo) < r, then C(O, oo) = C = (0, X) for some
Proof The set A � { (t, X) I Lg(t, X) > 0} is connected, nonempty and time in· dependent. Therefore by theorem 1 8. 1 , C is connected and nonempty. C is time independent because the problem is time independent. Therefore, the projection of C on the X-axis is a connected open set in the real line. The only such sets are open intervals. This ends the proof. Theorem 1 8.2 drastically simplifies the time independent problem. The value -y(O, X0 , oo) = -y(O, X0) must be the form: -y(O, X0)
= E[e- r rx(r) I X(O) = X0 ] = X E [e- r r I X(O) = Xo]
= XA1(X ; X0),
( 1 8. 1 7)
where T is the time of first passage from X0 to some barrier X, and Af(X; X0) is the Laplace transform of the time of first passage from X0 to X. Much is known about 1H(X; X0). Furthermore, it is clear that if we find X that solves max XA-!(X; X0),
( 1 8 . 1 8)
X
then we have found the optimum barrier and the value function. This makes our problem much easier to solve. For if the tree is of size X and we plan to cut it down when it reaches size Y, then its expected present discounted value is
H(X, Y) = YE[e-
r Ty
I X(O) = X] = Y.(}f (Y; X).
(18. 19)
For the moment put [(X, t) = a(X, t), o(X. t) = vfb (X, t). We want to use f for something else now. Notice that H is the product of Y and the Laplace trans· form of first passage time from X to Y. Let us suppres� Y for a moment and con· sider the value of the tree as a function of the tree's current size alone. We saw above that this valuation function, f(X), satisfies the linear second-order differ ential equation
r{(X) - a (X)['(X) - � b (X)f"(X) = 0.
(18.20)
Put M '( Y ; X) = a.(}t/aX M also satisfies (18.20) in the X argument. Further more, H(X, Y) satisfies ( I 8.20) even if Y is not optimal. Fix Y and suppose
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Applications in economics
X(t) = X is strictly less than Y. Then if At is small, X(t + At) < Y w.p.l. and the tree will not be cut down before t + At. Thus, [(X) = e- rAtEf(X(t + At)).
Use stochastic calculus to expand the r.h.s. of this last equation in a Taylor's series to obtain [(X) = e- rA r E [f(X) + f'(X)AX + � f"(X)AX2 + ... ] = [ 1 - rAt + ... ] [[(X) + f'(X)a(X)At + �f"(X)b(X)At + .. ]. Rearrange and discard all terms of order (At)2 to get {18.20). Since (18.20) is a second-order differential equation, two boundary condi tions arc required to determine a particular solution. One condition is given by observing that if a tree were cut down and sold for Y when its size is Y, then its value at Y must be Y or (18.21) [(Y) = Y. The other boundary condition is not so easy to pin down. We discuss some al ternative specifications, all of which have different economic interpretations. The first possibility is that there is a particular size Q such that if the tree ever becomes as small as Q, it stays at Q. In this case a tree of size Q is worth Q, so our second boundary condition is simply that (18.22a) f(Q) = Q. We call (18.22a) an absorbing ba"ier at Q. Another possibility is that there is no natural lower bound to a tree's size. In this case we require that f(X) con tinues to make mathematical sense when X approaches oo Since .
-
.
[(X) = Y E[e- r Ty I X(O) = X],
and e- rTy is always less than unity, it must be that f(X) remains bounded or that lim I f(X) I < oo (18.22b) A third possibility is to suppose that there is, as with the absorbing barrier, a size Q below which the tree cannot sink, but that, in contrast to the absorbing X --+ -oo
.
Stochastic methods in economics and finance
200
barrier, when it reaches size Q it does not stay at Q but instead bounces back to life - and away from Q. This specification, called a reflecting barrier at Q, is, admittedly, not very plausible for trees, but it is a useful case for other problems. A fourth boundary condition is when f(Q) = 0, but when Q =1= 0 this discontinuity in value causes technical problems. Cox and Miller (1965, pp. 231-232) show that if a diffusion process has a reflecting boundary at Q, then its Laplace transform must satisfy M'(Y; Q) dft1/dX = 0. (18.22c) From (18.22c) andf(X) = YM(Y; Q) we conclude that f'(Q) 0. Notice that if (X) satisfies any of the three boundary conditions (18.22) so docs K (X), where K is a constant. Hence it is natural to search for solutions H(X, Y) to {18.20) of the form H(X, Y) K(Y)<J>(X). We usc the boundary condition H(Y, Y) = Y to determine K. Let (X) be a solution to (18.20) satis fying (18.22), then H(X, Y) = K(Y)(X) satisfies (18.21) only if f/(Y, Y) = Y or if K(Y) = Y 1(Y). Thus, the value of a tree of size X which will be cut down when it reaches Y is H(X, Y) = ( Y/¢(Y)}<J>(X) and clearly the optimum cutting size should be chosen to maximize Y1(Y). Let X* be the optimal cutting size and define (X) (18.23) V(X) f/(X, X*) = X* {. \<J>(X*) =
=
=
=
)·
Notice that V(X) also satisfies (18.20) because X* is independent of X. No tice also that the smooth pasting condition V' (X *) = 1 is satisfied. To see how eq. (1 8.20)and these boundary conditions determine X* and V(X) it is instructive to begin by analyzing the case where a (X) and b (X) are constant and equal to a and b, respectively. Most techniques and results carry over to the more general variable coefficient case. With constant coefficients the tree's growth process is simply Brownian motion with drift a and infinitesimal variance b. Then solutions of (18.20) are of the form (18.24) f(X) = A eAX + Bell X where -a + (a2 + 2 rb)Y2 (18.25) >O X= b and
201
Applications in economics
Jl =
-a- (a2
+ 2 rb)lh. b
{ 1 8.26)
= -A
are the roots of the characteristic equation of ( 1 8.20). If B e
f(Q) =
Q=
and the optimal cutting size
X* maximizes
X
(eA.X - e
X*
V(X).
It is not easy to analyze the effects of parameter changes on and For general the absorbing case is complicated. In the remaining we discuss only 0. the case I f f( · ) is bounded at -oo, things are much simpler. Note that f( · ) can remain finite only if B = 0. Thus, and maximizes xe - A.X or I X -I eA. . Effects of changes in the process work entirely through Since I -• ) 0. It is straightforward to calculate - , that dA/db 0. Thus, increasing the instantaneous variance of the process gov erning the tree's growth increases both the tree's value and the time at which the tree is cut down, since dA/da 0 and dA/dr 0. The effects of changes in the growth rate and the interest rate are similarly straightforward. 0 A reflecting barrier is not so simple. For a solution to {18.24) to satisfy it must be that B = -A(A!Jl)e(A.- J.t)Q, so that must be chosen to maximize X/(e'AX - CNJJ.)e('A- J.t)QeJ.tx). It is difficult to analyze the effects of changes in parameters on cutting size and value. We show below in proposition 1 8.4, summarizing our results, that the reflect ing barrier case is qualitatively the same as the case when f( ·) is bounded at The absorbing barrier case is considerably more subtle. To analyze more completely the constant coefficient case for reflecting and absorbing barriers, it is helpful to examine the function
Q, Q=
X* = A-t ,
X* A X < X* = A d V(X)/dA = (X- A V(X) < < < >
X*
V(X) = A.
f' (Q) =
-oo.
(18.27) g(X) = f'(X) ct>'(X) . [(X) ¢ (X) One should note that g is the rate of change of the logarithm of the Laplace transform of first passage to a fixed barrier Y with respect to X. It is independent of Y. Call g the logarithmic rate of change of the Laplace transform with respect to X. The value of a tree of size X which will be cut down at Y > X is =
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Stochastic methods in economics andfinance
H(X, Y) = Y
(X) <J>( Y)
--
= Y exp
(
y
-
J
X
)
g(u)d u .
It follows from (1 8.20) that g satisfies a first-order differential equation
g'(X) rfb - (a/b)g(X) - g 2 (X).
(18.28)
=
Since g is a first-order differential equation, only one boundary condition is re quired to identify a particular solution. Thus, in principle, to calculate H(X, Y) in a particular case, use a boundary condition of the form (18.22) to identify a particular solution of (1 8.28) and integrate along that solution from X to Y to get - log(H(X, Y)/Y). This observation suggests a way of getting comparative statics results. Sup pose g and g are two solutions to (1 8.28) corresponding to different parameters. For simplicity we call the tree which grows according to the process which de termines g( · ) the g tree and that which grows according to the process which determines g( · ) the g tree. Suppose also that
g(X) < g(X) Then exp
( J g(u)du ) > ( y
-
X
(1 8.29)
for all X.
exp
y
-
J
X
K lu )d u
)
and
H(X, Y) > h(X, Y).
This means that for any cutting size Y the g tree is worth more than the g tree. If X* is the optimal cutting size for the g tree and X* is the optimal cutting size for the g tree, then in an obvious notation
V(X) = H(X, X*) �H(X, X*) > fl(X, X*) = V(X), so that the g tree is worth more than the g tree. It also follows that X* > X*. The optimal cutting size for the g tree, X*, must maximize X/<J>(X). First-order conditions for maximization imply X * must satisfy (X*) = X*¢' (X*) or that g(X*) =
1/X*.
(18.30)
Second-order conditions are that g(X) intersect 1/X from below. Thus, if g(X) < g(X), it must be that if g(X*) = X*-1 , g(X*) < X*-1 ; at X* the value of the g tree is still increasing. Thus, X* > X*.
Applications in economics
203
To use this method of analysis it is necessary to translate the boundary con ditions on f given by (18.22a, b, c) into boundary conditions on g. Since solu tions to (18.20) are of the form (18.24), in general .ellX • g(X) = A Xe'J...'J...X + Bp.p.X
Ae X
+
Be
The solution 'YQ (X) corresponding to a reflecting barrier at Q for [, satisfies 'YQ(Q) = 0 and is given by 'YQ (X) =
1
exp[(X - p..)(X - Q)] -
I
(18.3 1)
1 - exp[(X - p..)(X - Q)] - X p..
The solution cxQ (X) corresponding to an absorbing barrier at Q = 0 for /is given by ex
Q (X) = _
X exp[(X - p..) X] - p.. exp[(X - p..) X] -
The solution to ( 18.20) which is bounded at the corresponding g = f' /f is given by f3(X) = X.
(18.32)
1
-oo
is of the form f(X) = A e'J... X ;
( 18.33)
We now have enough information to draw a phase diagram for g. Eq. (1 8.28) can be rewritten as
g' = (X - g)(g - p),
(1 8.34)
where X and p.. are the roots of the characteristic equation off( · ) and are given by {18.25) and (18.26). For g > X, g'(X) < 0; for X > g > p, g'(X) > 0, and for g < p, g' (X) < 0, so the phase diagram looks as in fig. 18.1. In this figure we have drawn cxQ (X), 'YQ (X) and f3(X) solutions corresponding to an absorbing barrier at Q, a reflecting barrier at Q, and bounded behavior at - oo, respectively. Since solutions to (18.28) cannot cross, a (X) > (3(X) > 'YQ (X). It is easy to Q calculate from (18.3 1 ) and (18.32) that
{18.35)
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Stochastic methods in economics and finance
and lim o:Q(X) = lim (3Q(X) Q -.. Q -.. - oo - oo
=
(18.36)
(3(X) = X.
Also, if Q > Q, it is easy to see that O!Q (X) > o:0 (X) > (3Q (X) > (3Q (X). We may sum up these observations in two propositions. Proposition 18.1. As Q � -oo, trees with reflecting barriers at Q become indis tinguishable from trees with absorbing barriers at Q or trees which are simply bounded at oo. Proposition 18.2. Trees with reflecting barriers are worth more than trees with absorbing barriers. The value and the optimal cutting size of a tree with a reflect ing barrier is an increasing function of the barrier. For trees with absorbing bar riers, the opposite is true. g(X)
\ (t(X) A
/
Figure
1 8. 1 .
Applications in economics
205
The phase diagram makes it easy to do comparative statics, to analyze the ef fects of changes in parameters on value and cutting size. The parameters of our problem are a, b and r. Differentiating (1 8. 28) we see that dg'(X)/dr = 1/b > O and dg'(X)/da = - (1/b)g(X) which is negative since we are only interested in positive g(X). This proves the rather obvious Proposition 1 8.3. Increasing the infinitesimal mean growth rate of a tree or de creasing the interest rate increases the value and the optimal cutting time of a tree. The effects of increasing2 the variance are only slightly more complicated. Note that dg'(X)/d b = - (I /b )[r - ag(X )] which is negative whenever g(X) < r/a. It is straightforward to calculate that A < rfa. If g( ) corresponds to a reflecting barrier at g(X) � A for all X, it follows that increasing the variance increases the cutting size and value of the tree. Proposition 1 8.4. If a tree's growth process is a diffusion with constant coeffi cients which is bounded at infinity or which has reflecting barriers, then increas ing the variance increases the value and cutting size of the tree. The effects of increasing the variance on the absorbing case arc more complex and the reader is referred to Brock, Rothschild and Stiglitz (1 979). We conclude this analysis of the stationary case by observing that under uncertainty the opti mal cutting size is greater than it would be under certainty. ·
-oo,
19. Miscellaneous applications and exercises
(1 )
Consider the stochastic differential equation of economic growth derived in section 2, dk = [sf(k) - (n-a2 )k ]dt - akdz,
(19.1)
with initial random condition k(O) = k0 > 0. Find a set of sufficient conditions for the existence of a unique solution k(t), t [0, oo) and use theorem 6.1 of Chapter 2 to establish the existence and uniqueness of k(t). (2) Equation (19 . 1 ) may be generalized by assuming that s and a are no longer constants but instead are functions of k, written as s(k) and a(k). With the new functions s(k) and a(k) the generalized stochastic differential equation of growth becomes E
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Stochastic methods in economics and finance
dk
=
[s(k)f(k) - (n-- o2 (k))k] d t - o(k)kdz,
(19.2)
with initial condition k (O) = k0 > 0. Find a set of sufficient conditions for the existence of a unique solution k (t), t E [0, oo) and use theorem 6.1 of Chapter 2 to establish the existence and uniqueness of k (t). (3) Suppose that eq. (19 .2) has a unique solution k (t). Discover sufficient conditions such that k (t) is bounded w.p.l , i.e. such that
P[w: k(t, w) < oo ] = I . (4) Assume that the coefficients of ( 1 9 .2) vanish for the equilibrium solu tion k * , where k * is a nonrandom, nonzero constant; that is, assume that
[s(k*)f(k*) - (n-o2 (k*))k *] = o(k*)k*
=
0.
Also, assume that there exists an T] > 0 such that o(k)k > 0 for 0 < I k-k * I < TJ . Is the equilibrium solution k* stable? (5) Money, growth and uncertainty: Tobin ( 1 965) was one of the first econ omists to introduce money in an economic growth model. The Tobin model has been studied by several authors. In this application we will first present briefly the deterministic Tobin model following Hadjimichalakis (1971) and then pro ceed to propose an extension by introducing uncertainty. We begin the model by describing its equations: homogeneous production functions of degree I
Y = F(K, L),
( 1 9.3)
p
( 1 9.4)
("1)
perfect foresight
-p = q ,
investment function
l = K +- -
saving function
s;s
labor growth
L (t) = L (O) en , L (O) > 0,
(19.7)
money supply growth
lv/(t) = Jltt(O) e8 1 ,
(1 9.8)
.
[
d dt
y+
p
'
:t (�) l t
/v/(0) > 0.
(19.5) (1 9.6)
Note that PJP denotes actual inflation and q denotes expected inflation. It is assumed that the two are equal under the assumption of perfect foresight .
Applications in economics
207
Under equilibrium, saving must equal investment. Thus,
which yields
K=s
[ Y + dt (A1P�)� - d( (Afp) · d
d
This last equation is called Tobin 's fundamental equation. On the basis of the above, the differential equation of money and growth, where we define m = M/P 1/L, i.e. m is per capita real money balances, is given by
k
=
s[(k) - (I - s) (8 - q)m - nk.
( 19.9)
Note that if 8 = 0 = q, then we obtain as a special case the Solow equation (2.1 ). Suppose that uncertainty is now introduced in Tobin's monetary growth model by postulating randomness in the growth of the money supply and de scribed by the stochastic differential equation
dA1 = 8JV!
dt + J.Livldz.
(19. 10)
Usc eq. ( 1 9 . I 0) and the appropriately modified model of cqs. ( I 9 .3)-( 1 9 .7) to study the effects of uncertainty in the money supply growth. (6) Random demand functions: Let a consumer solve max U(X,
Y)
(19.1 1 )
subject to
(19.12) Let
solve (19 .I I ) and processes:
(19 .1 2).
Now, let iH, p 1 and p 2 be random and follow the Ito
{19. 13)
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Stochastic methods in economics and finance
dp1 = JJ.p d t + aP. dzP , . 1 dp 2 = JJ.p d t + aP dzP . 2
.
2
2
(19.14) {19.15)
Assume that each instantaneous mean and variance of (19.13)-(19.15) is a function of (l'l1, p1 , p2 ). Assume that { zM }, {zp , } , {zP, } are Wiener processes that satisfy the formal rules dzM · dzP = pd t; dzM · dzP = pd t; dzp · dzP = pd t, 2 1 , , dzP ·dzP = dt; d zP dzP = dt; d zM ·dzM = dt. , . . Use Ito's lemma to write down the stochastic differential equations of the demands X = g(M, p1 , p2 ), and Y = h (M,p 1 , p2) for the case 2
U(X, Y) = AXa yP,
•
0 < a < 1 , 0 < {3 < 1 , 0 < a + {3 � l .
This is a random world when the individual cannot store commodities, i.e. it is a stochastic world without stock variables just as standard static demand theory is a deterministic world with no stock variables. (7) Edgeworth's random boxes: This is an Edgeworth Box with random en dowment. Let consumer i = 1 , 2 solve max X� y{J I
I
where e 1 = (a, 0), e2 = (0, b ),p · e; is a scalar product. Note that a and {3 do not vary across consumers. Use problem (6) to solve for the demand functions Determine the relative price p 1 /p 2 by supply equals demand: h 1 + h 2 = a, g1 + g2 = b.
Now let da = P.ad t + aad za, d b = p.bdt + abdzb ,
Applications in economics
209
determine the endowments, a and b, of the two consumers. Use Ito's lemma to find the stochastic process that equilibrium relative prices p 1 /p2 must follow. (8) Marshall with a2 : Consider the following algebraic version of the Marshal Han Cross: p
= D(q) = Aq - a = S(q) = BqiJ, A > 0, B > 0, a > 0, 13 > 0.
Find the stochastic processes generating equilibrium p (t) and q(t) when
= JlA d t + aA dzA , dB = Jln d t + a8 dz8 ,
dA
where we assume dzA dz8 = 0 for simplicity. Does the old proposition about random supply and demand leading to high price variation and low quantity variation when supply and demand are inelastic hold up in this new framework? (9) Monopoly vs. competition in exploring for oil: A monopolist solves max
E0
00
J
0
e-Pt (R(q) - I)dt
{19.16)
subject to
dE(t) = -q(t)dt + b(E(t))da(t),
(19.17)
where
da(t) = 1 da(t) = 0
with probability A.(/(t))dt, with probability
1 - A.{/(t))dt.
Here R (q) = D(q)q = total revenue, and /(t) = investment in enlarging the stock of oil, E(t). Investment takes the form of allocating money /(t) to "dig" where with probability A.(I(t))dt of "find" of size b (E(t)) will be discovered. Write down the Hamilton-Jacobi-Bellman equation for {19.16) and (19.17). Let competition solve the same problem with R (q) replaced by CS(q) where
CS(q) =
q
f D(y)dy,
qo
and q0 is a small positive lower limit. Note that provided that
q0
could be taken to be zero
Stochastic methods in economics and finance
210 q
J
D(y )dy for q > 0, is finite.
Write down the Hamilton-Jacobi-Bellman equation for competition. Point out the difference from monopoly. Also, attempt to say if monopoly or competition will invest more in finding new wells. The above model may be looked upon as a proxy for the amount of innova tion in the sense of divesting profits into projects to enlarge the stock of salable resources. One would expect that monopoly would invest less than competition. Any hints from this model? Finally , try to find a form of D(q), b (E(t)), 'A(l(t)) that permits a closed form solution for the Hamilton-Jacobi-Bellman equation. ( 1 0) Exhaustible resource problem : Consider the problem max E0
00
J e- rt R (q)dt
0
subject to dE =
-
qdt + a0Edz,
for the special case where
As in the previous problem R (q) is total revenue for the firm, q is quantity of output and E(t) is the stock of the resource at time t with E(O) = E0 > 0. Solve this problem for a closed form solution for the cases: (i) a0 = 0 and (ii) a0 > 0, and compare your solutions. (1 1) Write the quantity theory of money equation MV = PO,
where M is money supply, V is velocity of money, P is price level, and 0 is real output. Let
and
Applications in economics
211
where, E1dz � = dt, E1dz � = d t , E1dz 1 dz 2 = pd t. Let V be constant over time and nonrandom. Write out the formula for dPfP in percentage terms. ( 1 2) Consider the quadratic problem presented in section 1 2:
- JV(x (t)) = mJn E,
00
J
e-P(s- t ) { a(x (s))2 + b (v(s))2 } ds.
s=t
Solve this problem for each of the following laws of motion:
(i)
dx(s) = v(s)dt + a0 x(s)dz(s).
where a0 is a constant independent of time and of x; (ii)
dx(s) = [v(s) + Q0] dt + a0x(s)dz(s),
where Q0 is a nonrandom constant; and (iii) d x(s) = [ax(s) + v (s)] dt + a0x(s)dz (s), where a is a nonrandom constant. Furthermore, in each case check that your solution satisfies the transversality condition, lim E 0 e-P'q (t) ·x(t) = 0. l-+00
{13) Exchange rate in a two-country stochastic monetary model: We follow Lau (1977) to formulate a simple international monetary model. Let there be
two countries: country I , say England. and country 2, say France. Let there be two goods: good 1 cloth, and good 2, wheat . This is not a production model, so we assume that both goods are perishable and that at the beginning of each peri od England is endowed with a constant amount of cloth, y 1 , and France is en dowed with a constant amount of wheat, y 2 • We use q 1 to denote the price of good I , cloth, and q 2 to denote the price of good 2, wheat. We choose q 1 = I , i.e. cloth is the numeraire. Let c;i be the real consumption by country i of good j and Jet M;i be country i's holdings of money j. Denote by S; the discount rate for country i. Let P1 be pounds per unit of cloth and P2 francs per unit of cloth and let E be the exchange rate. By purchasing power parity, P1 = EP2 or E = PtfP2 , i.e. E is pounds per franc. We assume that the money stocks, il1f follow the stochastic differential equations. ,
Stochastic met/rods in economics and finance
212
and
where J.L1 , J.L2 , a 1 and a2 are constants, and z 1 and z 2 are normalized Wiener pro cesses such that
With the above information we now formulate the problem: England solves max E0
00
Je
0
-
t> ,
t
[u1 (c 11 , c 12 ) + vi (Mu/PI , M12 /P2 )] dt
subject to
,
1 } ;: 1 , { cf2 } ;: {Mf1 } ;:
for { cf
max E0
00
1
1
2 } ;: , while France solves
and {Mf
1
J e - 6 2 t [u2 (c21 , c22 ) + v2 (M2. /P1 , M22/P2 )l d t
0
subject to
for {c�1 } �= 1 , {c� };: 1 , {M�1 };: 1 and {M� };: 1 • Then {P 1 } �= 1 and {P };: 1 is 2 2 2 a monetary perfect foresight equilibrium if for every t E [0, oo) all of the follow ing hold:
Observe that we postulate separable utility functions u; and V; and also note that the superscript d denotes quantities demanded. Make any additional assumptions that are needed to study in this model the behavior of the exchange rate E . (14) Stochastic search theory: Consider a representative individual who searches for a higher wage. This search takes the fonn of allocating time to in fluence the probability of arrival of a Poisson event. The more time that he allo-
Applicatio11s in economics
213
cates to search activity the more frequently this Poisson event will arrive. If the Poisson event arrives the searcher's wage increases by a jump. The searcher faces a stochastic differential equation that determines the evaluation of his wage. This equation consists of an exogenous rise in the nominal wage plus a Brownian motion term plus a Poisson term. Only the Poisson term is influenced by the searcher's search activity. Let us get into the details. Consider the following model: max E0
00
J e-Pt y(t)dt =J(W(O), 0) 0
subject to
d W(t) = rW(t)dt + aW(t)dz(t) + g(W(t))dq(t), W0 given, where W(t), y(t), {z(t)};: 0 and {q(t)}�= o are nominal wage , flow income, discount rate on future income, standardized Wiener process, and Poisson pro cess, respectively. The probability that the Poisson event occurs, i.e. q(t + �t) q(t) = 1 , is given by X(21 )�t + o(�t) and y(t) = W(t) (1-21 (t)), where 21 (t) is the percentage of time devoted to wage augmentation activity, i.e. search activity. Furthermore, E, denotes expectation conditioned at t. The numbers r and a do not depend on t or W. Here g( W(t)) is the amount of the wage jump if the Pois p,
son event occurs. The idea is that the amount of a better job offer should depend upon current wage. Form
c/>(21, W, t) = e- Pt (l-2.)W +11 +lwrW + � lwwa2 W2 + X[J(W +g(W), t) -J(W, t)]. Assume an interior solution for 21 : a c�> dX - = 0; 0 = -e-P' W + - [J(W + g(W), t)-J(W, t)). d21 a 21
(19.18)
(19.19)
Notice that eqs. (19 . 1 8) and (19 .19) are the standard partial differential equa tions of stochastic control. Make any assumptions that seem economically reasonable and proceed to study the effects of search on nominal wages.
214
Stochastic methods in economics and fir�ar�ce
20. Further remarks and references
We would like to note that there is some arbitrariness on the part of the author in the selection of applications and in distinguishing them either as economics or finance applications. This chapter and the next are not intended to be exhaustive nor is it possible to have an empty intersection. An important book of readings and exercises which supplements this chapter is Diamond and Rothschild ( 1 978). The use of continuous time stochastic calculus in macroeconomic growth under uncertainty first appeared in the papers of Bourguignon (1 974), Merton {1 975a) and Bismut ( 1 975). These papers build on several earlier papers on economic growth under uncertainty such as Brock and Mirman ( 1972), Levhari and Srinivasin { 1 969), Mirman ( 1 973), Stigum (1 972), Leland ( 1 974) and Mir man and Zilcha (1975). among others. The main unresolved issue in continuous time economic growth under uncertainty is the stochastic stability of the sta tionary distribution. Stochastic point equilibrium is discussed in Malliaris ( 1 978). For discrete time growth models under uncertainty several results are presented in Brock and Majumdar ( 1 979) and section 1 5 of this chapter. Two recent con tributions on optimal saving under uncertainty are Foldes ( 1 978a, 1978b ). Our analysis of growth in an open economy under uncertainty is very limited. However, there has been great interest recently in introducing uncertainty in in ternational economics with several papers, such as Batra ( 1 975), Mayer ( 1 976), Baron and Forsythe ( 1 979), and the recent book by Helpman and Razin (1 979). Sec also the international monetary model under uncertainty n i exercise 1 3 of section 1 9 due to Lau (I 977). In section 8 we presented the concept of rational expectations to illustrate the use of stochastic methods and more importantly to familiarize the reader with the defmitional aspects of this concept. Rational expectations as a concept has found numerous applications and for a survey article we suggest Shiller (1 978) and Kantor ( 1 979). We use rational expectations in sections 9 and 1 1 . The cost of adjustment type of investment theory developed by Lucas (1967b, 1967), Gould (1 968) and Treadway ( 1 969, 1 970), among others, was extended under uncertainty in the paper of Lucas and Prescott {1971 ). Elements of the Lucas and Prescott ( 197 1 ) paper are presented in section 9 to illustrate the use of stochastic methods in microeconomic theory. Another microeconomic application is found in section 1 6. Some basic references in the analysis of the firm under uncertainty, other than Sandmo ( 1 97 1 ), are Mills ( 1959), Nelson ( 1 96 1 ), Pratt (1 964), McCall (1 967), Stigum (1 969a, l 969b), Baron ( 1 970, 1 97 1 ), Zabel ( 1 970), Leland ( 1 972). Ishii (1 977), Wu ( 1979) and Perrakis ( 1980), just to mention a few . Note that there is a considerable bibliography on problems of general equilibrium under uncertainty. We have not presented any
Applications in economics
215
applications in this area primarily because the techniques are not similar to the ones presented in this book. Techniques of general equilibrium under uncertainty usually involve arguments of a topological nature and/or arguments of functional analysis. A representative paper in general equilibrium under uncertainty is Bewley (1 978). However, we remark that stochastic stability techniques have been used in general equilibrium in some specific fonnulations such as Turnovsky and Weintraub (1971). In sections 1 0 and 1 1 we attempt to develop a general approach to continuous time stochastic processes that arise in dynamic economics from maximizing be havior of agents, as in Brock and Magill (1979). The analysis considers a class of stochastic discounted infinite horizon maximum problems that arise in economics and uses Bismut's ( 1 973) approach in solving these problems. It is shown that the idea of a competitive path, introduced in the continuous time deterministic case in Magill (l 977b ), generalizes in a natural way in the case of uncertainty to a competitive process. Theorem 1 0.1 shows that under a concavity assumption on the basic integrand of the problem a competitive process which satisfies a transversality condition is optimal under a discounted catching-up criterion. Next, we consider the sample path properties of a competitive process. If for almost every realization of a competitive process the associated dual price pro cess generates a path of subgradients for the value function, we call the process McKenzie competitive, since it was McKenzie ( 1976) who first recognized the importance of this property in the detenninistic case. Theorem 1 0.2 shows that two McKenzie competitive processes starting from distinct nonrandom initial conditions converge almost surely if the processes are bounded almost surely and if a certain curvature condition is satisfied by the Hamiltonian of the system. The problem of finding sufficient conditions for the existence of a McKenzie competitive process remains an open problem .. Business cycles and macroeconomic stabilization methods in an environment where uncertainty prevails are areas of research in which stochastic calculus tech niques are quite appropriate . However, the research has just begun; we note that Lucas' ( 1 975) paper represents a methodological advance in business cycle theory. In the Lucas (I 975) paper, discrete time stochastic techniques arc used to show that random monetary shocks and an accelerator effect interact to generate serially correlated cyclical movements in real output and procyclical movements in prices, in the ratio of investment to output, and in nominal interest rates. Also, note the Slutsky ( 1 937) paper in which it is shown that a weighted sum of independent and identically distributed random variables with mean zero and finite variance leads to approximately regular cyclical motion. Slutzky's (1 937) ideas have not yet been fully utilized. Magill (I 977a) has a brief analysis in which he shows that the introduction of uncertainty imbeds the short-run study
216
Stochastic methods in economics and finance
of the business cycle into the long-run process of optimal capital accumulation. Tinbergen's (1952) classic work on static stabilization has been extended to al low for uncertainty by Brainard (1 967) and in section 1 7 we give a simple illus tration. See also Poole (1 970), Chow ( 1970, 1 973) and Tumovsky (I 973). In section 14 we illustrate the use of stochastic calculus techniques in a macroeconomic model with rational expectations, following Gertler ( 1 979). The Gertler { 1 979) paper explores the consequences for price dynamics of imperfect price flexibility and it demonstrates that the same condition which ensures sta bility in the deterministic system also ensures that the distribution of the state variables converges to a stable path in the stochastic case. In section 1 4 we illus trated the stochastic case; for a comparison between the deterministic and the stochastic case see Gertler (1 979). Section 1 8 follows Brock, Rothschild and Stiglitz (1 979) to illustrate various methods of continuous optimal stopping to stochastic capital theory. This sec tion treats the time independent case. Similar results continue to hold in the gen eral case when the instantaneous mean and the instantaneous variance of the dif fusion process are functions of the tree's current size. For details see the Brock, Rothschild and Stiglitz ( 1 979) paper. See also the paper by Miller and Voltaire ( 1 980) which treats the repeated or sequential stochastic tree problem, i.e. the problem of deciding when to harvest and replant trees given the knowledge of the process of each tree's growth history. Miller and Voltaire (I 980) show that the results for the nonrepeated and for the repeated cases are qualitatively similar. The reader who is interested in further applications of optimal stopping methods in economics should consult Samuelson and McKean ( 1965), Boyce ( 1 970) and Jovanovic (1 979a).
CHAPTER 4
APPLICATIONS IN FINANCE
There is no need to enlarge upon the im portance of a realistic theory explaining how individuals choose among alternate courses of action when the consequences of their actions are incompletely known to them. Arrow ( 1 97 1 , p. 1 ).
1 . Introduction In this chapter we present several applications of stochastic methods in finance to illustrate the techniques discussed in Chapters 1 and 2 . We also include some applications which use additional techniques to familiarize the reader with a suf ficient sample of stochastic methods applied in modem finance.
2. Stochastic rate of inflation In this application we illustrate the use of Ito's lemma in determining the solu tion of the behavior of prices and real return of an asset when inflation is described by an Ito process. The analysis follows Fischer ( 1 975). Suppose that the rate of inflation is stochastic and the price level is describable by the process dP
p
= fl dt + sdz.
(2. 1 )
Stochastic: methods in economics and finance
218
The stochastic part is d z with z being a Wiener process. The drift of the process, n, is the expected rate of inflation per unit of time. It is defined by . n _ hm -
ll - 0
E,
}_ Il
{
P(t + Jz) - P(t) p (t)
}
(2.2)
,
where E, is the expectation operator conditioned on the value of P(t). The vari ance of the process per unit of time is defined by 1 s2 = hm E1 h-o h •
-
{t
P(t +
lz) - P(t)
P(t)
]}
2
-
n/1
(2.3)
A discrete time difference equation which satisfies II and s2 as defined in (2.2) and (2.3) is
P(t + h) - P(t) = 11/z + sy(t) (h) tf2 ' P(t)
(2.4)
where y(t) is a normal random variable with zero mean and unit variance which is not temporally correlated. The limit as h 0 of (2 .4) then describes a Wiener process for the variable sy(t) (h) 11 2 and eq. (2 . 1 ) can be written as --+
dP p
= fl d t + sy(t) (h) 11 2 = n d t + sdz,
where dz = y(t) (h) 1 12 . Note that (2 . 1 ) says that over a short time interval the proportionate change in the price level is normal with mean n d t and variance s2 d t. Rewrite (2.1) as dP = Plldt + Psdz and let
y (t) = P(O) exp
(2.5)
[(
n
-
;) /
s
t+s
]
(2.6)
dz
We use Ito's lemma to show that y (t) satisfies eq. (2.5). Let
F(t , z ) = P(O) exp
[ ( s; ) / n-
t +S
]
dz
Applications in finance
219
and compute oF/ot, o F/oz and o 2 F/oz2 as below:
{ ( n ; } I ] (n - ; } [ (n - ;} I ] { ( ;} I ]
oF or
P(O) ex
oF oz
P(O) exp
::{
=
P(O) ex
-
n-
•
•
t +s
dz
•
t+s
dz
s = sy(t),
•
t+s
dz
2 s = s2y (t).
= y ( t)
( n ;} , -
•
Thus, applying Ito's formula and by making the necessary substitutions we end up with dy
oF
= at d t + = y (t)
oF I o2 F (dz)2 dz + 2 oz2 oz
( ;} II -
•
d t + sy(t) dz + 4 s2 y(t) (dz)'
s2 = y(t) I ld t - y(t) 2 d t + sy(t) d z + i s2 y (t) dt
= y(t) n d t + y (t)sdz. Thus, (2.6) satisfies eq. (2.5), which is what we wanted to show. We continue with a further application. Consider the two Ito processes
dP
p = lid! + sdz
and
dQ
Q=
rd
t.
(2.7)
We usc Ito's lemma to compute the stochastic process describing the variable q = u (P, Q) = QJP. Recall that Ito's formula for this case is:
oq or
oq oP
oq oQ
dq = - d t + - d P + - d Q
Stochastic methods in economics and finance
220
Computing the various terms and using the multiplication rules in (4.12) in Chap ter 2, the result is: dq
= =
Finally, dq q
-
=
� dP � d Q +
- pl
Q
+
� (!�) (Ps)2 dt
(OPdt + Psdz) +
p
1
(rQdt) +
p
Q
s2 d t.
(r - n + s2 ) d t - sdz,
which describes the proportional rate having a nominal return as in (2 .7).
of change of the real return of an asset
3. The Black-Scholes option pricing model In this section we follow Black and Scholes ( 1 973) and Merton (1 973a) to de velop an option pricing model. Consider an asset, a stock option for example, denoted by A , the price of which at time t can be written as
W(t) = F(S, t),
(3.1)
where F is a twice continuously differentiable function. Here S(t) is the price of some other asset, denoted by B, for example the stock upon which the option is written. The price of B is assumed to follow the stochastic differential equation dS(t) = f(S(t), t) d t + 17 (S(t), t) dz (t),
(3.2)
S(O) = S0 given. Consider an investor who builds up a portfolio of three assets, A, B and a riskless asset denoted by C. We assume that C earns the competitive rate of re turn r(t). The nominal value of the portfolio is
P(t) = N1 (t)S(t) + N2 (t) W(t) + Q(t),
(3.3)
where N1 denotes the number of shares of B, N2 the number of shares of A, and Q is the number of dollars invested in the riskless asset C. Assume that B pays no dividends or other distributions. By Ito's lemma we compute
Applications in finance
221
(3.4) =
where
a b
=
=
adt + bdz, F1 + Fsf + � Fss TJ2 = aw W,
(3.5)
Fs TJ = aw W.
(3.6)
Here we follow Black and Scholes (1973) and assume as a simplifying special case that f(S, t) = aS and that TJ(S, t) = aS, where a and a are constants. Next we write the dynamics for S(t) in this special case of (3.2) in percentage terms as
dS
= ad t + adz.
S
(3 7) .
Now for a portfolio strategy where N1 and N2 are adjusted slowly relative to the change in S, W and t we may assume that dN1 = dN2 = 0 and proceed to study the change in the nominal value of the portfolio, dP, as follows:
N1 (dS) + N2 (dW) + dQ = (ad t + adz)N1 S + (aw d t + ow dz)N2 W + rQdt.
dP
=
(3.8)
(3.9) Design the proportions wl and w2 so that the position is riskless for all t � 0: var,
(�) =
var, (W1 adz +
W2 owdz) = 0,
where var1 denotes variance conditioned on choose (W1 , W2 ) = ( W1 , W2 ) so that
(3.10)
(S(t), W(t), Q(t)). In other words, (3.1 1 )
Then from (3.9)
( ) dP
-
E, p = [aW1 + aw W2 + r( 1 - W1 - W2 )] d t = r(t) dt -
-
-
(3.12)
Stochastic methods in economics and finance
222
since the portfolio is riskless. Eqs. (3.1 I ) and (3 . 1 2) yield the famous Black Scholes-Merton equations:
W1
=-
°w
(3. 1 3)
and (3. ) 4) which simplify to -
-
(3. 1 5 )
--
Eq. (3. 1 5) says that the net rate of return per unit of risk must be the same for the two assets. For a further special case
a(S, t) = a0 ; a(S, t) = a0 ; r(t) = r0 ,
(3.16)
where a0 , a0 and r0 are constants that are independent of (S, t), by using eq. (3.15) and making the necessary substitutions from (3.5) and (3.6) we obtain the partial differential equation
� a� S2 F88 (S, t) + r0SF8 (S, t) - r0F(S, t) + F, (S, t) = 0.
(3. 1 7)
Its boundary condition is determined by the specifications of the asset. For the case of an option which can be exercised only at the expiration date T with exercise price £, the boundary condition is
F(0, r) = 0,
r=
T-t,
F(S, T) = max [O , S- E].
(3.18)
Let W(S, r; E, r0, a�) denote the solution F, subject to the boundary con dition. This solution is given by Black -Scholes (I 97 3) and Merton ( 1973a) as (3.19) where
Applications in {i11ance
1
223
(3.20)
(v) = 2 7T)l/2 (
-
i.e. the cumulative normal distribution, with
1
(3. 2 1 )
4. Consumption and portfolio rules Another useful application o f stochastic calculus techniques is found i n Merton (1971 ). Assume that in an economy all assets are of a limited liability type, that there exist continuously trading perfect markets with no transaction costs for all assets, and that prices per share P;(t) arc generated by Ito processes, i.e.
(4. 1 ) where ex; is the instantaneous conditional ex pected percentage change in price per unit of time and oi is the instantaneous conditional variance per unit of time. In the particular case where the geometric Brownian motion hypothesis is as sumed to hold for asset prices, ex.; and O; will be constants and prices will be sta tionarily and log-normally distributed. To derive the correct budget equation it is necessary to examine the discrete time formulation of the model and then to take limits to obtain the continuous time form. Consider a period model with period length h , where all income is generated by capital gains. Suppose that wealth, W(t), and P;(t) are known at the beginning of period t. The notation used is,
N;(t) = number of shares of asset i purchased during period t, i.e. between t and t + h, where h > 0� and C(t) = amount of consumption per unit of time during t. The model assumes that the individual comes into period t with wealth invested in assets so that
W(t)
=
n
L N;(t - h)P;(t). I
(4.2)
Stochastic methods in economics and finance
224
Note that we write N1 (t -lz) because this is the number of shares purchased for the portfolio during the period t-h to t, evaluated at current prices P1(t). The decisions about the amount of consumption for period t, C(t), and the new port folio, N1(t), are simultaneously made at known current prices: n
- C(t) h =
(4.3)
[N;(t) -N;(t- h)]P;(t).
� 1
(4.2) and (4.3) by h to eliminate backward differences and thus
Increment eqs. obtain
n
- C(t + h)h =
�
l n
= � 1
+ and
[N;(t + h) - N1 (t)] P1(t + h) [N1(t + h) - N1(t)] [P1(t + h) -P1 (t)] n
� I
[N;(t + h) -N;(t)]P1(t)
(4.4)
"
W(t + lz) = :E N;(t)P;(t + h).
(4.5)
I
Take limits in (4.4) and n
(4.5) as h � 0 to conclude that n
- C(t) d t = � dN;(t) dP1(t) + � dN1 (t)P;(t) 1 and, similarly,
W(t) =
J
n
� I
N;(t)P;(t).
Use Ito's lemma to differentiate
d W(t) =
(4.6)
"
� 1
N1(t) dP;(t) +
(4.7) W(t) to obtain n
� 1
dN1(t)P1(t) +
"
� 1
dN;(t) dP1(t).
(4.8)
The last two terms, �� dN1P1 + �;' dN1 dP1, in (4.8) are the net value of addi tions to wealth from sources other than capital gains. If dy(t) denotes the in stantaneous flow of noncapital gains., i.e. wage income, then
dy - C(t) d t =
"
� I
dN1P1 +
which yields the budget equation
"
� 1
dN1 dP1,
225
Applications in finance n
dW= � J
N;(t)dP; + dy - C(t) dt.
(4.9)
Define the new variable w;(t) = N;(t)P;(t)/W(t) and usc (4. 1 ) to obtain, dW
=
n
� 1
W; Wa;dt - Cdt + dy +
n
� I
W; Wa; dz;.
(4.10)
Merton (1971) assumes that dy = 0, i.e. all income is derived from capital gains and also that an = 0, i.e. the nth asset is risk free. Therefore , letting an r, =
d W=
7
n-J
W;(a;- r) Wdt + (rW- C) d t +
t
n- 1
W;a;Wdz;
becomes the budget constraint. The problem of choosing optimal portfolio and consumption rules for the individual who lives T years can now be formulated. It is the following max subject to
Eo [/ u(C(t), t) dt + B(W(T), T)J
W(t) � 0 dW=
n- 1
7
for all
t
E
(4.1 1 )
(4.12}
[0, T] w.p.l,
W;(a;- r) Wdt + (r W-C) dt +
Here u and B are strictly concave in C and
n-l
7
W; a;Wdz;.
(4. 1 3)
W.
To derive the optimal rules we use the technique of stochastic dynamic pro gramming presented in Chapter 2. Define (4.14) and also define
<J> (w, C, W, P, t) = u(C, t) + Y [J), where !f
=
]
:r { f w, <>; w - c
a
aw
(4.15)
+
�
a
,
P
,
a
�
;
Stochastic methods in economics and finance
226
+
,
n
� � P.1 Ww-I o1.1. 1
I
a2 3P; 3 W
---
(4. 1 6)
Under the assumptions of the problem there exists a set of optimal rules w* and C* satisfying
0 = { max {¢ (C, w; W, P, t)} C, w } = ¢ (C*, w* ; W, P, t) for t E [0, T) .
(4 .17 )
In the usual fashion of maximization under constraint we define the Lagrangian L = ¢ + "A [ I - ��� W;] and obtain the first-order conditions:
(4.18) + � 1; w II
I
0
=
ok;
P; hi,
L>-. (C*, w*) = l
-
k = I , ... , n, ,
� wr I
(4.19} (4.20}
Merton solves for C* and w* and inserts the solutions to (4.17) and (4. 1 5) to obtain a complicated partial differential equation. See Merton (197 1 , p. 383). If this partial differential equation is solved for J, then its solution, after appro priate substitutions, could yield the optimal consumption C* and portfolio rules w*. 5. Hyperbolic absolute risk-aversion functions
In this application we specialize the analysis of Merton ( 1971) as presented in the previous section for the class of hyperbolic absolute risk aversion (HARA} utility functions of the form u (C, t) = e - P t v(C),
where
( 1 --y
I - -y
v(C) =
Applications in finance
{3C
--
'Y
+ 11
Note that the absolute risk (v" jv') is given by v
"
A (C) = - , = v
)
'Y
(5. 1 )
•
aversion
----
c
1 --y
227
11 + -
denoted by A (C) and defined to be A (C) =
> 0,
(5.2)
{3
provided -y =1- I ; (3 > 0; ({3C/(l - -y)) + 11 > 0; and TJ = 1 if -y = - oo. This family of utility functions is rich because by suitable adjustments of the parameters one can have a utility function with absolute, or relative, risk aver sion that is increasing, decreasing or constant. Without loss of generality assume that there are two assets, one risk-free and one risky. The return of the risk-free asset is r and the price of risky asset is log normally distributed satisfying dP - = a d t + odz. p
(5 .3)
The optimality equation for this special case is
0=
{1 - -r? 'Y
� +
e- p r 11
[ ]
eP'Jw 'Y /('Y- 1 >
{3
J
0 - 'Y) fi + r W lw tJ
1
2
lw
-
ww
+ 1,
(a -r?
(5.4)
2 a o2
subject to J ( W, T) = 0. For simplicity we assume that the individual has a zero bequest function. A solution of the partial differential equation in (5 4) is given by Merton ( 1 973c, p. 2 1 3) .
[:
+
; { 1 -cxp [-r(T-1)]�
')'
,
p -- f> v
(5.5)
Stochastic methods in economics and finance
228
S
-r)2/2S a1 (W, t) -t))]/{3r.
r
S
S
where = 1 -')', v = + (a and is assumed positive. If < 0, and � in (5 .5) will hold only when 0 � therefore "Y > 1 , the solution J ("Y- 1 ) 71 [ 1 -exp(-r (T The explicit solutions for optimal consumption and portfolio rules are given below:
C* (t)
=
6
and
w* (t)
-j;{I - { �
[p-')'v) [w(t) +
= a-r2 + Sa
ex
1
{ 1 -exp
p
71(a-r)
W(t) {3ra2
')'V
[r(t- T))
(t -
T�}
{ 1 -exp
�
-
S 71
{3
-
[r(t- T)]}.
W(t)
(5.6)
(5.7 )
The basic observation obtained from the above two equations is that the de mand functions are linear in wealth. It can be shown that the HARA family is the only class of concave utility functions which imply linear solutions.
6. Portfolio jump processes We follow Merton ( 1 97 1 ) to discuss an application of the maximum principle for jump processes in a portfolio problem. Consider a two-asset case with a com mon stock whose price is log-normally distributed and a risky bond which pays an instantaneous rate of interest r when not in default, but in the event of de fault the price becomes zero. The process which generates the bond's price is as sumed to be given by d P = rPd
t
-
Pd q ,
(6. 1 )
q(t)
with being an independent Poisson process. The new budget equation that replaces (4. 1 3) is d
W = {w W(a-r) + r W - C} dt + waWdz - (l - w) Wdq.
(6.2)
Note that (6.2) is an example of mixed Wiener and Poisson dynamics. An appli cation of the generalized Ito formula and the maximum principle for jump pro cesses in section 1 2 of Chapter 2 yields the optimality equation
Applications in finance
0
=
229
u(C*, t} + J, (W, t) + A. [J(w* W, t) - J (W, t)] + lw (W, t)[(w * (a-r)+r) W - C*] + � lww (W,t)a2 w*2 W2 , (6.3)
where the optimal assumption implicit equations
C* and portfolio w * rules are determined by the
0
= uc(C*, t) - Jw (W, t)
(6.4)
0
= A.Jw (w * W, t) + lw (W, t) (a-r) + lww (W, t) a2 w * W.
(6.5)
and
There is one additional novelty in this Merton problem that is not present in the pure Brownian motion case. That is, for a HARA utility function you must not only conjecture a solution from J(W, t) = g(t) wa and solve for the exponent by the method of equating exponents, you must also conjecture a form for the demand function w W = d W + e, where the term d is dependent of wealth. This is a natural conjecture to make for the form of the demand function for the risky asset given the separation theorem which says that for utility functions of the hyperbolic absolute risk-aversion class that the proportion of wealth held in the risky asset should be independent of the investor's wealth level and independent of his age . It is natural to conjecture the same sort of separation theorem for the Poisson case as well. At any rate, the philosophy is to try it and see if it works. Furthermore, conjecture that the term e = 0. This is natural because if wealth is zero there would be no demand for the risky asset. Next equate exponents on wealth in the partial differential equation for the state valuation function, J, to solve for the unknown exponent on wealth. I t will turn out to be the same exponent as that on consumption in the utility function. Cancel all terms involving wealth off the partial differential equation (6.3) for the state valuation function. That will give an ordinary differential equation for the unknown function g(t). Furthermore , examine the necessary condition (6.5) to determine the unknown proportion d. Cancel off all terms involving wealth and all terms involving the unknown function g(t} in the necessary condition (6.5} to get the relationship
d=
A. a-r + o2 (1 --y) a2 ( 1 --y)
tJY- J
·
(6.6)
This last is the same as Merton's (80') in Merton ( 1 97 1 , p. 397) with w = d. Thus, conjecturing a linear demand function for the risky asset worked.
Stochastic methods in economics and finance
230
The moral of this exercise is that for the Poisson case it was necessary to con jecture a form of the demand function for the risky asset in order to obtain a closed form solution. But the conjecture of the appropriate form for the demand function was motivated by the form of the demand function that was derived in the pure Brownian motion case. In other words, in the pure Brownian motion case when the utility function was hyperbolic absolute risk averse then the demand function for the risky asset was linear and the proportion of the investor's port folio held in the risky asset turned out to be independent of the investor's wealth level and of his or her age. This independence is called the separation theorem. The name separation theorem derives from the fact that the consumption deci sion and the portfolio diversification decision turn out to be determined inde pendently of each other in this particular case. Merton's (
1 97 1 ) paper contains several other examples of closed form solution
determinations for Poisson processes. Furthermore, it contains closed fo rm solu tion determinations for more general processes as well.
7.
The demand
for index bonds
Consider, as Fischer
( 1975)
does, a household with three assets in its portfolio:
a real bond, a risky asset and a nominal bond. Assume that the portfolio can be adjusted instantaneously and costlessly . We also assume that the rate of inflation is stochastically describable by the process dP - = fl d t + sdz.
(7 . 1 )
p
The real bond pays a real return of r1 and a nominal return o f r 1 plus the realized
rate of inflation. Note that d
Q1 -Q.
dP = r1 d t + - = (r1 + fl ) d t + s dz = R 1 d t + s 1 dz 1
7 )
( .2
p
is the equation describing the nominal return on the "index bond. The nominal return on equity is d
2
Q Q, = R2 d t + s2 dz2 , where R2 is the expected nominal return on equity per unit of time and
(7 .3) s�
is the
variance of the nominal return per unit time. Using the results of application
2,
Applications in finance
231
if we let d Q3 /Q3 = R 3 d t describe the deterministic nominal return of the nomi nal bond, then the real return on the nominal bond is (7.4) Let w 1 , w2 and w3 be the proportions of the portfolio held in real bonds, equity and nominal bonds, respectively. Obviously, w1 w2 w3 I . The flow budget constraint, giving the change in nominal wealth, W, is similar to (4.1 0): 3 2 d W = .r1 W; R; Wdt - PCdt + .r1 w; s; Wd z; , (7.5) +
+
=
where C is the rate of consumption. Uncertainty about the change in nominal wealth arises from holdings of real bonds and equity. Since w3 = 1 - w1 - w2 , may rewrite eq. (7 .5) as 2 2 d W = .r w;(R; - R3) Wd t (R3 W - PC) d t + .r W; S; Wdz; (7.6) We are now in a position to formulate the household's choice problem: (7.7) max E0 J u [C(t), t] d t {C,w;} subject to (7 .6) and we
+
I
.
I
00
o
W(O) = W0 ,
where u is a strictly concave utility function in C, and E0 is the expectation con ditional on P(O). The first-order necessary conditions of optimality are: 0 = uc(C, t) - Plw, (7.8) (7.9) (7.10) where, as before, J(W, P, t) = max E, J u(C, s) ds {C,w,. } 00
t
Stochastic methods in economics and finance
232
and p is the instantaneous coefficient of correlation between the Wiener processes dz 1 and dz2 and I p I < 1 . It is now possible to solve for asset demands from the two equations (7 .9) and (7 .1 0) and the fact that L W; = 1 , to obtain
(7 . I I ) (7 .12)
From (7. 1 1 )-(7.13) Fischer ( 1 975) studies the complete properties of the demand functions for the three assets. In particular consider the demand func tion for index bonds in (7 . 1 1 ) Observe that the coefficient -JwlJww W is the inverse of the degree of relative risk aversion of the household. If we make the simplifying assumption that p = 0 in (7 .1 1 ), then (7 . l l ) says that the demand for index bonds depends on (i) the degree of relative risk aversion, (ii) the dif ference between expected nominal returns on the two types of bonds, R 1 R3 , and (iii) the variance of inflation , s � . But how about the term lwpPflww W in (7 .1 1 )? This term can be related to the degree of relative risk ave·rsion as follows: .
-
(7 .J 4)
To obtain (7 .14) differentiate (7 .8) first with respect to P to get
(7 . 1 5 ) and with respect to
W to get
(7 . 1 6) Finally, note that since consumption is a function of real wealth
ac aP
w ac
= - P aw ·
(7 .17)
Applications in finance
233
Combining (7 .1 5)-(7 .17) we get (7 .14). There are many other valuable insights that this analysis uncovers. We mention just one more. Consider the yield differentials in terms of real returns when 1 = 0, i.e. when the household has no index bonds in its portfolio, given by w
(7 .18) w2
Suppose that = I , i.e. the net quantities of real and nominal bonds are zero. If there is positive covariance between equity returns and inflation then from (7 .18) we obtain that r1 -r3 > 0, which means that index bonds will have to pay a higher return than the expected real yield on nominal bonds. In other words, if equity is a hedge against inflation, then index bonds do not command a pre mium over nominal bonds. Conversely, if equity is not a hedge against inflation then index bonds will command a premium over nominal bonds. 8. Term structure in an efficient market
Methods of stochastic calculus similar to the ones used by Black and Scholes (1973) and Merton (197 1 ), which were presented in earlier sections, have been used by Vasicek (1 977) to give an explicit characterization of the term structure of interest rates in an efficient market. Following Vasicek (1977) we describe this model below. Let P(t, s) denote the price at time t of a discount bond maturing at time s, with t � s. The bond is assumed to have a maturity value, P(s, s), of one unit, i.e. (8.1)
P(s, s) = 1 .
The yield to maturity, R(t, T), is the internal rate of return at time t on a bond with maturity date s = t + T, given by R(t, D = -
1 logP(t, t + T), T> O. T
(8.2)
From (8.2) the rates R(t, T) considered as a function of Twill be referred to as the term structure at time t. We use (8.2) to define the spot rate as the instan taneous borrowing and lending rate, r(t), given by r(t) = R(t, 0) = lim R(t, n. r-o
(8.3)
234
Stochastic methods in economics and finance
It is assumed that r(t) is a continuous function of time described by a stochastic differential equation of the form
dr =f(r, t)dt + p(r, t)dz,
(8.4)
where, as usual, z(t) is a Wiener process with unit variance. It is assumed that the price of a discount bond, P(t, s), is determined by the assessment, at time t, of the development of the spot rate process (8.4) over the term of the bond, and thus we write
P(t, s) = P(t . s, r(t)).
(8.5)
Eq. (8.5) shows that the spot rate is the only state variable for the whole term structure, which implies that the instantaneous returns on bonds of different maturities are perfectly correlated. Finally, we assume that there are no trans actions costs, information is available to all investors simultaneously, and that investors act rationally; that is to say, we assume that the market is efficient. This last assumption implies that no profitable riskless arbitrage is possible. From eqs. (8.4) and (8.5) by using Ito's lemma we obtain the stochastic dif ferential equation
dP = Pp(t, s, r(t)) dt - Pa(t,s, r(t)} dz, which describes the bond price changes. In (8.6} the functions J.l and as follows:
p(t' s, r) -
1 P(t, s, r)
a are defined (8.7)
a(t. s,r) = - --- p 0 P(t, s , r). P(t,s,r) r 1
(8 .6)
a
(8.8)
Consider now the quantity q(t, r(t)) given by
q (t, r) =
JJ(t, s, r) -r , r <;. s, a(t,s,r)
(8.9)
which is called the market price of risk and which specifies the increase in ex pected instantaneous rate of return on a bond per an additional unit of risk. Sub stitute the expressions for J.l and a from (8.7)and (8.8) into (8.9), make the neces sary rearrangements and obtain the term stn.tcture equation given by
Applications in finance
- - rP = O.
oP oP 1 o2 P - + (f + PQ) - + -2 P 2 3r 2 ot or
235
(8.1 0)
Observe that (8.10) is a partial differential equation whose solution P may be ob tained once the spot rate process r(t) and the market price of risk q(t, r) are specified. The boundary condition of (8.1 0) is
(8.1 I )
P(s, s, r) = l .
Knowing P(t, s, r) as a solution of (8.1 0) subject to (8.1 I ) allows us to obtain the term structure from (8.2). Vasicek ( 1977) uses techniques presented in Friedman ( I 975) to write a rep resentation for the bond price as a solution to the term structure equation (8.10) subject to (8 . 1 1 ), given by
(
s
P(t, s) = E, exp - J r(u)du - 4 +
s
t
J q(u, r(u)) dz(u) t
},
t�
s.
s
J q2 (u, r(u)) du t
(8 .12)
To obtain some economic insight in eq. (8. 1 2), construct a portfolio con sisting of a bond whose maturity approaches infmity, called a long bond, and lending or borrowing at the spot rate, with proportions A(t) and 1 - A(t), respec tively. Here we define A(t) as
A(!) =
J-L(t, oo )-r(t) a2 (t ,oo) '
(8.1 3)
i.e.
A(t) a (t, oo) = q(t,r(t)).
(8.14)
The price Q(t) of such a portfolio satisfies the equation
dQ = AQ(J-L(I, oo)d t - a(t, oo)dz) + ( I -A)Qrdt.
(8. 1 5)
(8. 1 5) can be integrated by evaluating the differential of Jog Q and using (8.14) to yield Eq.
d (log Q)
=
AJ-L(t, oo)dt - AG(t, oo) dz + ( 1 -A)rdt - � A2 a2 (t, oo) d t
=
rdt + � q2 d t - qdz,
Stochastic methods in economics and finance
236
from which we obtain
Q(t) = exp Q(s)
(- J r(u)du - � J q2 (u, r(u)) du s
s
t
t
Using this last equation we may rewrite (8 .12) as
P(t, s} - E1
s
+
J q(u , r(u})dz t
)
.
Q(t) � , t � s, Q(s)
which means that the price of any bond measured in units of the value of a port· folio Q follows a martingale,
P(t, s) Q(t)
_ _
E1
P(u, s) Q(u)
'" 10r
..,..,
..,..,
{ """'. U """'. S.
Therefore, we conclude that if the bond price at time t is a certain fraction of the value of the portfolio Q, then the same will hold in the future . 9. Market risk adjustment in project valuation
In this application we follow Constantinides ( 1978) to develop a rule which re· duces the problem of valuation under market risk to a problem of valuation when the price of risk is zero. Let V(x, t) denote the market value of a project where such a project can be chosen to be an investment, an option, a claim on a firm, etc. The project is as· sumed to generate a stream of cash flows and V(x, t) represents the time and risk adjusted value of these cash flows. The market value function V(x, t) is specified by the state variable x and time t, where we assume that x changes according to the stochastic differential equation
dx = J.L(X, t} d t + a(x, t)dz = J,Ldt + adz,
(9 . 1 )
where for notational convenience we write J.L = J.L(x, t} and a = a(x, t). I n (9.1), as earlier, z is a Wiener process with unit variance. The cash return generated by the project during the time interval (t, t + dt) is assumed to be nonstochastic given by edt, where c = c(x, t).
237
Applications in finance
Consider now the return on the project in the time interval (t, t + d t); it is the sum of capital appreciation d V(x, t) and cash return edt. To obtain d V(x, t) we use Ito's lemma to conclude (9.2) assuming V(x, t) is twice continuously clifferentiable with respect to x and once continuously differentiable with respect to t. Using (9 .2) we may write the rate of return on the project as
d V(x, t) + cdt V(x, t)
=
( V 1
c
+
a2
)
v, + J..L Vx + 2 Vxx d t +
a Vx
V
dz.
(9.3)
From (9 .3) we write the expected value per unit of time, ap, and the covariance with the market per unit of time, aPM , as (9.4) and (9.5)
where p = p (x, t) is the instantaneous correlation coefficient between dz and the market return. In (9.5) aM denotes the positive square root of the variance of the market portfolio. At this point Constantinides (1 978) uses a result stated in Merton ( 1 973b) and proved in Merton ( 1 972). Before we state this result we indicate the notation. Let O:; denote the expected rate of return of security i per unit of time and let a;i denote the covariance of returns per unit of time. The riskless borrowing lending rate is denoted by r and the subscript M refers to the market portfolio. The result is this: under certain assumptions, which lead to Merton's ( 1973) intertemporal capital asset pricing model, the equilibrium security returns must satisfy the equation (9.6) where f3; = a;M /a�. Note that (9 .6) is the continuous time analogue of the secu rity market line of the classical capital asset pricing model. For example, see Francis and Archer ( 1979, p. 1 58). For our purposes we rewrite (9 .6) as
Stochastic methods in economics and finance
238
(9.7) where A = (aM - r)/aM . After this disgression we return to our model. Substi tute (9.4) and (9.5) in (9.7) to get (9.8) which is a partial differential equation for which, under certain boundary con ditions, we may obtain a solution V(x, t) giving the market value of the project. To proceed a step further with the analysis let
J..L * = J..L *(x, t) = J..L (x, t) - Ap (x, t) a (x, t) and rewrite (9 .8) as c
02
- r v + vt + J.1 * vx + 2 vxx = 0 .
(9.9)
(9.10)
Next, we want to compare (9 .I 0) with a similar equation describing the value of the project in a capital market which pays no premium for market risk, i.e. aM r = 0. Set aM - r = 0 in (9.6) and usc (9.4) and (9.5) to obtain -
(9. 1 1 ) where V(x, t) denotes the value of the project in a capital market which pays no premium for risk. The boundary conditions of (9 .I I ) arc identical to those im posed on V in (9 .I 0) because these conditions are independent of the market risk premium. We conclude this application by comparing eqs. (9 .I 0) and (9 .I I ). A com parison shows that V(x, t) may be considered as the market value of the project in a capital market which pays no premium for market risk provided J.1 * (x, t) replaces J..L (x , t). This observation leads Constantinides ( 1978) to suggest the fol lowing rule for determining the market value of a project. First, replace the drift J..L (x, t) by p. * (x, t) as in (9 .9). Secondly, discount expected cash flows at the risk less rate.
10. Demand for cash balances The various models that have been developed to explain the demand for money can be classified into two categories. Some models, such as Baumol ( 1 952) and
Applications in finance
239
Tobin ( 1 956), assume that transactions occur in a steady stream which is perfectly foreseen, while other models, such as Olivera ( 1 971) and Miller and Orr ( 1 966), assume that net cash flows are completely random. In this application we follow Frenkel and Jovanovic ( 1 980) to incorporate various aspects of these two cate gories. Assume that changes in money holdings follow an Ito stochastic differential equation: dM(t) =
-
J.Ld t + adz (t),
1H(O) = M0 , J.1. ;a: 0,
(10.1)
where z is a Wiener process with unit variance, M0 is the optimal initial money holdings, and J.L is the deterministic part of net expenditures. From ( l 0 . 1 ) upon integration we obtain
1H(t) = M0 -J.Lt + az(t),
(1 0.2)
where M(t) is normally distributed with mean A10 - J.Lt and variance a2 t. The optimal level of money holdings is determined by minimizing the cost of finan· cial management. We distinguish two such sources of cost: first, forgone earn· ings which depend on the interest rate r and on the money holdings .M(t), and secondly, the cost of adjustments which depends on the frequency of adjustment and on the fixed cost C per adjustment. It is assumed that an adjustment of the money stock is necessary whenever money holdings reach a lower bound. This lower bound is assumed to be zero. Note that costs from each source are random because at each period t money holdings as described by ( I 0.2) are random. Therefore the optimal size of cash balances is determined by minimizing expected cost. It is analytically convenient to separate the expected cost into two parts: first, the expected cost that was incurred prior to the period of the first adjust ment and, secondly, the expected cost that is incurred thereafter. The period when holdings reach zero and adjustment is necessary is random. For the analy sis of the first part we write at period t the instantaneous forgone earnings as rt.1(t) and their present value as rt.1(t) e-rt. Denote by h(A1, t I 1H0 , 0) the prob ability that money holdings A1(t) which at period t = 0 were at the optimal level l�10 , have not reached zero prior to period t at which time money holdings are A1. Thus, the present value of expected forgone earnings up to the first adjust ment may be written as
Stochastic methods in economics and finance
240
( 1 0.3) Frenkel and Jovanovic ( 1 980) show that ( 1 0.3) may be simplified to be written as (I 0.4) where "' = exp
{-::
[(!12 + 2ra2) 1 / 2
-Ill } .
(J 0.5)
Putting aside (1 0.4) for a moment we analyze next the expected cost which is incurred following the first adjustment. Denote by G(M0 ) the present value of total expected cost and by f(M0 , t) the probability that cash holdings reach zero at time t, having been the optimal level k/0 at t = 0 . Thus, the present value of the expected cost following the first adjustment is given by l2 (Mo ) =
00
J
e- rt [C + G(Mo)l f(Mo , t) dt.
( 1 0.6)
0
Note that G(Jl.-10 ) excludes the current fixed cost of adjustment which explains why C is added to G(M0 ) in (10.6). Frenkel and Jovanovic ( 1 980) argue that (1 0 .6) can be simplified to be written as J2 (M0 ) = a [C + G(M0 )) .
( 1 0.7)
Using ( 1 0 .4) and (10.7) we may write the present value of total expected cost as J.1.
G(A'10 ) = M0 - (1 - a) - + a [C + G(M0)], r
which upon rearrangement reduces to G(M0 ) =
p.
M0 + aC
1 -a
-
- . r
(10.8)
Minimizing the expected cost of financial management G (M0) with respect to the optimal level of cash balances !v/0 we obtain the necessary condition (1
-
a) + (Mo
+
C)
aa
a M0
= 0.
(1 0 .9)
Applications in finance
24 1
Expand (1 0.9) in Taylor series around M0 and after terms of third and higher order are ignored solve for M0 to obtain
(10.10) Eq. (1 0.1 0) satisfies the homogeneity postulate with a rise of a given proportion in a, C and p. resulting in an equiproportional rise in 1\10 . Two special cases are of interest. In the first case if we assume a2 = 0, we ob tain by expanding the bracketed term in the denominator of eq. (10.10) around
a2 = 0:
(}.12 + 2ra2 ) I /2 = J.1. +
2ra2 l9 (a4). 2 J.1. + l
(10. 1 1 )
In ( l 0.1 1 ), (!J (a4 ) denotes terms of order a4 or higher. Putting ( 1 0. 1 1 ) in ( 10.1 0) we get Jl1o
=
(
2Ca2 (ra2 fp.) + (' (a4)
)1/2
Finally, taking the limit in ( 1 0 . 1 2) as a2
.
hm 1\10
ol .-. o
=
( 2 CiJ. ) I/2 r
(10 . 1 2) �
0 we conclude that
,
which is the result obtained by the Baumol -Tobin formulation of optimal trans action balances. In the second special case we let p. = 0 and evaluate (10.10) to get A10 =
(
)
1 /2 2Ca2 , (2ra2 ) 1 /2
(10.13)
which is similar to the results of the Miller- Orr model. Thus, using stochastic calculus techniques and following the Frenkel - Jova novic (1980) model an extension of some of the existing models of the demand for money has been achieved. In such an extension the implications of the two special cases are clear. In the first case, corresponding to the Baumol- Tobin framework, it is assumed that the process governing net disbursements is deter ministic, i.e. a2 = 0. In the second case, corresponding to the Miller-Orr frame work, it is assumed that the process governing net disbursements is stochastic without any drift, i.e. p. = 0.
242
Stochastic methods in economics and finance
The price of systematic risk ln 1976 Steve Ross produced a theory of capital asset pricing that showed that the assumption that all systematic risk free portfolios earn the risk free rate of re turn plus the assumption that asset returns are generated by a K-factor model leads to the existence of prices Ao, A1 , A.2 , , AK on mean returns and on each of the K-factors (Ross, 1976). These prices satisfied the property that expected returns EZ1 on each asset i was a linear function of the standard deviation of the re turns on asset i with respect to each factor k, i.e. K (1 1 .1) a; = A0 + k1; Akbki ' i = 1 , 2, . , N, =l 11.
•••
=
a1
.
where the original model of asset returns is given by K ( 1 1 .2) Z; = O; + k1; bk i 8k + f;, i = 1 , 2, ... , N. =l Here Z; denotes random ante anticipated returns from holding the asset one unit of time, S k is systematic risk emanating from factor k, €1 is unsystematic risk specific to asset i, and and bki are constants. Assume that the means of S k and €; are zero for each k and i, that € , , €N are independent, and that S and €; are uncorrelated random variables with finite variances for each k and i. Ross proved that Ao , A1 , , AK exist that satisfy (1 1 .1) by forming portfolios E RN such that ex
a;
1
k
•••
•••
11
N
L i
(1 1.3) and constructing the 11; such that the coefficients of each 8k in the portfolio returns, =l
11; = 0,
N
· � i=l 11·Z ' ' =
K � TJ·O· + � i=J l l k = 1
N
(
N
)
N
� bk·fl· I I 8k + i=� fl·f· I I, i= 1 1
(1 1 .4)
are zero, and requiring that N
L
i= I
11·0· I I
=0
for all such systematic risk free zero wealth portfolios.
( 1 1 .5)
243
Applications in finance Here
(1 1 .3) corresponds to
the zero wealth condition. The condition
N 1: 0 = . bk;Tl;, k = I , 2, ..., K,
(11 .6)
1 ::: 1
corresponds to the systematic risk-free condition . Actually, Ross did not require that
(1 1 .3) hold
for all zero wealth systematic risk-free portfolios but only for
those that are well diversified in the sense that the 'fl; are of comparable size so random variable
r,{:
1
, eN to argue that the
'fl;f; was small and hence bears a small price in a world of
that he could use the assumption of independence of € 1 ,
•••
investors who would pay a positive price only for the avoidance of risks that could not be diversified away. Out of this analysis Ross argues that the condition: for all
N 1: Tl;bk; = 0, i= 1
N 1: Tl; = 0 ;
i= 1
implies that in
l: 17.a. ' '
N
k = I , 2, .. ., K
(1 1 .7)
equilibrium
=
i= J
Tl E R N ,
0,
(1 1 .8)
should hold. All that (1 1 .7) and ( 1 1 .8) say is that at zero wealth, zero systematic risk port.. folios should earn a zero mean rate of return. The condition in
( 1 1 .7) and ( 1 1 .8)
is economically compelling because in its absence rather obvious arbitrage op portunities appear to exist. Whatever the case , ( 1 1 .7) and
such that ( 1
( 1 1 .8) imply that there exists A0, A1 , A2 ,
•••
, AK
1 .1) holds and the proof is just simple linear algebra. Notice that Ross
made no assumptions about mean variance investor utility functions or normal distributions of asset returns common to the usual Sharpe- Lintner type of asset pricing theories which are so standard in the fmance literature. However, Ross's model, like the standard capital asset pricing models in fi nance, does not link the asset returns to underlying sources of uncertainty. Ap plication
15
of Chapter
3
will be used as a module in the construction of an in
tertemporal general equilibrium asset pricing model where relationships of the fonn
( 1 1 .2) are
detennined within the model and hence the
Ao , A 1 ,
•••
, AK
will
be determined within the model as well. Such a model of asset price determina tion will preserve the beauty and empirical tractabiHty of the Ross-Sharpe Lintner fonnulation but at the same time will give us a context where we can ask general equilibrium questions such as. What is the impact of an increase of the progressivity of the income tax on the demand for and supply of risky assets and the
Ao , A1 ,
•••
, AK ?
Stochastic methods in economics and finance
244
Let us get on with relating the growth model of section 1 5 of Chapter 3 to ( I l . l ). For simplicity assume all processes i are active, i.e. ( 1 5 . 1 1) of Chapter 3 holds with equality. We record (15.1 1) here for convenience
( 1 1 .9) Now ( 1 1 .2) is a special hypothesis about asset returns. What kind of hypothesis about technological uncertainty corresponds to ( 1 1 .2)? Well, as an example, put for each i = 1 , 2, . . . , N 1 o K2 f; (xu, r1) = (Ait + A;1 5 1 1 + A;, S 2 1 + .. . + Ait SK 1)f; (x;1)
( 1 1 . 10)
- 'u!,· (xu),
where A;� are constants and {S k 1 };o= 1 are independent and identically distributed random variables for each k. For each t the mean of S k t is zero, the variance is fmite, and 5s t is independent of i> k t for each s, k and t. Furthermore, assume that f( · ) is concave, increasing, twice differentiable, f' (0) = oo, f ' (oo) = 0 and that there is a bound e0 such that r1 > e0 with probability 1 for all t. These as sumptions are stronger than necessary but will enable us to avoid concern with technical tangentialities. Defme, for all t, S 0 1 = 1 , so that we may sum from k = 0 to K in ( 1 1 . 1 1 ) below. Insert ( 1 1 .1 0) into ( I 1 .9) to get for all t, k and i
{
(J
)
}
' u' (c, ) = iJ E, u (ct+ 1 ) o A � ilk, f;'(xit) K k ' = L ([Au f; (x;r)] E1 {{3 u (cI+ 1 ) S k 1 }). k =O '
�
(1 1 . 1 1)
Now set (1 1 . 1 1 ) aside for a moment and look at the marginal benefit of saving one unit of capital and assigning it to process i at the beginning of period t. At the end of period t, r1 is revealed and extra output
( 1 1 . 1 2) emerges. Putting
a; = A� f;'(x;1); bk; = A;� f;'(x;1); Sk 1 = bk ,
{ 1 1 . 1 3)
eq. { 1 1 .12) is identical with Ross•s ( 1 1 .2) with €1 = 0. We proceed now to gen erate the analogue to ( 1 1 .1 ). Turn to ( 1 1 .1 1) and rewrite it, using ( 1 1 . 1 3), as
Applications in finance
245
( 1 1 . 1 4) Hence, a; =
u ' (c,)
13 E1 {u' (ct+ 1 ) }
so that :\.0 , :\.1 ,
••.
K
-
kL =t
bk;
S ( E1 {u' (c t+ 1 ) k 1 } ) __ E {__ '} ( ), u ct+ 1
( 1 1 . 1 5)
, AK , defined by
u' (c, )
( 1 1 . 1 6)
yields
( 1 1 . 1 7) Here the subscript t is dropped to ease the notation. These results are extremely suggestive and show that the model studied may be quite rich in economic con tent. Although the model is normative, in the next section we shall turn it into an equilibrium asset pricing model so that the Ak become equilibrium risk prices. Let us explore the economic meanings of ( 1 1 . 1 6) in some detail. Suppose that K = 1 and that there is a risk-free asset N in the sense that ( 1 1 . 1 8) i.e. ( 1 1 . 1 9) Then by ( 1 1 . 1 9) we obtain ( 1 1 .20) so that for all i,j =F N ( 1 1 .2 1 ) The second part of eq. ( 1 1 .20) corresponds t o the security market line which says that expected return and risk are linearly related in a one-factor model. Eq. ( 1 1 .2 1 ) corresponds to the usual Sharpe-Lintner-·Mossin capital asset pricing
246
Stochastic methods in economics and finance
model result that in equilibrium the excess return per unit of risk must be equated across all assets. The economic interpretation of Ao given in ( 1 1 . 1 6) is well known and needs no explanation here. Look at the formula for 'Ak . The covariance of the marginal utility of consumption at time t + 1 with the zero mean fmite variance shock S k t appears in the numerator. Since output increases when Sk t increases and since c,+ 1 = g (y t+ 1 ) does not decrease when y t+ 1 increases, therefore this covariance is likely to be negative so that the sign of 'Ak is positive. We will look into the determinants of the magnitudes of Ao, 'A 1 , , 'AK in more detail later. Let us show how this model may be helpful in the empirical problem in estimat ing the 'A0 , A 1 , , AK from time series data. First, how is one to close Ross's model (1 1 .2) since the Z; are subjective? The most natural way to close the model in markets as well organized as U.S. securi ties markets would seem to be rational expectations: the subjective distribution of i; is equal to the actual or objective distribution of Z;. We shall show that our asset pricing model under rational expectations, which is developed below, gen erates the same solution as the normative model. Hence, the convergence theorem implies that {x,, c,, x 1 , , xu , ... , xN t };: 1 converges to a stationary stochastic process. Hence, the mean ergodic theorem which says very loosely that the time average of any function of a stationary stochastic process equals the average of over the stationary distribution of that process, allows us to apply time series methods developed for stationary stochastic processes to estimate Ao , X 1 , , AK . As is well known, time series data are useful for the estimation of 'A0 , A 1 , , AK . Let us next turn to the development of the asset pricing model. •••
••.
G
G
•••
•.•
1 2. An asset pricing model In this section we reinterpret the model of section 1 5 of Chapter 3 and add to it a market for claims to pure rents so that it describes the evolution of equilibrium asset prices. In this way we will not only generate a general equilibrium context in which to discuss the martingale property of capital asset prices, but also the model will contain a nontrivial investment decision, a nontrivial market for claims to pure rents, i.e. a stock market, as well as a market for the pricing of the physical capital stock. We believe that there is considerable benefit in showing how to tum optimal growth models into asset pricing models. This is so because there is a large litera ture on stochastic growth models which may be carried over to the asset pricing problem with little effort .
Applications in finance
247
We develop an asset pricing model much like that of Lucas ( 1 978). The model contains one representative consumer whose preferences are identical to the planner's preferences given in ( 1 5 . 1 ) of Chapter 3. The model contains N differ ent firms that rent capital from the consumption side at rate Rt+ 1 at each date so as to maximize (12.1)
i
Notice that it is assumed that each firm makes its decision t o hire xi t after r, is revealed. Here Ri, t+ l denotes the rental rate on capital prevailing in industry i at date t + I . It is to be determined within the model. The model will introduce a stock market in such a way that the real quantity side of the model is the same as that of the growth model in equilibrium. Our model is closed under the assumption of rational expectations. The quantity side of the model is essentially an Arrow-Debreu model, as is the model of Lucas ( 1 978). We introduce securities markets in such a way that there is a security for each state of the world. However, there is a separate market where claims to the rents in ( 1 2 . 1 ) are competitively traded. Recall that in an Arrow-Debreu econo my the rents are redistributed in a lump-sum fashion . The model i s in the spirit o f Lucas' (1978) model where each firm i has out standing one perfectly divisible equity share. Ownership of a% of the equity shares in firm i at date t entitles one to a% of profits of the firm i at date t + 1 . Equilibrium asset prices and equilibrium consumption, capital and output are determined by optimization under the hypothesis of rational expectations much as in Lucas (1 978). Let us describe the model. The representative consumer solves maximize E 1
00
L t
=I
1
(3'- u (ct )
(1 2.2)
subject to
c, +x, + P, · Z, � 1T, · Z,_ 1 + P, · Z,_ 1 + .l: I= 1 N
c, ;;?; 0;
x, ;;?; 0; Zt ;;?; 0; x;t > 0;
cI + XI
+ p1 • z0 1T
Z0 = 1 ;
=
1
•
i = I 2, . , N, all t, ,
N
l=
fl;1
(1 2.3) ( 1 2.4)
. .
z0 + p1 • z0 + _LI Ri I X;0 = y 1
R; 1 = f/ (x;0 , r0);
x 0 , { x;o } :-: 1 given.
R;r x; ,,_ 1 =y, ,
'
= f; (x;0, r0)-J; (.Y;0, r0)x;0;
( 1 2.5)
Stochastic methods in economics and finance
248
Note that c,, x1, P;,, Z;1, 1T;1 and Ru , all assumed measurable .'F1, denote con sumption at date t, total capital stock owned at date t by the consumer, price of one share of firm i at date t , number of shares of firm i owned by the individual at date t, profits· of firm i at date t, and rental factor, i.e. Ru is principle plus in terest obtained on a unit of capital leased to firm i. Here a dot denotes scalar product. Firm i is assumed to hire xit so as to maximize ( 1 2. 1 ). The consumer is as sumed to lease capital X;1 at date t to firm i before r1 is revealed. Hence R; , t + 1 is uncertain at date t. The consumer, in order to solve his problem at date 1 , must form expectations on {Pit};: 1 , {Rit} �= 1 , { 1T1} ;: 1 and maximize ( 1 2 .2) subject to ( 1 2 3) and ( 1 2.4). In this way notional demands for consumption goods and equities as well as notional supplies of capital stocks and capital services to each of the N firms are drawn up by the consumer side of the economy. Similarly for the firm side. We close the model with this definition: The collection of stochas tic processes � = ({P;,};: 1 , {R;,};: 1 , { 1T;r };: 1 , {:X;r};: 1 , {Zit };: 1 , i = 1 , 2, ... , N, {c1!;: 1 , {:X,Loo= 1 ) is a rational expectations equilibrium (R.E.E.) if facing r!l = ({Pit};: 1 , { Rit}�= 1 , { 1Tit}�= 1 ) the consumer chooses .
(1 2.6) and the i th firm chooses
(1 2.7) and furthermore (asset market clears)
zit � 1
(goods market clears)
c, + x1 = .I:1 f; (:X; 1_ . , rr_ 1 ) , a.e.,
(capital market clears)
.I: :X;r = :X,
N
l=
1
if zit < 1 , pit = 0, a.e., N
t=
(1 2.8) ( 1 2.9)
•
a.e.
( 1 2 . 1 0)
Here a.e. means almost everywhere. This ends the definition of R.E.E. that we will use in this section. It is easy to write down first-order necessary conditions for an R.E.E. Let us start on the consumer side first. We drop upper bars to ease notation. At date t if the consumer buys a share of finn i the cost is P;1 units of consumption goods. The marginal cost at date t in utils forgone is u '(c1)Pit. At the end of period t, rt is revealed and Pi , t+ 1 and 1Ti , t+ 1 become known. Hence , the consumer obtains
Applications in finance
249
(1 2.1 1 )
extra utils at the beginning of t + 1 if he collects Tr;. t+ 1 and sells the share ex dividend at P;,t+ But these utils are uncertain and are received one period into the future. The expected present value of utility gained at t + 1 is 1 •
(3E r { u , (cr + 1 ) ( P;. t + 1 + Tr;. t + 1 ) }
·
(12.1 2)
Consumer equilibrium in the market for asset i requires that the marginal oppor tunity cost at date t be greater than or equal to the present value of the marginal benefit of dividends and ex-dividend sale price at date t + 1 : (I 2.13) � (3E1 { u ' (ct + 1 ) (Tr;. 1+ 1 + P;. t + 1 )}, a .e., P;1u ' (c1 ) (1 2.14) P;1u ' (c1)Z;r = (3E1 { u '(c,+ 1 ) (Tr;, t + 1 + P; , t+ l )}Z;r ., a.e. Similar reasoning in the rental market yields u ' (c1)
(12.15) ( 1 2.16)
It would be nice if the first-order necessary conditions (1 2.13)-(1 2.1 6) char acterized consumer optima. But it is well known that a transversality condition at infinity is also needed to completely characterize optima. Recent work by Benveniste and Scheinkman (1977) allows us to prove Lemma 1 2. 1 . Suppose A. I of section 15 in Chapter 3 holds and assume that #' is such that W(),,, t) -+ 0 as t -+ 0, where W()11, t) is defined by W(y1, t) = maximum E1
00
1
Lt {f- u (cs)
s=
subject to ( 1 2.3)- (12.5) with t replaced by s and 1 replaced by t. Recall that y1 denotes the r.h.s. of (12.3). Then, given { P;, } ;: 1 , { Tr;1}�= 1 , { R;r}�= 1 , i = 1 , 2, . . . , N, optimum solutions { Z;, }�= , { x;r}�= I , i = 1 , 2, ..., N, {c1};'= 1 , { x1};'= 1 characterized by ( 1 2 . 1 3) and ( 1 2 . 1 I4) and TVCoo (equity market) lim E1 { (3 1 - 1 u '(c1) P1 · Z1 } = 0, ( 1 2. 1 7) 1 TVCoo (capital market) 1lim (12.18) 1 { (3 - 1 u '(c1 )x1} = 0. E -+oo f-+ oo
Stochastic methods in economics and finance
250
Proof. Suppose {Z1 }, { c1 } and {.X1 } satisfy {1 2.13)-(12.17) and let { Z1 }, {c1} and {x 1 } be any other collection of stochastic processes satisfying the same ini tial conditions and (I 2.3)-(1 2.5). Compute for each T an upper bound to the shortfall: E1
{ f (31- 1 u(c1) - f (3' - 1 u(C,) } { J (3'- ' u '(C,) (c, - c,)} jl f1 (31- 1 u '(C1) [ rr1 Z1_ 1 + P1 Z1 _ 1 + -� R;rx; 1_ 1 1=
<;; E ,
= E1
t= 1
1
,
t=
•
I= 1
N
]}
+ P, . z, + x,
E1
(12.20)
•
- P1 · Z1 - x1 - rr1 · Z1_ 1 - P1 · Z1_ 1 - t= _1: R;,x; 1
=
(12. 19)
•
•
1_ 1
(1 2.21)
{(3T- t u'(Cr) [Pr (Zr - Zr ) +Xr - xr] } •
(1 2.22) (1 2.23)
Here eqs. (12.13) - {12.16) were used to telescope out the middle terms in the series of the r .h.s. of (12.21 ). The terms corresponding to date cancel each other because the initial con ditions are the same. Hence only the terms of the r .h.s. of (1 2.22) remain of all the terms of the r.h.s. of {12.20) and (12.2 1 ). That the r.h.s. of {12.22) has an asymptotic upper bound of zero follows from (12.17), (12.18) and the non negativity of Zr and X r . This shows that (12.13) - (12.18) imply optimality. Notice that no assumptions on W(y 1 , t) are needed to get this side of the proof. Now let {Z,}, {c1 } and {x1 } be optimal given { P1 , R1 , rr1 }. Since u ' (O) implies that c1 > 0 a.e. and W is differentiable at y1 we have by concavity of W and u � 0 that
l
= oo
W(y 1 , t) � W(y 1 , t) - W(y1/2 , t) � W' (y 1, t) y1/2 = {3
1 1 - u ' (c1)y1/ 2. (12.24)
Hence,
(1 2.25) But
251
Applications in finance
(1 2.26) so that by the first-order necessary conditions
(1 2.27) because in more detail
( 1 2. 1 3)- (12.16) imply (1 2.28)
X; ,1_ 1 u'(c, _ 1 ) = f3E1_ 1 [u' (c, )R;,l x;.r - 1 , 1 ( r x, _ 1 u'(c1 _ 1 ) = E,_ ,
[
u '(c, )
{f
Ru x; , r- •
�
( 1 2.29)
•
Pi,t- l u' (c,_ 1 ) Z;,,_ 1 = f3E,_ 1 [u' (c1 ) (1Tit + P;1 ) Z;,,_ 1 ] ,
Hence, because P1_ 1
(12.30)
� 0, z,_ 1 � 0, (12.25) implies (12.32) ( 1 2.33)
as was to be shown.
(1953) and the second part is ( 1977). Lemma 1 2 . 1 is important be
The first part of this argument follows Malinvaud taken from Benveniste and Scheinkman cause it characterizes consumer optima. The assu mption that
E 1 W(y 1, t) -+- 0,
A general sufficient condition on 9 for
as
t -+-
oo ,
restrains� It requires thatY'
1
be such that along any path in9utils cannot grow faster than (3 on the average.
E 1 W(y1 , t) � 0
can be given by what
should be a straightforward extension of the methods of Brock and Gale ( 1969)
and McFadden
(1973) to our setup.
An obvious sufficient condition is that the utility function be bounded, i.e. there are numbers B < B such that for all c
� 0, !l � u (c) � ii.
We note that the method used here of introducing a stock market into this
Stochastic methods in economics and finance
252
type of model where an investment decision Scheinkman ( 1 977) in the certainty case. Next, we establish a basic lemma. Lemma 12.2. (i) Let X = ({ c1 }�= 1 , { x;1};: 1 , ( 1 5 . 1 ) of Chapter 3 and defme
is
present was first developed by
{ x1 };: 1 ) solve the optimal growth problem ( 1 2.34)
Then let { Pit};: 1 , i = 1 , 2, ... , N, satisfy {1 2.30) and ( 1 2.32). Put
( 1 2.35) Then ({ P;r }�= 1 , { R ;r };: 1 , { 7T;1};: . , { x;1};:p {Z;,}�= 1 , i = 1 , 2, ... , N, {c1 };: 1 , { :X1 }�= 1 ) = .:jf is an R.E.E. (ii) Let .Jf be an R.E.E. Then X solves the optimal growth problem ( 1 5 . 1 ) of Chapter 3. Proof. Let X solve the optimal growth problem ( 1 5 .1 ). It is obvious that Ji satis fies the fust-order necessary conditions for an R.E.E. by its very defmition. What is at issue is the TVCoo { 1 2. 17) and { 1 2. 1 8). Put 00
s 1 V(x1_ 1 , t - 1 ) = maximum E1 � (3 - u (cs) s= t
( 1 2.36)
subject to N
Cs + Xs = .�1 J:,. (x,· s- l , rs- 1 ), N
J=
•
{ 1 2 . 37)
.� x,.s = xs; x,.s � O, i = l , 2, ... , N, cs � O, xs � O, J= 1 s=
t, t + 1 , ... , x1_ 1 given.
{ 1 2.38)
Then following a similar argument as that in ( 1 2.24)- ( 1 2.33) we have, since u is bounded, that for any x1 � O, V(x 1, t) � 0 as t � oo and
V(x1, t) � V(x 1 , t) - V(x1/2, t) � V' (x1, t)x1/2 1 = E1 [.6 u' (c1+ 1 ) f;' (x;1 , r1) x1/2 ] 1 1 = E 1 [.6 - u' (c1)x1/2] � 0. ( 1 2.39)
253
Applications in finance
Since the l.h.s. of ( 1 2.39) must go to zero the r.h.s. must also. Hence E 1 [p 1 - 1 u' (c1) x1 ]
-+
0
as t -+ oo
( 1 2.40)
along any optimum program. This establishes ( 1 2. 1 8). What about ( 1 2 . I 7)? Here the stochastic process { Pu}�= 1 was assumed to have been constructed from the quantity side of the model by use of ( 1 2.30) so that the TVCoo ( 1 2.32) was satisfied. Hence TVCoo ( 1 2 . 1 7) is satisfied by the very construction of { P;, };: 1 . This establishes that (i) implies (ii). In showing that (ii) implies (i) it is clear that the first-order necessary condi tions for the quantity side of an R.E.E. boil down to the first-order conditions for the optimal growth problem. What must be established is the TVCoo ( 1 2.40). But this follows from ( 1 2. 1 8) of lemma 1 2. 1 . This ends the proof of lemma 1 2.2. Lemma 1 2.2 shows that the quantity side of any competitive equilibrium may be manufactured from solutions to the growth problem. This fact will enable us to identify the Ross prices. Furthermore, it will be used in the existence proof of an asset pricing function which is developed in the next section. We now return to the discussion of the relationship between the growth model of section 1 5 of Chapter 3 and the risk prices of Ross. This will facilitate the economic interpre tation of an R.E.E. stochastic process ({ R;t r;: 1 , { P;r };: I { 7T;r };: 1 ). Drop the upper bars off equilibrium quantities from this point on in order to case the notation. Assume that conditions are such that all asset prices are posi tive with probability I in equilibrium. Then Zu = I w.p.l and from ( 1 2 .30) we get for each t , '
( 1 2.4 1 ) Next, profit maximization implies
f/ (x;, , r1) = R;, t+ 1 , 7T;, t+ 1 = f; (x;1, r1) - f;' (x;, r1 ) x;, .
( 1 2.42)
u' (c1) = (j E, [u' (ct+ 1 ) f;' (x;1 , r1) ] .
( 1 2.43)
Turning to the rental market suppose that all processes are used w.p.l. Then (1 2.42) and ( 1 2.28) give us for each i and r:
Observe that we are not entitled to write the returns Z;r defined by ( 1 2.41) in the linear Ross form ( 1 1 . 1 ) unless P;(y,+ 1 ) is linear inyt+ l ' An example is pre sented in section 1 6 where P; (y1 + 1 ) turns out to be linear iny1+ 1 • But first we must show that an asset pricing function exists. This is done in the next section.
Stochastic methods in economics and finance
254 13.
Existence of an asset pricing function
Since in equilibrium the quantity side of the asset pricing model of section 1 2 is the same as the N process growth model of section 1 5 of Chapter 3, we therefore may use the facts collected in section 1 5 to prove the existence of an asset pricing function P(y) in much the same way as in Lucas ( 1 978). We begin by making an assumption. Assumption 1. Assume for all r E R: (a) f;'(O, r) = + oo, i = 1 , 2, ... , N. (b) 7T;(x, r) =f; (x, r) - f;'(x, r)x > 0 for all
x > 0.
Assumption 1 (a) implies that ( 1 2. 1 5) holds with equality in equilibrium. Also, assumption 1 (b) implies ( 1 2. 1 3) holds with equality in equilibrium. Let us search, as does Lucas { 1 978), for a bounded continuous function P;(y) such that in equilibrium the following holds:
( 13.1) Convert the foregoing problem into a flXed point problem. Note first from section 1 5 of Chapter 3 that
u' (c,) = U'(y1), t = 1 , 2, ... , rr;,
( 1 3 .2 )
t+ f;(x;, , rt) - f;' (x;, , r,) X;, = 1T;(Xu, r1) = 7T;(11;(X,) x1, r1) = 7T; [11;(h(y,))h (y1), r, ] 1
= l;(y,, r,), yt+ 1 =
N
.:r
J=l
N
Put
=
( 1 3.3)
. :r
J= I
!.1· (111- (x,) xt , r,) J:1.(111· (h(yt)) h (y,). r1] = Y(y1, r,).
{ 1 3.4)
G;(y1) = (3 f U' [ Y (y,, r)] l;(y,, r) JJ(dr), R
F;(y,) = P;(y,) U' (y,),
( 1 3.5)
(T1 F1) (yt) = G;(yc) + (3 f F; [ Y(y,, r)] JJ(dr). R
{ 1 3.6)
Applications in finance
255
Then for each i, ( 1 3 . 1 ) may be written as
(13.7) Problem ( 1 3.7) is a fixed point problem in that we search for a function F; that remains fixed under operator T;. In order to use the contraction mapping theorem to find a fixed point Fj we must show first that T; sends the class of bounded continuous functions on [0, oo), call it C[O, oo}, into itself. The results of section 1 5 established that all of the functions listed in ( 1 3 .2) -( 1 3 .6) are continuous inyt. We need Lemma 13.1. If U(y) is bounded on [0, oo) then G;(Y) is bounded.
Proof.
First by concavity of U we have
U(y) - U(O) � U' (y) (y - 0} = U ' (y)y.
( 1 3.8)
Hence, there is B such that U'(y)y � B
for all y E [0, oo) .
(1 3.9)
Secondly, I U' [ Y(y , r)] J;(y , r) J.L (d r}
R
=
I { U ' [ Y(y , r)] Y(y , r) J; (y, r)/Y(y , r) }p (dr)
R
� B I [ J;(y, r}/ Y(y, r)]J.L(d r) � B, since f;' � 0 implies
f; - f;'Xit = J; � f; ;
N
Y = . � J;1.; J;/ Y � J . = J J
Thus, G; is bounded by (jB. This ends the proof. Next, we show that if
( 1 3. 10) is chosen to be the norm on C[O, oo) , then Lemma 13.2.
T; is a contraction with modulus (3.
T;: C[O, oo) � C[O, oo) is a contraction with modulus (3.
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Stochastic methods in economics and finance
Proof.
We must show that for any two clements, F and G in C[O, oo ],
( 1 3. 1 1 )
Now for y E [0, oo) we have
I T;F(y) - T;G(y) I
=
13 1 f(F[ Y(y, r)] - G[ Y{y, r)]}p(dr) I
� t3 f I F(y') - G(y') I p(dr) � t3 J , sup I F(y' ) - G(y' ) I p(dr) y E ( O . oo) =
t3 11 F - G II .
( 1 3 . 1 2)
Take the supremum of the l.h.s. of ( 1 3 . 1 2) to get
( 1 3 . 1 3) This ends the proof. Theorem 1 3. 1 . For each i there exists exactly one asset pricing function of the form P; {y), where P; E C[O, oo).
Apply the contraction mapping theorem to produce a f1Xed point F;{y) E C[O, oo ). Put Proof.
P; (t) = F; (y)/ u ' (y).
{ 1 3 . 1 4)
It is clear that P; (y) satisfies { 1 3 . 1 ). Furthermore, the very definition of T; and P; (y) that satisfies ( 1 3 . 1 ) is such that P; ( ) U ' ( ) = F; ( ) is a flXed point of T; . This ends the proof. ·
·
·
Note that assumption J (a) is not needed for the existence theorem. However, assumption I (b) is needed in the theorem so that { 1 3 .1) holds with equality. 14. Certainty equivalence formulae
What we propose to do in this section is to use the asset pricing model of section 1 2 to construct a version of the price-equals-present-value-of-dividends (P = PVD)
Applications in finance
257
formula for the pricing of common stocks. In equilibrium our formula must hold. The formula will be derived from the following special case of the model of section 1 2 :
( ) 4.1) In eq. ( 14.1} observe that {cS 1 }�= t is an independent and identically distri buted sequence of random variables with zero mean and finite variance a2 • The numbers A ? and A / and the random variables 5 , are assumed to satisfy: there is e0 > 0 such that for each t
( 14.2) Optimum profits arc given by
(1 4.3) In order to shorten the notational burden in the calculations below, put
( 1 4.4) ( 14.5) rr; (Xit , r,) = bi t + V;, 5 ,,
( 1 4.6)
where
( 14. 7) All quantities will be evaluated at equilibrium levels unless otherwise noted. The notation is meant to be suggestive with D; , denoting average dividends or profits expected at date t, V;, denoting the coefficient of variability of profits with re s�cct to the process {S 1};': 1 , and so forth. For a specific parable think of the {cS t}�= 1 process as the market. Then production and profits in all industries i = I , 2, ... , N are affected by the market. High values of cS t correspond to booms and low values correspond to slumps. Industries i with A/ > 0 are procyclical. Those with A/ < 0 are countercyclical, and those with A / = 0 are acyclical.
Stochastic methods ill economics and finance
258
Assumption I . There is at least one industry, call it N, that is acyclical. The Nth industry will be called risk free. For emphasis we will sometimes say that N is systematic risk free.
In order that all industries be active in equilibrium and that output remain bound· ed we shall assume the following: Assumption 2. (i)
J; (0) = oo , i = I , 2, ..., N.
(ii)
J; (oo) = 0, i = I , 2, ..., N.
Assumption 2(i) guarantees that all x; 1 > 0 along an equilibrium. Assumption 2(ii) implies there is a bound B such that xi t � B w.p.l for all i and t. Although concavity of f(x) and f(O) = 0 imply that optimum profits are non· negative we shall require that profits are positive for each x > 0, i.e. Assumption 3. For all x > 0, 1l'; (x) =l;(x) -J;' (x) x > 0.
Assumption 3 will be used to show that equity prices are positive in equilibrium. By the first.order necessary conditions of equilibrium ( 1 2 . 1 3) - (12.16), ( 14.2), assumption 2(i) and assumption 3, it follows that (14.8)
u' (c1) = (3 E1 [u' (ct+ 1 ) R ;, t+ 1 ] = (3E,u'(c1+ 1 ) JJ;r + f3E1[u'(ct+ 1 ) 81 ] a;,, a.e.
(14.9)
The r.h.s. of (14.9) follows from (14.5). It is clear from assumption (3) that equity prices are positive since 1l'; , t+ 1 is positive w .p.l. Hence, both ( 14.8) and (14.9) are equalities and Z;, = 1 . The P = PDV formula will be derived from (1 4.8) and (1 4.9) by recursion. Use (14.3) and (14.6) to get (14.10) In order to shorten notation put (14.10) we get
u'(c,+ 1 ) = u�+ l for all
t. From (1 4.8) and
(14.1 1 ) Notice that
JJ;1 , a;1 D;t ' V;r are, in theory at least, observable. Hence, if we re·
259
Applications in finance
curse ( 1 4. 1 1 ) forward by replacing t by {14.1 1), we can use (14.9) to solve for
Etmt = mt =
,3 Etu�+ 1 u ,t {3u�+ 1 ut,
in terms of J.L;, we get
;
a ,
Etnt llt =
=
t + 1 in { l 4.8)and insert the result into
f3Et(u�+ 1 � t) u,
,
(14. 1 2)
,
ut
and build up a P = PDV formula for P;,. To do so from (14.1 1 )
P;r = Etmtl5it + Etnt Vit + !3Et[u�+ 1 Pi,t+ 1 ]/u� =
=
E,m,l5it + E,n, vit + E, {m,[Et+ 1 mf+ l l5i, t+ I + E,+ 1 nf+ 1 V;, t+ 1 + f3Et+1 (u�+2 Pi,t+2)/u�+ 1 ]} E,m,J5it + E,n, vit + E, [mt E,+ m,+ 1 D;, t+ 1 + m, E,+ 1 n,+ I vi,t+ l ] + ... J
+ E1(mtEt+1 m,+ 1 . .. Et+r(mt+TDi,t+ T)) +
+
E, [mt Ef+ 1 mt+ l . .. EI+ T- 1 mt+T- 1 EI+ T nt+ T vi,t+ T] Et [mtEt+ 1 m,+ 1 Et+ r (mt+ rPi,t+T+ )].
( 1 4 . 1 3)
1
•••
Assumption 4. The utility function u ( ) is such that for all {Pit ' rru, R;t };: 1 , i = 1 , 2, . .., N, the TVCoo is necessary for a consumer's maximum. Now the TVCoo implies that ·
( 1 4. 14) By ( 14.9) we get for each t, { 1 4.15) Therefore if a'tv, = 0, ( 14. 1 5) implies (14.16) Note here that
At is the classical risk adjusted rate of return from portfolio theory.
260
Stochastic methods in economics and finance
Also, p.�1 is principal plus interest obtained by employing a marginal unit in process N. It is important to observe that A1 is independent of i. Furthermore, the Ross risk price "A1 is determined by "A1 = {3A1• This follows from ( 1 1 .1 6) and (14.16). Turn now to the K-factor case. In the K-factor case put
f; (xu , r1) =l;(x;1 )
(f
k=O
A;k B�+ 1
)
, B�+ 1 = 0, A? > 0.
(14.17)
Put the same assumptions on the data as in section 13. Then, as in (1 4.4), ( 1 4.5) and (14.6), we may write K + 1: = f;(xit, rt ) P.u k = l Ok;t 0- kt+ 1 ,
(14. 1 8)
K
f;' (x; t • r,) = P.;t + 1: O;t B t+ l , k=l 1r;(X;1 , r1)
=
K
+ 1:
jjit k=l
(14.19)
Jlj, o kt+ l - rri, t+ l , k -
=
(1 4.20)
where the entities in (14.18)-(14.20) are defined as in (1 4.7). Keep the same assumptions as above. Then (14.8) and ( 14.9) become
(14.2 1 )
( 14.22) Define
(14.23) Then letting N be a risk-free process, i.e. a�� = 0 for all k and t, we get
(14.24) and for each i
( 14.25)
Applications in finance
261
Hence, from (1 4.25) it follows that K
(JJ.';v, - J.I.;1)!JJ.';v1 =k=1;l a;: (E1n� ) , i =
I , 2, ... , N.
Here it is assumed that a unique solution of ( 1 4.26) for fined by
k-
E, nt
= -
Akf
u,
1
J.I.Nr·
( 14.26)
E1n�
exists and is de
(14.27)
Note also that the Ross price of systematic risk k, A.k 1, satisfies A.k 1 = f3 !:l� which follows from ( 1 1 .16) and ( 14.27). Let
then (14.8) becomes
Hence,
( 1 4.28) Eq. (1 4.28) says that investing Pit in the stock market must give the same ex
pected return after paying for the services of risk bearing as investing it in the risk-free process. It states that the stock market is a fair game taking into account the opportunity cost of funds and the cost of risk bearing. Clearly �it = 0, and p.';v1 = 1 is necessary for a martingale. For specific preferences eq. (14.28) is test able and its violation would signal market inefficiency in our model world. It is worth pointing out here that if the random variable
( 14.29) is independent of Pi.t+ t at date t, then (14.16) implies that (14.28) may be re written as
( 14.30)
262
Stochastic methods in economics and finance
Eq. (14.30) is directly testable since it contains no subjective entities, unlike (1 4.28). The problem of deriving equations like (14.30) that contain no entities that are subjective and hence are directly testable remains open. The assumption of independence is way too strong. Perhaps weaker assumptions exist that will produce a formula analogous to O 4.30) that holds in at least an approximate sense.
15. A testable fonnuJa In what follows a formula is developed that is directly testable under the hypo thesis of linearity of the asset pricing functions P;(y). An example where P;(y) is linear is given in the following section. Theorem 15.1. Assume assumptions 1 -4 of section 14. Furthermore, assume that there are constants K; and L; such that
P;(y) = K;y + L;, i = 1 , 2, .., N. .
(15.1)
Then, for each t and i
(15.2) must hold.
(15.3) (15.4) where E,P; , t+ I ' E, 1T;, t+ I , st and vt do not depend upon yt+ I but depend on (xu, . , xN1) only. In order to establish (15.2) it must be shown that (15.3) and (15.4) hold. By (14.17) and the defmition of yt + 1 we have ..
(15.5)
263
Applications in finance
Hence, yt+ 1 is a linear combination of the stocks S :+ 1 with weights that depend only upon (xit ' ... , xN1). So also is Pi, t+ l ' Thus, ( 1 5 .3) holds for appropriate st since P; . t+ I is linear in yI+ I . Eq. ( 14.20) is identical to ( 1 5 .4). Divide both sides of (I 4.2 1 ) by u� to get ( 1 5 .6) Put, using {14.23), -
k k m1 - Et m1, -n1 - E1 n , . =
(15.7)
=
By ( 1 5 .6) and ( 1 5 .7) we have
- �K sk -k k -k 11 P Vit t + nz, + k� i t n, =l ir
·
But (14.24) and ( 14.27) imply
This ends the proof. It is worth pointing out that although ( 1 5 .2) contains no subjective entities and, hence, is directly testable it was derived under the strong hypothesis of linearity of the asset pricing function P;(y t+ 1 ). The linearity hypothesis was needed to be able to write the one-period returns Z;t to holding asset i in the linear form (1 1 .2) of Ross. The linear form of Zu was used, in turn, to derive ( 1 5 .2). We sus pect that strong conditions will be required on utility and technology to be able to write equilibrium asset returns in the form ( 1 1 .2). Hence ( 1 5 .2) is not gen eral; it holds only as a linear approximation. The economic content of ( 1 5 .2) is compelling. It is a standard no arbitrage profits condition. The price of risk bearing over the time interval [t, t + 1 ] sells for ll: per unit of risk of type k. At date t risk emerges from two sources: (i) 1T;. t+ 1 and (ii) P; t+ 1 • Profits contain vt units of risk of type k. The price of . stock i at date t + 1 contains S;� units of risk of type k. Hence, the total cost of risk bearing from all sources of risk for all types of risk is
264
Stochastic methods in economics and finance
Thus, ( 1 5 .2) just says that the risk-free earnings from an investment of Pit must equal the sum of risk adjusted sale value of stock i at date t + 1 and risk adjusted profits. 16. An example
In this section we present a solved example that illustrates the ideas presented in sections 1 1 - 1 5 . Let the data be given by u (c) = log c,
(16.1)
f; (x;, r) = A;(r)xf, i = 1 , 2, . . , N, .
0
(1 6.2)
We shall assume that for all i,
A; (r) > 0
for all r E R,
and A ; (r) is continuous in r. Since R is compact each A;(r) has a positive lower bound �; > 0. First-order necessary conditions become, for all t,
( 1 6.3)
i
=
1 , 2, . . ., N,
( 16.4)
(16.5)
Conjecture an optimum solution o f the form N
Cr = ( l - A)yt ; Xr = Ayt; Xu = fl;Xt;
where
. �1 f/1 = 1 ,
•=
( 16.6)
265
Applications in finance
A > O ; 17; � 0,
i = 1 , 2 , ... , N.
( 16.7)
Insert (1 6.6) into ( 1 6.3) and (1 6.4), solve for A, {171 }� 1 and check that (16.5) is satisfied. Doing this we get
1 � (ja E, ( 1 - A)y1 if and only if
{
{ \
A; (r1) -1 x ,) A (r � i i�t
I - � (ja E, ----N Yt
{
if and only if
1
- � (j a: E,
Y,
17f - I X� - I
yt
� (ja: E,
if and only if
{
l x -
�
}
A (rt) i
_ _ _ _ _ _ _ N _
_1; J= 1
A1- (r1) 171� x�
if and only if _ I
A; (r1)x�- J
(1 - A Yt+ t
N
1;1 j=
A;(r ) A . (rt) 17� I
,
I
}
11o: - t ;x
t
}
},
( 1 6.8)
,
,
-
f3a:171� - 1 rl.
---x,
( 16.9) Set ( 16.9) aside for the moment. From (16.4), following the same steps that we used to get (16.9), we obtain xI. ,
- =
Yt
0:
f3a:TJ·I r.,I
(16.10)
if and only if
(16. 1 1 ) Hence (1 6.9) holds with equality for all t and i. Since it is weU known and is easy to see that for N = I
Stochastic methods in economics and finance
266
'A = {3a,
it is natural to conjecture for N � 1 that 'A = f3a ;
fir- • r; = 1 .
;=
1, 2
• ...
( 1 6. 1 2)
, N,
and test ( 16.5). If (16.12) satisfies ( 16.5) then we have found an optimum solu tion and hence the unique optimum solution. Continuing, we have -
_
.,.., 1 - a .
' lj
'
N
�
i= l
fl,·
= 1.
(16. 13)
It can be shown that (16.13) has a unique solution {f/;};: 1 • It is easy to check that (1 6.6) with 'A = a{3, fl; = fl; , i = 1 , 2, ... , N, generates a solution that not only satisfies (1 6.3) and (16.4) by construction but also satis fies ( 16 .5). We leave this to the reader. Let us use the solution to calculate an example of an equilibrium asset price function from the work of section 1 2 . From ( 1 2.35) and ( 1 2.32) we get
(16.14)
(16.15) Hence, the first-order necessary condition for an asset pricing function of the form P;, = P; (y,) becomes for u (c) = log c, using c, = (1 - 'A.)y, : (16.16) Eqs. ( 1 6.15) and 06.16) give us
P; (y,) !y,
=
tlE,
{(I
- a) A; (r,) 'I;' X�
I
[ i¥.
J
A; (r,) 11/ x�
Applications in finance
+ P; (y t+l ) fyt+ =
Here by ( 1 6 . 1 3) 1i1 =
E,
�;
26 7
I}
{3(1 - o:) 17; + 13 E, [P; (yt+ 1 )/yt+ 1 ), i = 1 , 2, ... , N. (16.17)
j Ci Ai (r,) ilj) J
(r,) 11;"
·
(16.18)
The system of equations (16.17) is in a particularly suitable form for the ap plication of the contraction mapping theorem to produce a unique ftxed point P(y) = (P.(y), ... , PN (y)) that solves ( 1 6 . 1 7). Rather than do this we just con jecture a solution of the form
P; (y) = K;y, i = 1 , 2, ..., N,
( 1 6.19)
and fmd i( from ( 1 6.19) by equating coefficients. Obviously from (16.19) K; satisfies
K; = f3 ( 1 - a) 11; + 13 K;, i = 1 , 2, ..., N,
( 16.20)
so that
( 1 6.2 1 ) Since the r.h.s. of(16.19) is a contraction o f modulus {3 on the space of bound ed continuous functions on [0, oo ) with values in R N , the solution ( 1 6.19) is the only solution such that each P; (y)fy is bounded and continuous on [0, oo ). We now have a solved example. It is interesting to examine the dependence of P; (y) on the problem data from ( 1 6.19) and ( 16.21). First, in the one-asset case we fmd 11N = 1 from ( 1 6. 1 8) so that
P(y) =
{3 ( 1 - o:)y. 1 - {3
( 16.22)
Hence, (i) the asset price decreases as the elasticity of output with respect to capital input increases; (ii) the variance of output has no effect on the stock price; and (iii) the asset price increases when (3 increases. Result (i) follows because profit's share of national output is inversely re lated to a. One would expect (ii) from the log utility function. One would ex-
Stochastic methods in economics and finance
268
pect (iii) because as (3 increases the future is worth more relative to the present and hence savings should increase thereby forcing asset prices to rise. Furthermore, (16.22) says that asset price increases as current available in come y increases. Secondly, in the multi-asset deterministic case we have
P; (y) =
(3 I
_
(3
( 1 - o:) 11;Y, i = I , 2, ... , N.
(1 6.23)
We can see that if the coefficient A; measures the productivity of firm i using the common technology x (k so that output of i is A;x(k, then firms that are relatively more productive bear higher relative prices for their stock. Absolute productivity does not affect relative prices. This is so because 11; is homogeneous of degree zero in (A 1 , ••• , A N). This is again one of those results that looks intuitively clear after hindsight has been applied. The consumers in this economy have no other alternative but to lease capital or to invest in stock in the N firms. Hence, if the productivity of all of them is halved the constellation of asset price relatives will not change al though output will drop. This type of result is specific to the log utility and Cobb-Douglas production technologies. 1 7. Miscellaneous applications and exercises ( I ) In the Black-Scholes application of section 3 suppose that the stock price
S upon which the option is written is given by
(17.1) instead of eq. (3 .7). Observe that in ( 17.1) we have two correlated noises with Derive the Black-Scholes equation in this case and conjecture what would hap pen to the option as p 1 or p � - 1 . (2) More on Black-Scholes: The Black-Scholes option pricing model of section 3 shows that the price of an option F is a function of the stock price, S, upon which the option is written, the exercise price E, the time to maturity of the option ; , the riskless rate of interest r and the instantaneous variance rate on the stock price a2 • Intuitively we would expect: �
Applications in /itUJnce
269
> 0, 0, > 0, > 0, > 0,
(a) aF;as i.e. as the stock price rises so does the option price, (b) aFjaE < i.e. as the exercise price rises the option price falls, (c) a Fja; i.e. as the time to maturity increases the price of the option rises, i.e. as the riskless rate of interest rises so does the option (d) aFja r price, (e) aF;aa2 i.e. as the variance rate rises so does the price of the option. Check to see if (a)-(e) are true. See Merton ( 1 973a) and Smith (1 976). (3) Merton's costate equation: Consider Merton's model of section 4 in the special case of one risky asset and one risk-free asset, where max E0
{J
u (C,
0
t)dt + B( W(T))
}
a r) Wd t + (rW C)d t + Wt wa. dz l ' dPtfP1 = a1 d t a dz 1 , W0 , P1 (0) given. Here a1 and a1 are independent of P1 and t, and denotes the percentage of W in the risky asset and the risk-free asset pays interest rate r. Let p(t) denote the current value of the costate where the state is W(t). Use the Hamiltonian and d w = Wt (
.-
-
+
1
w1
the costate equation from Chapter 3 plus the necessary conditions that the con trols C and w 1 must satisfy and write the costate in the form: (17.2)
a1
Find the functions J.l.p and aP . What do you get if and a 1 are dependent on time? Stochastically integrate (17 .2). Can you fmd the solution for in terms ofp0? How is the value determined? Does p (t), given P o , depend upon (C, t), W0 , T and B(W(T))? Does homogeneity of beliefs on r and a1 imply homo geneity of behavior of the investors regardless of their tastes, age and initial wealth? How would you qualify this statement? (4) Consider a specific case developed by Vasicek (1977) as an illustration of his general model presented in section 8 . Assume that the market price of risk q is constant and independent of time and the level of the spot rate. We follows the process write q = q. Next assume that the spot rate
p(D
(t, r)
(t, r) dr = a('y-r)dt + pdz, a > 0,
p(t)
a1 ,
U
r(t)
as a specific case of (8.4). Eq. (17 .3) is called the
(17 .3)
Ornstein-Uhlenbeck process
Stochastic methods in economics and finance
270
and when et > 0 it is called in the finance literature the elastic random walk. The great advantage of this process is that it has a stationary distribution. Under the above specific assumptions find the solution of the term structure equation (8.10) and study its economic content. See Vasicek (1977).
(5) A simple generalization of Constantinides' (1978) model presented in section 9 is as follows. Suppose that the cash flow generated by the project in the time interval ( t, t + d t) is stochastic given by cd t + s dz 2 , with c = c (x, t) and s = s(x, t) and z 2 is a Wiener process. Rewrite (9 .1) as dx = 11dt + odz1 and assume that z 1 and z2 are correlated. Under these assumptions find the new equations which replace (9.3) and (9.8) and interpret the resulting rule. (6) Consider the following tableau of asset returns: dx;/x; = et1 d t
+
o;dz,
i = 1 , 2, . . , N. .
(17 .4)
Here dx;Jx1 is instantaneous return on stock i. Note that the shock dz is com mon to all stocks. Use the hypothesis that all dz-risk-free portfolios earn the risk free rate of return, r, to show
(I 7 .5) for all i ,j. Assume that oN
=
0 and etN = r.
Now introduce inflation. Let the pric,e level, P, follow dP/P = 1rdt + od w. Let the returns in ( 1 7 .4) above be nominal. Assume that E,d w dz out, using stochastic calculus, the real return d(x;/P) X;/P
=
( 1 7 .6)
= pdt. Work
( ) d t + ( )dz + ( ) d w.
:= A ; d t + B;dz + C1 dw,
(17.7)
i.e. flll in the ( ) . Next, use a hedging and equilibrium argument analogous to that of Black Scholes to derive a relationship between the coefficients A1, B1 and C1 that must
Applications in finance
271
hold in equilibrium. Feel free to assume existence of dz and dw risk-free assets. It is usually hypothesized that stock returns are negatively related to anticipated inflation. Can you find a { o:1, a1 }� 1 1T, a, p structure consistent with this hypo thesis in the equilibrium relationship among A ; , B1 and C1? (7) Consider the following continuous time analogue of the model presented in section 1 5 of Chapter 3 and sections 1 1 - 1 6 of this chapter. A representative consumer solves ,
max E0
00
J c-f3t u (c (t))dt
0 N
(17.8)
1: x1(t) = x(t) 1
subject to N
N
N
1
1
I
c(t)dt + dx + 1: P1dE1 = 1: d7T;E; + � dR1x; , x (O) = x0 given; E;(O) = I , i = l , 2, ... N, where dRI. = A I.dt + B.dz, I
(17.9)
d7T; = a1 d t + b1dz, i = 1 , 2, . ., N. .
{17 .10)
The firm's problem is given by max d7r.I = max [J; l".(xI. ) - xI.AI. ] d t + [al. (xI. ) - xI.B.] I dz
(17.1 1 )
to yield f;' (x1 ) = A1 ; a;(x; ) = B; ,
( 1 7 . 1 2}
a1 = f; (x; ) - x1 J; (x; ); b; = a; (X; ) - x1a; (x; ).
( 1 7 . 1 3)
Here E1 (t) denotes the number of shares of firm i at t, x(t) is real output at time t and c(t)dt is amount of consumption during (t, t + dt). Ownership of E; (t) shares at t gives right to d7T; units of output at t + dt. Investment of x1(t) in pro cess i at t gives right to dR1 new output at t + d t as well as the original x1(t). At each date t, x (t) amount of output is allocated across i = 1 , 2, . . . , N firms with
Stochastic methods in ecotlomics andfinance
272
rentals dR; to be received at t + d t conditioned upon what happens at t + dt. Equilibrium is defined as in section 1 2 of this chapter. Recall that in equilibrium the representative consumer faces d7T;, dR; and P; parametrically and solves (17 .8) for c, x, {x; 1 and { £; 1 • Approximate problem (17 .8) by its discrete time analogue, take limits and drop higher-order terms to derive the following conditions of optimality:
}�
}�
(3 - E , - E,
( t ) (dr) ( u ) ( t ) = ( Z.); ( : ; ) du ' u d I
1
du u d I
=E , E,
dR;
d
+E
+E
,
du ' dR;
,
I
dt
du .z. u d
,
(17.14) (17.15)
,
In equilibrium, the instantaneous means and standard deviations of dR; and d7T; can be written as time independent functions of x(t) at each t. Explain why this is so. Similarly, explain why P; (t) may be written as P;(x(t)). Suppose next that the Nth process is risk-free , i.e. aN(x N) = 0. Put r(x) =flv(xN) in equilibrium. Use Ito's lemma on P; (x) to derive the Lintner- Sharpe certainty equivalence formula
r(x)P;(x) = P;x (x)J.l (X) + � Pixx (x)a2 (x) + a;(x) +
[P;x (x)a(x) + b;(x)J E,
( ) du'dz u' d 1
,
(17. 16)
where dx = J.l (x) d t + a(x)dz. Explain why (17.16) makes economic sense. 18. Further remarks and references
In this chapter we have presented a number of applications to illustrate various stochastic techniques. The number of research papers in finance that usc such stochastic techniques has increased significantly over the last two decades and there are several recent books of readings which have collected major contribu tions such as Szego and Shell (1972), Ziemba and Vickson (1975), Levy and Sarnat (1 977) and Bicksler (1 979), among others. Papers from these books may be used to supplement the applications in this chapter. The Black-Scholes theory of option pricing and the subsequent modifications of this theory by several authors have generated great interest with a rapid litera-
Applications in finance
273
ture growth in this area. For an overview of the major results in option pricing we suggest the review articles by Smith ( 1976, 1979). ln the original Black-Scholes (1973) paper several assumptions were made such as: ( 1 ) there are no penalties for short sales; (2) there are no taxes and no transactions costs; (3) the market operates continuously and the stock price fol lows an Ito process; (4) the stock pays no dividends and the option can only be exercised at the terminal date of the contract; and (5) the riskless rate is known and constant. Under these assumptions Black and Scholes show that a riskless hedge can be formulated using proper proportions of call options and shares of the underlying stock. Such an instantaneously riskless hedge yields a rate of re turn equal to the known constant riskless rate. Since 1973 several authors have modified these assumptions and, more importantly, the Black-Scholes option pricing theory has found many applications in various areas of fmance. Some modifications of the original assumptions include the following. Merton (1973a) analyzes the option model with a stochastic interest rate. Ingersoll (1976) has included a differential tax structure. Cox and Ross (1 976) have priced options for alternative stochastic processes. Merton (1976) has studied the problem of a discontinuous return structure. Smith (1 976) discusses some additional modifications. For purposes of illustration we follow Smith (1979) to illustrate some appli cations of the Black-Scholes theory of o ption pricing which was initially devel oped for a European call. Recall that a European call is an option to buy a share of a stock at the maturity date of the contract for a stated amount called the exercise price. Section 3 and exercise (2) of this chapter show that the solution of the price of a call option may be denoted as F(X, T , £, r, a2 ). Merton (1973a) studies a European put, i.e. an option to sell a share of stock at the maturity date of the contract for a given exercise price. He finds that when borrowing and lending rates are equal, then the price of a European put is equal to the value of a portfolio of a European call with the same terms as the put, riskless bonds with a face value equal to the exercise price of options and a short position in the stock. The Black-Scholes call pricing model can also be used in pricing the debt and equity of a firm. Assume that: (1) the firm issues pure discount bonds with no dividends until after the bonds mature at which time the bondholders are paid, if possible , and the residual, if available, is paid to the stockholders; (2) the total value of the firm is unaffected by capital structure, i.e. a Modigliani-Miller (1958) world applies; and (3) there are homogeneous expectations about the dynamic behavior of the value of the fum's assets with a log-normal distribution having a constant rate of return. Under these assumptions and a given constant riskless rate, the Black-Scholes theory provides the correct valuation of the
274
Stochastic methods in economics and finance
firm's equity. In this case , issuing bonds is equivalent to the stockholders selling the assets of the firm to the bondholders for the proceeds of the issue plus a call option to repurchase these assets with an exercise price equal to the face value of the bonds. Thus, the equity of the firm is like a call option. Next suppose that instead of having a bond contract calling for only one pay ment of principal plus interest at the maturity date, we consider convertible bonds. A convertible bond offers the bondholder at the maturity date the option to either receive the face value of the bond or new shares equal to a fraction of the value of the assets of the firm. Applying the Black-Scholes theory it has been shown that a convertible bond is equivalent to a nonconvertible bond plus a call option; see Ingersoll ( 1 977) and his references. The Black-Scholes theory can also be applied to the pricing of various other contingent claims, such as the pricing of underwriting contracts, the pricing of collateralized loans, the pricing of leases, and the pricing of insurance, among others. Smith (1 979) summarizes the results of such pricings and provides appro priate bibliographical references. A simplified approach to option pricing has been suggested recently by Cox and Ross (1 979). They present a simple discrete time option pricing formula in which the fundamental economic principles of option valuation by arbitrage methods are clear and intuitive. Furthermore, their approach requires only ele mentary mathematics, yet it contains as a special limiting case the Black-Scholes model. Specifically, Cox and Ross (1 979) suggest that whenever stock price movements conform to a discrete binomial process or to a limiting form of such a process, options can be prices solely on the basis of arbitrage methods. To price an option by arbitrage methods there must exist a portfolio of other assets which exactly replicates in every state of nature the payoff received by an opti mally exercised option. The basic result of Cox and Ross may be stated as follows. Suppose that markets are perfect, that changes in the interest rate are never random, and that changes in the stock price are always random. In a discrete time model a neces sary and sufficient condition for pricing options of all maturities and exercise prices by arbitrage methods using only the stock and bonds in the portfolio is that in each period (1) the stock price can change from its beginning of period value to only ex-dividend values at the end of the period, and (2) the dividends and the size of each of the two possible changes are presently known functions depending at most on (i) current and past stock prices, (ii) current and past val ues of random variables whose changes in each period are perfectly correlated with the change in the stock price, and (iii) calendar time. For a verification of this result see Cox and Ross (1 979). Having briefly indicated some of the modifications and applications of the
Applications in finance
275
Black-Scholes theory we inform the reader that two papers, by Harrison and Kreps (1979) and Kreps (1980), consider some foundational issues that arise in conjunction with the arbitrage theory of option pricing. The important point to consider is this: the ability to trade securities frequently can enable a few multi period securities to span many states of nature. In the Black --Scholes theory there are two securities and uncountably many states of nature, but because there are infmitely many trading opportunities and because uncertainty resolves nicely, markets are effectively complete. Thus, even though there arc far fewer securities than states of nature, nonetheless markets are complete and risk is al located efficiently. The question of what is important in determining the number of securities needed to have complete markets and an evaluation of the robust ness of the Black-Scholes theory are presented in Kreps (1 980). Merton's (1971) continuous model discussed briefly in section 4 provides a general equilibrium framework for the analysis of consumption and investment decisions under uncertainty. Two earlier papers that studied a similar problem in discrete time are Fama (1 970a) and Hakansson (1 970). Some other important papers are Sharpe (1 964), Lintner (196Sa, 196Sb) and Mossin (1 966). The ad vantages of using continuous analysis are discussed in Merton (197Sb) who claims that the continuous time solution is consistent with its discrete time counterpart when the trading interval is sufficiently small and that the assump tions required are descriptive of capital markets as they actually function. Fur thermore , the continuous time analysis has all the advantages of simplicity and empirical tractability found in the classic mean-· variance model but without its objectionable assumptions. Merton (1973b) uses methods similar to Merton (1971) to develop an inter temporal capital asset pricing model. The model for the capital market is de duced from the portfolio selection behavior of an arbitrary number of investors who act to maximize the expected utility of lifetime consumption and who trade continuously in time. Explicit demand functions for assets are derived and it is shown that current demands are affected by the possibility of uncertain changes in future investment opportunities. The equilibrium relationships among expected returns are derived by aggregating demands and requiring the market to clear. It is shown that, contrary to the classical CAPM, expected returns on risky assets may differ from the riskless rate even when they have no systematic or market risk. Merton {1973b) has been extended by Breeden (1979). The exten sion is achieved by using the same continuous time framework as Merton ( 1 973b) However, Breeden ( 1979) shows that Merton's multi-beta pricing equation can be collapsed into a single-beta equation, where the instantaneous expected ex cess return on any security is proportional to its beta, or covariance, with respect to aggregate consumption alone. The result is shown by Breeden to extend to a .
276
Stochastic methods in economics and finance
multi-good world with an asset's beta measured relative to aggregate real con sumption.
(197 1 , 1973b), just to mention a few, include Magill and Constantinides (I 976) who introduce transaction costs in the port folio selection problem, Constantinides (I 980) who introduces personal tax con Other extensions of Merton
sideration in the consumption investment problem, and Litzenberger and Ramas wamy
(1 979) who, in addition to personal taxes, also include dividends.
A useful survey paper that presents a general equilibrium approach for the analysis of consumption, investment and portfolio selection, is Merton
(1981).
In this paper Merton provides a comprehensive and unified survey of the con sumption-saving choice and the portfolio selection choice under uncertainty. Both these choices are integral parts of the microeconomic theory of investment under uncertainty. This survey supplements our brief discussion in section
4 of
this chapter. Note that the consistency of the pricing of equity by the Black Scholes option theory with the continuous time capital asset pricing model is discussed in Smith In section
(I 979, pp. 89 - 90).
7 we follow
Fischer
(1 975) to illustrate the use of stochastic calcu
lus techniques on the topic of indexed bonds. For a further, recent reference we
(1977). A tion is presented in Humphrey (1974).
indicate Liviatan and Levhari
brief history of the concept of indexa
The term structure theory of interest rates has attracted much attention for a long time . For a general overview the books by Meiselman
(I 962), Malkiel
(1966), Nelson (1972) and by Michaelsen (1 973) can be found useful. Vasicek's (1977) paper is used in section 8 to illustrate the popularity of stochastic calcu lus techniques. A stochastic calculus approach is also used by Cox, Ingersoll and Ross (1978) to develop a general theory of the term structure of interest rates. Cox, Ingersoll and Ross use the continuous time CAPM framework developed by
Merton
(1973b) along with the rational expectations equilibrium model of Lucas
(I 978) to develop a complete
intertemporal asset pricing model that is tractable
and does not contradict individual expectations as the equilibrium moves through time. From such a model, Cox, Ingersoll and Ross develop a theory of the term structure of interest rates and derive closed form solutions for the term structure and the prices of bonds which are potentially testable . For details see Cox, Ingersoll and Ross Dothan
(I 978).
Some other related papers are Richard
(1 978) and Brennan and Schwartz (1977).
(1978),
Stochastic calculus methods have also found applications in the area of fu tures pricing. In section
6
of Chapter
1
we briefly introduced the notion of fu
tures pricing. The reader who is interested in studying futures pricing should consult Black
(1976)
and Cox, Ingersoll and Ross
(1980) and some of the refer
ences cited in these papers. Futures markets and forward markets have been
Applications in finance
27 7
treated by the academic literature as if they were synonymous. Cox, Ingersoll and Ross (1 980) explain the fundamental differences and clarify the appropriate relationships between these two markets. The demand for cash balances and the related issues on cash management are well established areas of research. Some references, other than the ones indicated in application (10), are Eppen and Fama (1968, 1 969), Neave (1 970), Vial (1 972), Constantinides (1 976) and Constantinides and Richard (1978). Related issues that might be of interest to some readers are discussed in Daellenbach and Archer (1 969) and Crane (1971 ) In sections 1 1 - 18 we follow Brock (1 978) to develop an intertemporal gen eral equilibrium theory of capital asset pricing inspired by Merton (1973b) We note, however, that Merton's intertemporal capital asset pricing model is not a general equilibrium theory in the sense of Arrow-Debreu because technological sources of uncertainty are not related to the equilibrium prices of risky assets. This is done in these sections by integrating ideas from modern finance and sto chastic growth models. Basically what is done is to modify the stochastic growth model of Brock and Mirman (1 972) in order to incorporate a nontrivial invest ment decision into the asset pricing model of Lucas (1 978). This is done in such a way as to preserve the empirical tractability of the Merton formulation and at the same time determine the risk prices derived by Ross (1976) in his arbitrage theory of capital asset pricing. Ross's price of systematic risk k at date t, denoted by 'Ak t' which is induced by the source of systematic risk Bk t' is determined by the covariance of the marginal utility of consumption with o�,- In this way Ross's 'Ak , are determined by the interaction of sources of production uncer tainty and the demand for risky assets. Furthermore, this model provides a con text in which conditions may be found on tastes and technology that are suffi cient for equilibrium returns to be a linear function of the uncertainty in the economy. Linearity of returns is necessary for Ross's theory. More advances should be expected along the lines of introducing imperfect information and inquiry into what rules frrms should follow in order to maxi mize equilibrium welfare of the representative consumer when some contingency markets are absent. Finally, a more difficult and perhaps more interesting prob lem would be to introduce heterogeneous consumers so that borrowing on fu ture income could be introduced and the impact of this on the price of risk could be investigated. .
.
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Dreyfus, S.E. ( 1 965), Dynamic Programming and the Calculus of Variations, Academic Press, New York. Dvoretzky, A. ( 1 967), "Existence and Properties of Certain Optimal Stopping Rules", in: Proc. Hfth Berkeley Symposium Math. Statist. Prob. , University of California Press, Los Angeles, pp. 4 4 1 -452. Dynkin, E.B. (1 965), Markov Processes, Springer-Verlag, New York. Dynkin, E. B. and A.A. Yushkevich ( 1 969), Markov Processes: Theorems and Problems, Ple num Press, New York. Eppen, G.D. and E. F. Fama (1 968), "Solutions for Cash-Balance and Simple Dynamic-Port folio Problems", Journal of Business, 4 1 , 94 - 1 1 2. Eppen, G.D. and l':.F. Fama ( 1 969), "Cash Balance and Simple Dynamic Portfolio Problems with Proportional Costs", International Economic Review, 10, 1 1 9 - 1 3 3 . Fama, E. ( l 970a), "Multiperiod Consumption-Invest ment Decisions", 11ze American Eco nomic Review, 60, 163 - 174. Fama, E.F. ( l 970b), "Efficient Capital Markets: A Review of TI1eory and Empirical Work", 11Ie Journal of Finance, 25, 383-4 1 7. Fama, E.F. ( 1 972), "Perfect Competition and Optimal Production Decisions Under Uncer tainty", Bell Journal of Economics and .Management Science, 3, 509-530. Fama, E. F. ( 1 975), "Short-Term Interest Rates As Predictors of lnflation ",American Eco nomic Review, 65, 269-· 282. Fama, E. ( 1 976), Foundations of Finance, Basic Books, New York. Fama, E.F. and M. Miller ( 1 972), Theory of Finance, Holt, Rinehart and Winston, New York. Feller, W. ( 1 954), "Diffusion Processes in One Dimension", Transactions of the American Mathematical Society, 91, 1 - 3 1 . Feller, W. ( 1 968), A n Imroduction to Probability 11zeory and its Applications, \'Ol. 1 , John Wiley & Sons, New York. FeUer, W. ( 1 9 7 1 ), An Introduction to Probability Theory and its Applications, vol. 2, John Wiley & Sons. New York. Fischer, S. ( 1 97 2), "Assets, Contingent Commodities and 1l1e Slutsky Equations", Eco nometrica, 40, 37 1 - 385. Fischer, S. ( 1 975), "The Demand for Index Bonds", Journal of Political Economy, 83, 509- 534. Fleming, W.H. ( 1 969), "Optimal Continuous-Parameter Stochastic Control", SIAM Review, 1 1 , 4 70- 509. Fleming, W.H. ( 1 97 1 ), "Stochastic Control for Small Noise Intensities", SIAM Journal on Control, 9, 4 7 3 - 5 1 7 . Fleming, W.H. and R.W. Riche! ( 1 975), Deterministic and Stochastic Optimal Comrol, Springer-Verlag, New York. Foldes, L. ( 1 978a), "Optimal Saving and Risk in Continuous Time", Review of Economic Studies, 45, 39-65. Foldes, L ( 1 9 7 8b), "Martingale Conditions for Optimal Saving-Discre te Time", Journal of Mathematical Economics, 5. 83- 96. Francis, J.C. and S.ll. Archer ( 1 979), Portfolio Analysis, 2nd edn., Prentice-Hall, Lnc., New Jersey . Frenkel, J.A. and B. Jovanovic ( 1 9�0), ..On Transactions and Precausionary Demand for Money", Quarterly Journal ofEconomics, 95, 25 -- 4 3. Friedman, A. ( 1 975), Stochastic Differential Equations and Applications, Academic Press, New York. Friedman, M. and L.J. Savage ( 1 948), ..The Utility Analysis of Choices Involving Risk", The Journal of Political Economy, 56, 279- 304. Gal'perin, E. A. and N.N. Krasovskii ( 1 963), ..On the Stabilization of Stationary Motions in Nonlinear Control Systems", Journal of Applied Mathematics and Mechanics, 27, 1 52 1 1546.
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ofEconomic Studies, 35, 4 7-55 . Greenwood, P.H. and C.A. lngene ( 1 978), "Uncertain Externalities, Liability Rules and Re source Allocation", The American Economic Review, 68, 300- 310. Grossman, S. ( 1 976), "On the Efficiency of Competitive Stock Markets where Traders have Diverse Information", Journal of F i11ance, 3 1 , 573-585. Grossman, S. and J.E. Stiglitz (1976), "Information and Competitive Price Systems", American Economic Review, 66, 246-253. Grossman, S.J. and J.E. Stiglitz ( 1 980), The Impossibility of Informationally Efficient Markets", The American Economic Review, 70, 393-408. Hadjimichalakis, M. (197 1}, "Equilibrium and Disequilibrium Growth with Money: The Tobin Models", Review of Economic Studies, 38, 457-4 79. Hadley, G. ( 1 964}, Nonlinear and Dynamic Programming, Addison-Wesley publishing Co., ..
Reading, Massachusetts. Hahn, F.H. ( 1 970), "Savings and Uncertainty", Review of Economic Studies, 37, 2 1 -24. Hahn, W. ( 1 963}, Theory and Application of Liapunov's Direct Method, Prentice-Hall, New York. Hahn, W. ( 1 967}, Stability ofMotion, Springer-Verlag, New York. Hakansson, N.H. ( 1 970}, ..Optimal Investment and Consumption Strategies Under Risk for a Class of Utility Functions", Econometrica, 38, 5 87-607. Hale, J. ( 1 969), "Dynamical Systems and Stability", Journal ofMathematical Analysis and
Applications, 26, 39-59. Hall, R.E. ( 1 978), "Stochastic Implications of the Life Cycle-Permanent Income Hypoth esis: Theory and Evidence", Journal ofPolitical Economy, 86, 971 -987. Harrison, J.M. and D.M. Kreps ( 1 979}, "Martingales and Arbitrage in Multiperiod Securi ties Markets", Journal of Economic Theory , 20, 381 -408. Harrison, J.M. and S.R. Pliska (1981}, "Martingales and Stochastic Integrals in the Theory of Continuous Trading", Stochastic Processes and their Applications, 1 1 . Hartman, P. (1961}, "On the Stability in the Large for Systems o f Ordinary Differential Equations", Canadian Journal ofMathematics, 1 3, 480-492. Hartman, P. ( 1 964}, Ordinary Differential Equations, John Wiley & Sons, New York. Hartman, P. and C. Olech ( 1 962}, "On Global Asymptotic Stability of Solutions of Or dinary Differential Equations", Transactions of The American Mathematical Society,
1 04, 154- 1 7 8. Helpman, E. and A. Razin (1979}, A Theory of International Trade Under Uncertainty, Academic Press, New York. Hestenes, M. ( 1 966}, Calculus of Variations and Optimal Control Theory, John Wiley & Sons, New York. Hoel, P.G., S.C. Port and C.J. Stone ( 1 972), Introduction to Stochastic Processes, Houghton Mifflin Company, Boston. Hirshleifer. J. ( 1 965), "Investment Decision Under Uncertainty: Choice-Theoretic Ap proaches", Quarterly Journal of Economics, 19, 509-5 36.
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Hirshleifer, J. ( 1 966), ..Investment Decision Under Uncertainty: Applications of the State Preference Approach", Quarterly Journal of Economics, 80, 252-277. Hirshlcifer, J. ( 1 973), ''Where Are We In The Theory of Information?", American Economic Review, 63, 3 1 - 39. Houthakker, H. ( 1 961), ..Systematic and Random Elements in Short-Term Price Move ments", American Economic Review, 5 1 , 164-172. Humphrey, T.M. (1 974), ..The Concept of Indexation in the History of EconomicThought", The Economic Review of the Federal Reserve Bank of Richmond, 60, 3- 1 6. Ingersoll, J. ( 1 976), ..A Theoretical and Empirical Investigation of the Dual-Purpose Funds: An Application of Contingent Claims Analysis", Journal of Financial Economics, 3, 83-123. Ingersoll, J.E. ( 1 977), "A Contingent-Claims Valuation of Convertible Securities", Journal of FinancU:zl Economics, 4, 289-322. lntriligator, M.D. ( 1 978), Econometric Models. Techniques and Applications, Prentice Hall, Inc., New Jersey. Ishii, Y. ( 1 977), ..On The Theory of The Competitive Firm Under Price Uncertainty: Note", 17Je American Economic Review, 67, 768-769. Ito, K. ( 1 944), ..Stochastic Integral", Proceedings of the Imperial Academy, Tokyo, 20, 5 1 9-524. Ito, K. ( 1946), ··on A Stochastic Integral Equation", Proceedings of the Imperial Academy, 22, 32-35. Ito, K. (1 950), ''Stochastic Differential Equations In A Differentiable Manifold", Nagoya Mathematics Journal, 1 , 35-47. Ito, K. ( 1 95 1 a), "On A Formula Concerning Stochastic Differentials", Nagoya Mathematics Journal, 3, 5 5-65. Ito, K. (1 95 1b), ..On Stochastic Differential Equations", Memoirs of the American Math ematical Society, The American Mathematical Society, Rhode Island. Ito, K. (1961), Lectures on Stochastic Processes, Tata Institute of Fundamental Research, India. Ito, K. and H.P. McKean, Jr. ( 1 974), Diffusion Processes and Their Sample Paths, Second Printing, Springer-Verlag, New York. Jensen, M.C. ( 1 969), ..Risk, The Pricing of Capital Assets, and The Evaluation of Invest ment Portfolios", Journal ofBusiness, 42, 167-247. Jensen, M.C., ed. ( 1972), ..Capital Markets: Theory and Evidence", Bell Journal ofEconomics and Management Science, 3, 357-398. Jensen, M.C. and J.B. Long, Jr. ( 1 972), ..Corporate Investment Under Uncertainty and Pa reto Optimality In The Capital Markets·•, Bell Journal of Economics and Management Science, 3, 1 5 1 - 174. Jovanovic, B, ( 1 979a), ..Job Matching and the Theory of Turnover", Journal of Political Economy, 81, 972-990. Jovanovic, B. (1979b), "Firm-specific Capital and Turnover" ,Journal of Political Economy, 87, 1 246-1 260. Kantor, B. ( 1 979), "Rational Expectations and Economic Thought", Journal of Economic Literature, 1 7 , 1422-1441. Karlin, S. and H.M. Taylor ( 1 975), A First Course in Stochastic Processes, 2nd edn., Aca demic Press, New York. Kats, 1.1. and N.N. Krasovskii ( 1 960), ..On The Stability of Systems with Random Dis turbances", Joumal ofApplied Mathematics and Mechanics, 24, 1 225- 1 245. Kihlstrom, R.E. and L.J. Mirman ( 1 974), ..Risk Aversion with Many Commodities" , Journal of Economic Theory, 8, 361-388. Khas'minskii, R.Z. ( 1 962), "On the Stability of the Trajectory of Markov Processes", Ap plied Mathematics and Mechanics (USSR), 26, 1 5 54-1565. Kolmogorov, A.N. ( 1950), Foundations of the Theory of Probability, Chelsea Publishing Company, New York.
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Kozin, F. ( 1 972), ..Stability of the Linear Stochastic System", in: R . Curtain, ed., Stability ofStochastic Dynamical Systems, Springer-Verlag, New York. Kozin, F. and S. Prodromou (1971), ..Necessary and Sufficient Conditions for Almost Sure Sample Stability of Linear Ito Equations", SIAM Journal of Applied Mathematics, 2 1 , 4 1 3-424. Krasovskii, N.N. ( 1 965), Stability of Motion, Stanford University Press, California. Kreps, D.M. ( 1 980), "Multiperiod Securities and the Efficient Allocation of Risk: A Com ment on the Black-Scholes Option Pricing Model'', Technical Report 306, Institute for Mathematical Studies in the Social Sciences, Stanford University, California. Kunita, H . ( 1 970), "Stochastic Integrals Based on Martingales Taking Values in Hilbert Spaces", Nagoya Mathematics Journal, 38, 4 1 - 5 2. Kunita, H. and S. Watanabe ( 1 967), "On Square Integrable Martingales", Nagoya Mathe matics Journal, 30, 209- 245. Kushner, H.J. ( 1 965}, "On the Stochastic Maximum Principle: Fixed Time of Control", Journal ofMathematical Analysis and Applications, 1 1 , 7 8- 92. Kushner, H.J. ( 1 967a), Stochastic Stability and Control, Academic Press, Inc., New York. Kushner, H.J. ( 1 967b), "Optimal Discounted Stochastic Control for Diffusion Processes" , SIAM Journal On Control, 5, 520- 5 3 1 . Kushner, H.J. (197 1}, Introduction to Stochastic Control, Holt, Rinehart and Winston, New York. Kushner, II.J. ( 1 972), "Stochastic Stability", in: R. Curtain, ed., Stability of Stochastic Dynamical Systems, Springer-Verlag, New York. Kushner, H.J. ( 1 975), "Existence Results for Optimal Stochastic Controls", Journal of Optimization Theory and Applications, 1 5 , 347 - 360. Kussmaul, A. V. ( 1 977), Stochastic Integration and Generalized Martingales, Pitman, Lon don. Kwakernaak, H. and R. Sivan ( 1 972), Linear Optimal Control Systems, Wiley-lntersciencc, New York. Ladde, G.S. and V. Lakshmikantham ( 1 980), Random Differential Inequalities, Academic Press, New York. LaSalle, J.P. ( 1 964), "Recent Advances in Liapunov Stability Theory", SIAM Review, 6, 1 - 1 1. LaSaUe, J.P. and S. Lefschetz (1961}, Stability by Liapunov 's Direct Method, Academic Press, New York. Lau, M. ( 1 977), ..The Behavior of the Exchange Rate in a Two-Country Monetary Model", University of Chicago, Department of Economics, unpublished paper. Lee, E. B. and L. Markus ( 1 967}, Foundations of Optimal Control Theory , John Wiley & Sons, New York. Lefschetz, S. ( 1 965), Stability of Nonlinear Control Systems, Academic Press, New York. Leland, H.E. ( 1 9 7 2a), "On Turnpike Portfolios", in: G. Szego and K. Shell, eds., Mathe matical Methods in Investment and Finance, North-Holland Publishing, Amsterdam, pp. 24- 3 3. Leland, H.E. ( 1 9 72b}, "Theory of the Firm Facing Uncertain Demand" , The A merican Economic Review, 62, 278- 29 1 . Leland, H.E. ( 1 974), "Production Theory and the Stock Market", Bell Journal of Eco nomics and Management Science, 5 , 1 25- 1 44. Leonardz, B. ( 1 9 74), To Stop or Not to Stop : Some Elementary Optimal Stopping Prob lems with Economic Interpretations, Halsted Press, New York. LeRoy, S. ( 1 97 3), .. Risk Aversion and the Martingale Property of Stock Prices", Inter national Economic Review, 1 4, 436-446. Levhari, D. ( 1 9 7 2}, "Optimal Savings and Portfolio Choice under Uncertainty", in: G. Szego and K. SheU, eds., Mathematical Models in Investment and Finance, North Holland Publishing, Amsterdam, pp. 34-4 8.
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Radner, R. (1974), "Market Equilibrium and Uncertainty: Concepts and Problems", in: M.D. Intriligator and D. Kendrick, eds., Frontiers of Quantitative Economics, North Holland Publishing, Amsterdam, pp. 4 3- 90. Richard, S.F. ( 1 978), "An Arbitrage Model of the Term Structure of Interest Rates" , Jour
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Rishel, R.\V. ( 1970), "Necessary and Sufficient Dynamic Programming Conditions for Continuous Time Stochastic Optimal Control", SIAM Journal on Control, 8, 5 5 9 - 5 7 1 . Robbins, H . ( 1970), "Optimal Stopping", American Mathematical Monthly , 77, 333- 34 3. Rockafellar, R.T. ( 1970), Convex Analysis, Princeton University Press, Princeton, New Jersey. RockafeUar, R.T. ( 1 973), "Saddle Points of Hamiltonian Systems in Convex Problems of Lagrange", Journal of Optimization Theory and Applications, 12, 367-590. Rockafellar, R.T. ( 1 976), "Saddle Points of Hamiltonian Systems in Convex Lagrange Prob lems having a Nonzero Discount Rate'\ Journal of Economic Theory, 1 2, 7 1 - 1 1 3. Roll, R. ( 1 970), The Behavior of Interest Rates: An Application of the Efficient Market Model to U. S. Treasury Bills. Basic Books, New York. Ross, S.A. ( 1 975), "Uncertainty and the Heterogeneous Capital Goods Model", Review of
Economic Studies, 42, 1 3 3- 1 46 . Ross, S.A. ( 1 976), "The Arbitrage Theory of Capital Asset Pricing", Journal of Economic
Theory, 1 3, 34 1 - 360.
Rothschild, M. ( 1 9 73), "Models of Market Organization with Imperfect Information : A Survey", Journal ofPolitical Economy, 8 1 , 1283- 1308. Rothschild, M. and J.E. Stiglitz ( 1 970), "Increasing Risk: I. A Definition" , Journal of Eco
nomic 111eory, 2, 225- 24 3.
Rothschild, M. ( 1 974), "Searching for the Lowest Price when the Distribution of Prices is Unknown", Journal ofPolitical Economy, 82, 689- 7 1 1 .
Selected bibliography
291
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..
292
Selected bibliography
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AUTHOR INDEX*
Alchian, A., 6 1 Anderson, B ., 1 39 An tosiewicz, H., 136 Aoki, M., 1 32, 1 39 Araujo, A., I 3 9 Archer, S.H. , 237, 277 Arnold, L., 70, 72, 74, 75, 77, 79, 80, 8 1 , 96, 99, 1 00, I 02, 1 33 Arrow, K .J., l08, 1 38, 2 1 7 Ash, R. B. , 3, 10, I 1 , 1 2, 29, 32, 60, 6 1 Astrom, K.J., 66, 67, 70, 1 3 2, 1 3 3, 1 3 9 Athans, M . , 1 39 Bachclicr, L. , 6 1 Balakrishman, A. V., 1 3 2 Baron, D., 214 Batra, R.N., 2 1 4 Baumol, W.J., 238 Bellman, R., 108, 1 39 Benes, U.E., 1 3 9 Benveniste, L., 1 39, 1 86, 249, 25 1 Berkowit7., L.D., 1 3 9 Bernoulli, D., 6 2 Berbekas, D.P., 132, 1 3 9 Bewley, T. , 184, 2 1 5 Bharucha-Reid, A.T., 62. 70, 7 2 Bicksler, J.L., 272 Billingsley, P., 5, 1 3 , 15, 16, 19, 29, 32, 34, 37, 58, 60, 6 1 , 62 Bismut, J.M., l l 5, 1 1 8, 1 1 9, 120, 1 39, 150, 160, 163, 214, 2 1 5 Black, F., 220, 2 2 1 , 222, 233, 273, 276 Borg, G., 136 Bourguignon , F., 106, 1 4 1 , 2 1 4 Boyce, W.M., 2 1 6 Brainard, W , 192, 2 1 6 Breeden, D.T., 275 .
•
Also sec Selected Bibliography.
Breiman, L., 63 Brennan, J.J., 276 Brock, W.A., 53, 1 38, 1 39, 140, 167, 168, 182, 185, 187, 188. 205, 214, 2 1 5, 2 1 6, 25 1 , 277 Bryson, A.E., 1 3 9 Bucy, R.S., 136 Burmeister, E., 142 Cass, D . . 1 3 9 Cesari, L., 1 36 Chow, G.C., 67, 1 3 2 Chow. Y.S., 45, 5 1 , 5 2 , 53, 63, 2 1 6 Chung, K.L., 60 Ginlar, E., 62, 63 Constantinides. G.M., 236, 237, 238, 270, 276, 277 Cornell, B. , 6 2 Cox, D. R., 6 2 , 200 Cox, J.C., 273, 274, 276, 277 Crane, D. B., 277 Daellcnbach, H.G., 277 Danthinc, J. P. , 62 Davis, M.H.A., 1 3 9 DeGroot, M., 45, 4 7 , 48, 4 9 , 5 1 , 63 Dellacherie, C., 6 1 , 139 Diamond. P.A., 214 Dobell, A. R., 142 Doob. J.L., 1 9, 29, 6 1 , 62, 69, 1 33 , 1 39 Dothan, L. U., 276 Dreyfus, S. E., 1 3 9 Dvorctzky, A., 63 Dynkin, E.B., 62, 63 Eppcn, G.D., 277
296
Author index
Falb, P.L., 1 3 9 Fama, E., 1 8 , 2 2 , 23, 6 1 , 6 2 , 275, 277 Feller, W., 6 1 , 62, 105, 106, 1 3 8 Fermat, P., 6 1 Fischer, S., 2 1 7 , 230, 232, 276 Fleming, W.H. , I 08, I 39 Foldes, L., 29, 2 1 4 Forsythe, R., 2 1 4 Francis, J.C., 237 Frenkel, J .A., 239, 240, 241 Friedman, A., 78, 132, 235 Friedman, M., 59 Gale, D., 25 1 Gal'perin, E.A., 1 3 9 Gertler, M . , 1 8 2, 2 1 6 Gihman, J . , 62, 70, 72, 77, 79, 8 1 , 93, 94, 96, 97, 98, 101, 103, 1 04, 1 22, 1 32, 1 35 Gould, J.P., 2 1 4 Grossman, S., 6 1 Hadjimichalakis, M . , 206 Hadley, G., 1 3 9 Hahn, W., 136 Hakansson, N.H., 275 Hale, J ., 1 3 9 Hall, R.E., 28, 29, 6 1 Harrison, J.M., 62, 80, 275 Hartman, P., 1 36, 139 Hclpman, E., 2 1 4 Hestenes, M., 1 3 9 Ho, Y., 1 3 9 Hoel, P.G., 40, 6 2 Houthakker, H . , 6 I Humphrey, T.M., 276 Ingersoll, J.E., 273, 274, 276, 277 Intrilligator, M.D. , 1 3 2 Ishii, Y., 2 1 4 Ito, K . , 37, 69, s 1 . 132, 1 3 3 , 1 35 Jensen, M.C. , 23, 24 Jovanovic, B., 216, 239, 240, 241 Kantor, B., 2 1 4 Karlin, S., 6 1 , 62 Kats, 1.1., 1 36 Khas'minskii, R.Z., 106, 1 3 8 Kolmogorov, A.N., 6 1 , 62 Kozin, F., 1 3 1 , 137, 1 3 8 Krasovskii, N.N., 136, 1 3 9 Kreps, D.M., 275
Kunita, H. , 1 3 3 Kurz, M., 1 08, 1 3 8 Kushner, H.J 104, 1 0 5 . 1 2 2, 1 36, 1 3 7 , 1 39, 140 Kussmaul, A. V., 1 33 K W
Ladde, G.S., 70, 72. 8 1 , 1 06 Lakshmikantham, V 70, 72, 8 I. I 06 LaSalle, J.P., 136 Lau, M., 214 Lee, E.B., 139 Lefschetz, S., 136, 1 3 9 Leland, H.E., 2 1 4 Leonardz, B., 6 3 LeRoy, S., 24 Levhari, D., 139, 214, 276 Levy, H., 272 Liapunov, A., 136 Lippman. S.A. , 48, 5 1 , 60, 63 Lintner, J., 275 Litzenberger, R.H., 276 Liviatan, N., 139, 276 Loeve, M.M. , 1 , 60 Lucas, R.E. , 62, 157, 160, 169, 1 82, 2 1 4, 2 1 5 , 247, 254, 276, 277 .•
MaCurdy, T.E., 6 1 Magill, M.J.P., I 39, 140, 1 6 7 , 1 6 8 , 2 1 5 , 276 Majumdar, M., 102, 2 1 4 Malinvaud, E., 25 1 Malkiel, B.G., 276 Malliaris, A.G., 6 1 , 2 1 4 Mandelbrot, B .B . , 23, 6 1 , 62 Mandl, P. , I 06, 138 Mangasarian, O.L., I 39 Markov, A.A., 62 Markus, L., 1 3 9 Massera, J .L., 136 Mayer, W., 2 1 4 McCali, J.J., 48, 5 1 , 60, 6 1 , 63. 1 9 1 , 2 1 4 McFadden, D . , 25 1 McKean, Jr., H.P., 37, 78, 80, I 33, 2 1 6 McKenzie, L., 1 3 9 , 2 1 5 McNees, K., 62 McShane, E.J., 1 3 3 Meiselman, D., 276 Merton, R.C., 106, 1 38, 139, 140, 1 4 1 , 146, 147, 148, 182, 214, 220. 222, 223, 225, 226, 227, 228, 229, 230, 233, 237, 269, 273, 275, �76, 277 Metivier, M., 78, 80, 1 3 3 Meyer, P.A., 18, 29, 6 1 , 78, 80, 1 33, 1 3 5
Author index Michaelsen, J.B., 276 Miller, I I.D. , 62. 200 Miller, M . H., 239, 273 Miller, R.A., 2 1 6 Mills, E.S., 214 Mirman, L.J . , 1 38, 1 82, 185, 1 86, 187, 1 88, 214, 277 Miroshnichcnko, T.P., 195, 1 9 7 Modigliani, F., 62, 273 Moore, J., 1 39 Mortensen, D.T., 169 Mossin, J., 275 Mullinaux, D., 62 Muth, J. F. , 22, 28, 1 53 Nagel, E 65 Neave, E. H. , 277 Nelson, C.R., 214, 276 Neveu, J . 4, 29, 60, 6 1 .•
.
Ohlson, J.A., 24 Olcch, C., 139 Olivera, J H G 239 O'Neill, D.E., 6 1 Orr, D. , 239 .
.
.,
Padgett, W.J.,69, 1 3 2 Papoulis, A., 60 Pascal, B., 6 1 Perrak is, S., 2 1 4 Pellaumail, J . , 78, 80, 1 3 3, 134 Pliska, S.R., 62, 80 Pontryagin, L.S., 1 3 9 Poole, W., 216 Port, S.C., 40, 6 2 Prabhu, N.U., 6 2 Pratt, J .W , 2 1 4 Prescott, E., 1 5 7 , 160, 169, 2 1 4 Prodromou, S . , 1 3 1 .
Ramaswamy, K., 276 Razin, A., 2 1 4 Richard, S.F. , 276, 277 Rishel, R.W. , 108, 1 39 Robbins, H. , 44, 45, 5 1 , 52, 53, 63 Rockafellar, R.T. 1 1 8, 1 39, 163 Roll, R., 6 2 Ross, S.A., 1 8 2 , 273, 274, 276, 277 Rothschild, M. , 53, 59, 205, 2 1 4, 2 1 6 Roxin, F- , 1 39 Ruiz-Moncayo, A., 63 ,
297
Samuelson, P. , 2 1 , 22, 23, 24, 25, 28. 6 1 , 1 3 9, 2 1 6 Sandmo, A 188, 2 1 4 Sargent. T., 62, 1 3 2 Sarnat, � 272 Savage, L.J. , 59 Schcinkman, J.A., 1 39, 186, 249, 25 1 , 252 Scholes, M., 220, 2 2 1 , 222, 233, 273 Schuss, Z., 69 Schwartz, E.S., 276 Sharpe, W.F. , 275 Shell, K. , 139, 272 Shiller, R.J ., 62, 2 1 4 Shiryayev, A.N., 44, 63 Shreve, S.E., 1 3 2 Siegmund, D., 45, 5 1 , 52, 53, 6 3 Sivan, R . , 1 39 Simon, H. A. , 1 4 1 Skorohod, A.V., 62, 70, 72, 77, 79, 8 1 , 93, 94., 96, 97, 98, 1 0 1 , 103, 1 04, 1 22, 132, 1 35 Slutsky, E., 2 1 5 Smith, J r. , C.W. , 269, 273, 274, 276 Solow, R.M., 1 4 1 Soong, T.T., 69, 1 32 Srinivasin, T.N. , 2 1 4 Stigum, B .P. , 2 1 4 Stigliz, J.E., 5 3 , 59, 6 1 , 205, 2 1 6 Stone. C.J., 40, 6 2 Stratonovich, R.L.. 1 30, 1 34 Strauss, A. , 139 Syski, R. , 1 3 2 Szego. G., 272 .•
. •
Tanaka, H., 106, 138 Taylor, H.M ., 6 1 , 6 2 Tinbergcn, J ., 2 1 6 Tobin. J., 206, 239 Treadway, A.B., 2 1 4 Tsokos, C.P. , 69, 1 3 2 Tucke� H., 1 5 , 16, 29, 58, 60, 62 Turnovsky, S.J., 192, 2 1 5, 2 1 6 Van Moerbeke, P., 1 24, 1 28 Vasicek, 0., 233, 235, 269, 270, 276 Vial, J .P. , 277 Vickson, R.G. , 272 Ville, J . , 6 1 Voltaire, K., 2 1 6 \Vald, A . , 44
298
Watanabe, S., 1 33 Weintraub, E.R. , 2 1 5 Wong, E., 1 3 3 Wonham, W. M., 69, ·136, 1 39 Wu, S. Y., 2 1 4 Yahav, J .A., 6 3 Yamabe, H., 1 39
Author index Yoshizawa, T., 1 36 Yushcvidt, A.A., 63 Zabel, E., 2 1 4 Zakai, M., 1 33 Ziemba, \V.T., 272 Zilcha, l . , 1 85, 1 86, 187, 2 1 4
SUBJ ECT INDEX
Absolutely continuous measure, 1 2 Absolute risk aversion, 1 9 1 , 227
Complete probability space, 3, 1 6 1 Condition,
Absorbing barrier, 1 99
of compatability, 34
Absorbing states, I 07
of no arbitrage profits, 263
Adjustment-cost model. 1 5 7 Admissible process, 1 83
of symmetry, 3 3 - 34 Conditional distribution, 1 6
Almost everywhere (a.e.). 9
Conditional expectation, 1 4 , 22. 28, 59
Almost surely (a.s.). 9 Anticipated price process, 1 5 8
Conditional expected rate of return, 27 - 28 Conditional expected value, 14 Cond i tiona I mean. 67
Arbitrage theory o f capital asset pricing,
242, 277
Conditional probability, 1 2- 1 3
Arro\....- Dcbreu model, 24 7. 277
Consistency properties, 3 3
Auto-- covariance function, 1 28
Constant relative risk aversion, 24, 1 74, 1 9 1 Consumer surplus. 1 59
Average cost, 1 9 1 Axiom of expcxtation formation, 28
Continuation region, 1 26, 1 94 Continuous in the mean square. 1 28
Backward difference�. 1 29 Backwards heat equation, 1 26
Continuous parameter process. 3 2 Contral·tion mapping theorem, 255
Bayes' theorem, 59
Control variable, I 08
Bellman's principle of optimality, I 08, 1 1 5,
Convergence,
1 48 Binomial distribution, 6 Bismut approach to optimal stochastic control, 1 1 8- 1 2 1 , 1 25. 1 50- 1 5 2 Black - Scholes option pricing model,
220- 223, 268, 2 73 - 275 Boote's inequality, 58
almost sure, 1 6 7- 1 68 dominated, I 0, 59 in distribution, 8. 1 85, 1 8 8 in probability, 7 , 76-77 in the mean square, 77 monotone, I 0, 59 with probability one, 7
Borci -Cantelli lemma. 4. 93 Borel sets, 2. 57
Convertible bond, 274 Correlation coefficient, I I , 1 92, 232
Brownian motion process. 36 - 38, 6 1 -6 2 geometric, 3 8 (see also Wiener procc.;s) Bounded ness property of solutions, I 04
Costate variable, I I 0 Counting measure, 57
Capital asset prking model. 1 8 2, 237,
245 ·- 25 3. 275-217 Capital market dficicncy, 23, 6 1 , 234 Certainty equivalence, 1 93, 256-262 Chapman--Kolmogorov equation, 40- 4 1
Covariance, I I , 246
Covariam:c function, 1 28 Debt, 273 Decomposition, 8 Demand funl·tion. random, 207 linear in wealth, 228
Chebyshev's inequality, 58
Density funl·tion, 6. 1 89
Competitive process. 1 6 1 - 1 68
Differentiable in the mean S<)Uarc, 98
300
Subject index
Differential operator, 100 Diffusion process, 99 coefficient, 99. I 08 property, 99- 100, 145 Directly testable equations, 262-263 Discount on future utility, 108 Discrete parameter process, 32 Distribution, binomial, 6 common, 43, 183 initial, 40 joint, 5, 53 marginal, 5 normal, 6 of a random variable, 5, 189 Poisson, 6 stationary, 40, I 06- 108. 1 46- 14 7 steady state, 40, 1 06- 1 08 Disturbances. additive, 1 9 2 mul tiplicative, 192- 1 93 Dominated conver!!ence theorem, I 0, 59 Dominated measure, 1 2 Drift, 38, 99, 108 Dynamic deterministic model, 66-67
of optimal stopping, 196 Fundamental theorem of calculus for Ito's stochastic integral, 9 1 --92 Futures pricin!!, 2 1 Game, fair. 1 7 , 261 favorable, 1 7 unfavorable, 1 7 Generalized hamiltonian, 165 Generalized martingale, 1 9 Gem:rated o-field, 2 G ihman-Skorohod approach to stability,
103 Global solution, 94 Global asymptotically stable, 105 Growth under uncertainty, monetary, 206 open economy, 143- 144 optimal saving function, 148 properties of, 1 44- 146 stationary distribution of, 1 46- 148 steady state per capita output, 147 stochastic differential equation of,
142- 143 N-sector model of,
Edgeworth box, 208 Elastic random walk, 270 Environmental factor. 1 19 Equilibrium, I 02, 243 stable, 103, 1 37, 206 Equilibrium distribution, 1 06- 108 Equilibrium solution, I 02, 206 Equity, 273 European call, 273 European put, 273 Event, 2 Excessive function, 1 26 Exercise price, 222 Expectation, 8 Expected present discounted value. 24 Exponentially stable in the mean, 1 3 7 Extended integrand problem, 1 70 Fair game, 1 7. 26 1 Fatou's theorem, 10, 59 Final gain, 1 25 Finite decomposition, 8 Finite-dimensional distributions, 33 Fixed point problem, 254 Free boundary, I 26 Free boundary problem, 1 24- 1 28 Fundamental partial differential equation
1 82- 1 88
Hamilton -·J acobi-Bellman equation of stochastic control theory, 1 10, 1 14,
148, 1 7 2, 209
Heat equation, 1 26 Hilbert-valued processes, 1 33 Hyperbolic absolute risk-aversion unility functions, 226-228 Inaccessible, 107 Independent events, 4 Independent random variable, 6 Index bond, demand for, 2 30- 233 as a hedge against inflation. 233 Indicator function, 46 Inflation, expeeted rate. 1 5 2 inertia, 1 80 stochastic process, 1 80, 230 Information sets, semi-strong, 22 strong, 22-23 weak, 22- 24, 26 Initial distribution. 40 Integrability property, 1 8 I n tegrable, 9 , 1 8
Subject index Integrable stochastic sequence, 5 1 -5 2 Integral equation, 69 International monetary model under uncertainty, 2 1 1 Intertemporal stochastic optimization, 28 Interest rate process, 2 3 3- 2 34 Investment-output plan, 1 5 8 Ito's lemma, 80-92, 95, 1 22- 1 23, 1 30,
142, 144. 146, 175, 1 80, 208-209, 2 1 8- 2 1 9, 224, 234, 23 7, 272 Ito's stochastic differential equation, 68, 92-96, 1 33, 1 4 2- 1 4 3, 180, 2 1 7, 2 1 9, 220- 221 ' 223, 227, 230, 234, 236, 239, 268- 270 Ito's stochastic integral, 74, 76, 78, 1 3 3 1 35 Jensen's inequality, 1 5 , 30, 58 Jump process, 1 2 1 - 1 24, 1 39, 209, 228-
230 Kolmogorov's backward equation, 1 0 1 Kolmogorov's inequality, 30 Kolmogorov's theorem, 34 Lack of memory, 36 Laplace transform, 1 9 8 Law of a random variable, 5 Learning without forgetting, 1 8 Lebesgue integral, 9, 7 3 Lebesgue measure, 3 Left-stable, I 03 Liapunov function, 104 Liapunov- Kushncr approach to stability,
1 04- 1 06
Liapunov method, I 04 Life-cycle consumption model, 28 Limit in mean square, 72 Limit inferior of a sequance, 3, 8, 1 0 Limit superior o f a sequence, 3 , 8, 1 0 Lipschitz condition, 94, 1 6 2 Local asymptotic stochastic stability, 105 Long bond, 235 Lower semi-martingale, 6 1 Market price of risk, 234, 236-238 Markov chain, 39- 40, 62 Markov process, 39, 4 1 , 54, 67, 98, 1 5 8 Markov property, 36, 39, 4 1 , 99, 144 - 1 4 5 Martingale, 1 7, 6 1 , 80, 236, 2 6 1 Martingale property, 1 7 , 24, 28, 29, 6 1 -62 Martingale convergence theorem, 32 Maximum principle, 1 09 - 1 1 3 , 1 24
301
Maximum principle for jump processes,
1 2 3 - 1 24 McCall's theorem, 1 9 1 McKenzie competitive process, 166 Marginal cost, 190 Marginal value of capital, 1 1 9 Mean, I t Mean ergodic theorem, 246 Mean-square-differentiable, 98 Measurability property, 1 8 Measure, 2, 57 Measurable function, 4 Measurable set, 2 Measurable space, 2 Merton's costate equation, 269 Miroshnichcnko's theorem, 1 9 7 Moment, kth, 1 1 , 102 kth central, 1 1 , 102 Monetary perfect foresight e<]uilibrium, 2 1 2 Monotone case, 52-53 Monotone convergence theorem, 10, 59 Monotoneity property, 17, 58 Myopic behavior, 5 1 Myopic perfect foresight, 1 7 9 Negative part of a function, 9 Negligible sl!t, 3 Net rate of return per unit of risk, 222 Nominal bond, 2 3 1 Nonanticipating function, 7 3 Nonanticipating property, 1 3 3 Nonanticipating a-fields, 72-73 Non-differentiability of Wiener process, 37 Normal accumulation rate, 23 Normal backwardation, 25 Normal distribution, 6 One-st·ep transition probability, 39 Optimal random process, 164 Optimal reward, 1 25 Optimal saving, 1 4 8 Optimal strategy, 1 25 Optimal stopping, boundary, 1 26 rule, 43, 45, 57 existence of, 4 7 Ornste in-Uhlenbeck process, 269-270 Outcome, 1 Pairwise disjoint sets, 2 Parabolic partial differential e<JUation, 1 26 Parameter process,
Subject index
302 continuous. 32
Ramsey's ruk, 1 50
discrete, 32 Part of a function.
Random variable, 4 Random vector, 4
llt:!!ativc, 9 positive, 9
Rational expectation' C
1 7 1 . 1 78, 248
Partition, 70 Point equtllhrium ..olutton. I 02
Rational expcctattons hypothc)ois. 22. 28,
Poisson distribution, 6
Realizatton of a prOl·cs�. 3 3
1 5 3- 156, 1 59, 1 7 8- 1 82. 246
Poisson process, 4 2. 228
Reflecting barrier, 200
Policy function, I l l
Reservation wage, 49
Pontryagin stocha�tiL maximum pnnctple.
Restriction on growth c..onditton, 94-- 1 6 2
112
RC\\ ard function. 4 3 , 1 25
Positive part of a function, 9
average, 1 2 5
Present discounted-V<�lue rule of capitaliza
Riemann -St iclje� integral, 7 2
ti on, 25 Price,
Righ t-stable. 103 Rio;k aversion, 24. 1 5 1 , 189. 23 2
fu turc, 21 now-expected level of the terminal spot,
22 spot, 2 1 terminal 'pot, 22
Risk free, 2 2 1 . 225, 243, 258. 270 Risk ncu t wl. 48 Risk premium, 1 5 2 Sample path, 3 3 - 34 Sample point, I
Probability,
conditional, I 2
Sampk �lability, 1 3 1
13 converges in, 7, 77 mea!lure, 58
Sampling with recall. 48
spa<:c, 3
Sampling without recall, 48 Semi-martingale, 6 1 Semi-strong information. 22
Process, admissible, 1 83
Scparabk proce��. 35
binomial, 274 Browman motion, 36
Separation theorem, 229 Set, 1 Sharpc- Lintncr formulation, 243, 245, 272
38
competitive, 1 6 1 168 continuouo; parameter. 32 discre te parameter, 32
o-algcbra. 2 o-ficld, 2
Markov, 39, 4 1
o-finite measure, 1 2
Poisson, 42, 22S scparabh!, 35 spot intereo;t rate, 233 - 234
Smooth-fit equations. 1 27
time-homo�eneous, 38, 4 1 Wiener, 36 38
Solutions, 9 3
Solow neoclassical differential equation of growth, 6 7, 142 bounded ness of, I 04
Properties of �elution� of �tochao;tic differ
existence of. 93
ential equations, 92, 96 Property of boundcdnc�� of solu tton�. l 04,
uniqueness of, 9 3-94
206
Property of dependence of solutions on parameter' and inittal data, 9 7 Property o f differentiability o f soluttons, 9 8 Propertie' of, conditional expected value, 1 4 - 1 5 conditional probability. 1 3 Ito's integral. 7 8
80
Radon-Nikodym dl'rivative, 1 2 Radon -Nikody m tht'orcm, 1 2
14
94
Space, I measurable, 2 probability, 3 Spot prices, 2 1 Spot rate, 233 Stable in the mean, 13 7, 1 8 1 Standard deviation, 1 1 StandardiLcd Wiener procesc;, 36, 1 74 State space, 33
State variable, I 08 Stationary, distribution, 40, I 06
I 08, 146
147
Subject index increments,
37
39 Steady state distribution. 40, 1 06 Step function, 7 3 Stochastic capital theory, 53 -57. 1 94 -205 Stochastic Cauchy Sequence, 75-76 Stochastic control, 108- 1 1 8, 1 23 - 1 24, 1 38- 1 40, 223-228, 230-233, 27 1 Stochastic demand for money, 238- 241 , 277 Stochastic demand function, 207 Stochastic differential, 8 1 Stochastic initial value problem. 92 Stochastic integrable sequence, 5 1 -5 2 Stochastic integral, 74, 76, 78, 1 30, 1 331 35 Stochastic integration, 69-80 Stochastic linear equations, 94- 96, 1 351 36, 180 (see also lto's stochastic differential transition probability.
equation)
105
Stochastic local asymptotic stability,
303
Strong martingale hypothesis, Strong information, 22- 23
Subadditivity property, 58 Submartingale, 19, 23-24, 27, Supermartingale, 19. 57, 6 1 Sy11tcmatic risk, 242, 258
Terminal spot price,
22
Tobin's fundamental equation. Trajectory of a process,
32
1 25
Uniformly integrable, Upcrossing, 3 1
52
Upcrossing martingale- theorem,
Stochastic Stochastic Stochastic
Version,
213 1 36-
139
1 3- 14, 34
Wald's lemma,
44
Weak information, White noise,
Stochastic stability in the large, I 05,
6
11
Verification theorem,
210 Ramsey problem, 148- 150 rate of intlation, 2 1 7- 220, 230 search theory, 48-5 1 , 60, 2 1 2-
23- 24, 26
38
Wiener process,
36-38, 42, 6 2, 68, 7 1 , 1 1 5,
215
1 38 Stochastic variational problem,
164
covariance of,
37
Stopping region,
geometric, 38 nondiffercntiability of,
Stopping rule,
standardized,
Stochastically equivalent,
32
43
independent random. random. 4 Variance.
Stochastic quantity theory of money,
1 6 7, 249
Tranwcrsality condition, Tychonov condition,
Value o f the sequence,
Process)
207
35
Transition probability matrix, 40- 4 1
Vanable,
(see also
1 16
Term structure, 233, 269, 276 Term structure equation. 235
constraints, 1 18, 223- 225, 230- 233 Stochastic neoclassical differential equation, Stochastic process,
29, 6 1 , 80
Taylor's theorem, 82, 87-88, 1 09, Ttchnological uncertainty, 244
Stochastic maximum principle with
1 42- 143
23
34
1 26 43, 46 Stopping time, 43, 46 Stopping variable, 43, 46 Strasbourg approach, 80 Stratonovich integral, 1 30, 1 34
37
36, 1 74
whitt.• noise as derivative of. with drift, 38, 40 With probability l (w.p. l.),
9
38, 68
