Table Of ContentIntroduction to Stochastic Analysis
Introduction to
Stochastic Analysis
Integrals and Differential Equations
Vigirdas Mackevičius
First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as
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ISTE Ltd John Wiley & Sons, Inc.
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The rights of Vigirdas Mackevičius to be identified as the author of this work have been asserted by him
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____________________________________________________________________________________
Library of Congress Cataloging-in-Publication Data
Mackevičius, Vigirdas.
Introduction to stochastic analysis, integrals, and differential equations / Vigirdas Mackevičius.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-84821-311-1
1. Stochastic analysis. I. Title.
QA274.2.M33 2011
519.2'2--dc22
2011012249
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-84821-311-1
Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Chapter1.Introduction: BasicNotionsofProbabilityTheory . . . . . . . 17
1.1.Probabilityspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.Randomvariables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.Characteristicsofarandomvariable . . . . . . . . . . . . . . . . . . . . 21
1.4.Typesofrandomvariables . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.Conditionalprobabilitiesanddistributions. . . . . . . . . . . . . . . . . 26
1.6.Conditionalexpectationsasrandomvariables . . . . . . . . . . . . . . . 27
1.7.Independenteventsandrandomvariables . . . . . . . . . . . . . . . . . 29
1.8.Convergenceofrandomvariables. . . . . . . . . . . . . . . . . . . . . . 29
1.9.Cauchycriterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.10.Seriesofrandomvariables . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.11.Lebesguetheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.12.Fubinitheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.13.Randomprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.14.Kolmogorovtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
Chapter2.BrownianMotion . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.Definitionandproperties. . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.WhitenoiseandBrownianmotion . . . . . . . . . . . . . . . . . . . . . 45
2.3.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Chapter3.StochasticModelswithBrownianMotionandWhiteNoise . . 51
3.1.Discretetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2.Continuoustime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5
6 IntroductiontoStochasticAnalysis
Chapter4.StochasticIntegralwithRespecttoBrownianMotion . . . . . . 59
4.1.Preliminaries.Stochasticintegralwithrespecttoastepprocess. . . . . 59
4.2.Definitionandproperties. . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3.Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Chapter5.Itô’sFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.1.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Chapter6.StochasticDifferentialEquations . . . . . . . . . . . . . . . . . . 97
6.1.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Chapter7.ItôProcesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.1.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Chapter8.StratonovichIntegralandEquations . . . . . . . . . . . . . . . . 125
8.1.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Chapter9.LinearStochasticDifferentialEquations . . . . . . . . . . . . . 137
9.1.ExplicitsolutionofalinearSDE . . . . . . . . . . . . . . . . . . . . . . 137
9.2.ExpectationandvarianceofasolutionofanLSDE . . . . . . . . . . . . 141
9.3.Otherexplicitlysolvableequations . . . . . . . . . . . . . . . . . . . . . 145
9.4.Stochasticexponentialequation. . . . . . . . . . . . . . . . . . . . . . . 147
9.5.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Chapter10.SolutionsofSDEsasMarkovDiffusionProcesses . . . . . . . 155
10.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.2.BackwardandforwardKolmogorovequations. . . . . . . . . . . . . . 161
10.3.Stationarydensityofadiffusionprocess . . . . . . . . . . . . . . . . . 172
10.4.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Chapter11.Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
11.1.Additivenoise: Langevinequation . . . . . . . . . . . . . . . . . . . . 180
11.2.Additivenoise: generalcase . . . . . . . . . . . . . . . . . . . . . . . . 180
11.3.Multiplicativenoise: generalremarks. . . . . . . . . . . . . . . . . . . 184
11.4.Multiplicativenoise: Verhulstequation . . . . . . . . . . . . . . . . . . 186
11.5.Multiplicativenoise: geneticmodel . . . . . . . . . . . . . . . . . . . . 189
Chapter12.ExampleinFinance: Black–ScholesModel . . . . . . . . . . . 195
12.1.Introduction:whatisanoption? . . . . . . . . . . . . . . . . . . . . . . 195
Contents 7
12.2.Self-financingstrategies . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.2.1.Portfolioanditstradingstrategy . . . . . . . . . . . . . . . . . . 197
12.2.2.Self-financingstrategies . . . . . . . . . . . . . . . . . . . . . . . 198
12.2.3.Stockdiscount. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
12.3.Optionpricingproblem:theBlack–Scholesmodel . . . . . . . . . . . 204
12.4.Black–Scholesformula . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
12.5. Risk-neutral probabilities: alternative derivation of Black–Scholes
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
12.6.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Chapter13.NumericalSolutionofStochasticDifferentialEquations . . . 217
13.1.Memoriesofapproximationsofordinarydifferentialequations . . . . 218
13.2.Eulerapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
13.3.Higher-orderstrongapproximations . . . . . . . . . . . . . . . . . . . 224
13.4.First-orderweakapproximations . . . . . . . . . . . . . . . . . . . . . 231
13.5.Higher-orderweakapproximations . . . . . . . . . . . . . . . . . . . . 238
13.6.Example:Milstein-typeapproximations . . . . . . . . . . . . . . . . . 241
13.7.Example:Runge–Kuttaapproximations . . . . . . . . . . . . . . . . . 244
13.8.Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Chapter14.ElementsofMultidimensionalStochasticAnalysis . . . . . . . 251
14.1.MultidimensionalBrownianmotion. . . . . . . . . . . . . . . . . . . . 251
14.2.Itô’sformulaforamultidimensionalBrownianmotion . . . . . . . . . 252
14.3.Stochasticdifferentialequations . . . . . . . . . . . . . . . . . . . . . . 253
14.4.Itôprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
14.5.Itô’sformulaformultidimensionalItôprocesses . . . . . . . . . . . . 256
14.6.Linearstochasticdifferentialequations . . . . . . . . . . . . . . . . . . 256
14.7.Diffusionprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
14.8.Approximationsofstochasticdifferentialequations . . . . . . . . . . . 259
Solutions,Hints,andAnswers . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Preface
Initially, I wanted to entitle the textbook “Stochastic Analysis for All” or
“Stochastic Analysis without Tears”, keeping in mind that it will be accessible not
onlytostudentsofmathematicsbutalsotophysicists,chemists,biologists,financiers,
actuaries, etc. However,thoughaimingforaswide areadabilityaspossible, finally,
Irejectedsuchtitlesregardingthemastooambitious.
Most people have an intuitive concept of probability based on their own life
experience. However, efforts to precisely define probabilistic notions meet serious
difficulties;thisisseenlookingatthehistoryofprobabilitytheory—fromelementary
combinatorial calculations in hazard games to a rigorous axiomatic theory, having
a store of applications in various practical and scientific areas. Possibly, as in no
other area of mathematics, in probability theory, there is a huge distance from the
beginningandelementstothepreciseandrigoroustheory.Thisisfirstlyrelatedtothe
fact that the “palace” of probability theory is built on the substructure of the rather
subtle and abstract measure theory. For example, for a mathematician, a random
variable is a real measurable function defined on the space of elementary events,
while for practitioners—physicists, chemists, biologists, actuaries, etc.—it is some
quantitydependingon chance. Fora mathematician,therandomnessis externalized
by a probability measure on the measure space (space of elementary events with a
σ-algebra of its subsets), which, together with the latter, constitutes a probability
space, the primary notion of probability theory. On its basis, all the notions of
probabilitytheory,suchasrandomvariable,independence,expectation,variance,etc.,
aredefined. Fora practitioner,randomnessisrepresentedbya distributionfunction,
which is well understood intuitively and can be used to define many of the above-
mentioned notions, though with some loss of strictness and generality. Thus, every
author of a book on probability theory has to look for a middle ground between
strict abstracttheory and accessibility for researchersin other sciences and evenfor
mathematiciansworkinginotherareasofmathematics. Theauthorofthisbookisno
anexception.
9
10 IntroductiontoStochasticAnalysis
A relatively recent area of probability theory, stochastic differential equations
are receiving increasing attention by researchersand practitionersin variousnatural
and applied sciences. First of all, this is related to the fact that “ordinary”
(deterministic)differentialequations,whenmodelingvariousreal-worldphenomena,
usuallydescribeonlytheaveragebehaviorofonesystemoranother. However,real-
worldsystemsaremostofteninfluencedbymanydifferentrandomfactors,alsocalled
perturbations. Itappearsthatsuchperturbations,whentheyaresufficientlyintensive,
donotonly“disorganize”thesystemforcingittooscillateabouttheaveragebehavior,
but also qualitatively change the average behavior itself. It is clear that, in such a
situation,deterministicequationscannot,inprinciple,adequatelydescribethesystem.
However, in stochastic analysis (theory of stochastic integration and stochastic
differentialequations)theproblemofstrictness-to-simplicityratioarisesmuchmore
strongly. Modern stochastic analysis is closely related with rather abstract areas
of probability theory, such as general theory of random processes and theory of
martingales. How can we avoid them when teaching the basics of stochastic
differential equations? The author tries to present them in the spirit of “naive”
stochastic integration. Just as in applications, the notion of a random variable is
imperceptiblyreplacedbyitsdistributionfunction,herewereplacetheveryimportant
notion of filtration (an increasing system of σ-algebras of events) by the intuitively
easier notionofhistory, or past. Correspondingly,the notionof anadapted random
process1becomeseasiertounderstandifbyanadaptedprocesswemeantheonewith
valuesthat,ateverytimemoment,“belongtothehistory”,or,inotherwords,“depend
onlyonthepast”.
A result of all these methodicalsearches and“inventions”is that the textbookis
writtenontwo levels. Themainlevel, whichisdevoted“foreverybody”,containsa
simplified theory of stochastic integrationand stochasticdifferentialequations, with
some lack of rigidity and preciseness (though without any “cheating”). The author
hopesthat it will be understandableto everybodywho is acquaintedwith the basics
of probability theory in the scope of a standard elementary course, together with a
minimalmathematical “ear”. Thesecondlevelisdevotedtoarigorousintroduction
tostochasticanalysisforstudentsofmathematics.Thecomments,definitions,detailed
proofs,ortheirrevisionswritteninthislevelaremarkedbythesymbol (cid:13)M.
To apply stochastic analysis in a treatment of real-world random processes, we
have to construct one or another theoretical model and to know how to simulate it.
Therefore, in the book(Chapter 13) much attention is paid to numericalsolution of
stochasticdifferentialequationsor,inotherwords,totheircomputersimulation.
1.Asaprocesswhich,ateverytimemoment,ismeasurablewithrespecttothecorresponding
σ-algebrafromthegivenfiltration....
Preface 11
We essentially restrict ourselves to the one-dimensional case. Passing to the
multidimensionalcase is often related not with principaldifficulties, but rather with
technicalinconvenienceinusing,forabeginner,complicatednotationandformulas.
However,forthereader’sconvenience,inthelastchapter,wepresentanoverviewof
themaindefinitionsandstatementsinthemultidimensionalcase.
Weusethedoublenumberingofstatements(theorems,propositions,etc.):thefirst
numberdenotesthenumberofthechapter,andthesecondindicatesthenumberofthe
statementwithinthechapter.
Thoughanenormousnumberofbooksandpapershaveinfluencedthecontentsof
thisbook,theshortreferencelistincludesonlythosedirectlyusedbytheauthor.
Almost all the graphs in the book were drawn by using the TEX macro package
PICTEX, while the graph points were calculated by using Pascal programs in Free-
Pascalenvironment.
The book essentially is a translation of the author’s book Stochastic Analysis:
StochasticIntegralsandStochasticDifferentialEquations(VilniusUniversityPress,
2005) from Lithuanian, complemented by the chapter with an application example
in finance. The author would like to thank a large number of students of the
Department of Mathematics and Informatics of Vilnius University and, especially,
Ke¸stutis Gadeikis. Thanks to them, the book contains significantly fewer mistakes
andmisprints.
Vigirdas Mackevicˇius
Vilnius, May 2011
Notation
N Thesetofpositiveintegers 1,2,...
{ }
N N +
∪{ ∞}
N N 0
+
∪{ }
R Realline( ,+ )
−∞ ∞
R ExtendedreallineR ,+
∪{−∞ ∞}
R Thesetofnon-negativerealnumbers[0,+ )
+
∞
“forall”,“foreach”,“forevery”
∀
...:... Thereexists...suchthat...
∃
:= “Denote”,“isequalbydefinitionto”
Identicallyequal
≡
= “implies”
⇒ ⇒
, “ifandonlyif”,“necessaryandsufficient”,“equivalent”
⇐⇒ ⇔
Emptyset
∅
x A AnelementxbelongstoasetA
∈
x / A AnelementxdoesnotbelongtoasetA
∈
Ac ThecomplementofasetA;thecontraryeventtoaneventA
A B,B A AsetAisasubsetofasetB;aneventAimpliesaneventB
⊂ ⊃
x A Asequence x , n N ofelementsofasetA
n n
{ }⊂ { ∈ }
A B Theunionofsets(events)AandB
∪
A B Theintersectionofsets(events)AandB
∩
∞ An, ∞n=1An Theunionofasequenceofsets(events)An,n∈N
n=1
S S
∞ An, ∞n=1An Theintersectionofasequenceofsets(events)An,n∈N
n=1
T T
13
