STOCHASTIC CALCULUS AND DIFFERENTIAL EQUATIONS FOR PHYSICS AND FINANCE Stochastic calculus provides a powerful description of a specific class of stochas- tic processes in physics and finance. However, many econophysicists struggle to understand it. This book presents the subject simply and systematically, giving graduate students and practitioners a better understanding and enabling them to applythemethodsinpractice. The book develops Ito calculus and Fokker–Planck equations as parallel approaches to stochastic processes, using those methods in a unified way. The focus is on nonstationary processes, and statistical ensembles are emphasized in time series analysis. Stochastic calculus is developed using general martin- gales.Scalingandfattailsarepresentedviadiffusivemodels.FractionalBrownian motion is thoroughly analyzed and contrasted with Ito processes. The Chapman– Kolmogorov and Fokker–Planck equations are shown in theory and by example to be more general than a Markov process. The book also presents new ideas in financialeconomicsandacriticalsurveyofeconometrics. joseph l. mccauley is Professor of Physics at the University of Houston. Duringhiscareerhehascontributedtoseveralfields,includingstatisticalphysics, superfluids,nonlineardynamics,cosmology,econophysics,economics,andfinance theory. STOCHASTIC CALCULUS AND DIFFERENTIAL EQUATIONS FOR PHYSICS AND FINANCE JOSEPH L. McCAULEY UniversityofHouston cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,Sa˜oPaulo,Delhi,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521763400 (cid:2)C JosephL.McCauley2013 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2013 PrintedandboundintheUnitedKingdombytheMPGBooksGroup AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloguinginPublicationdata McCauley,JosephL. Stochasticcalculusanddifferentialequationsforphysicsandfinance/JosephL.McCauley, UniversityofHouston. pages cm ISBN978-0-521-76340-0 1.Stochasticprocesses. 2.Differentialequations. 3.Statisticalphysics. 4.Finance–Mathematicalmodels. I.Title. QC20.7.S8M39 2012 519.2–dc23 2012030955 ISBN978-0-521-76340-0Hardback Additionalresourcesforthispublicationat www.cambridge.org/9780521763400 CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublicationanddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. Forouryoungestones, Will,Justin,Joshua,Kayleigh,andCharlie Contents Abbreviations pagexi Introduction 1 1 Randomvariablesandprobabilitydistributions 5 1.1 Particledescriptionsofpartialdifferentialequations 5 1.2 Randomvariablesandstochasticprocesses 7 1.3 Then-pointprobabilitydistributions 9 1.4 Simpleaveragesandscaling 10 1.5 Paircorrelationsand2-pointdensities 11 1.6 Conditionalprobabilitydensities 12 1.7 Statisticalensemblesandtimeseries 13 1.8 Whenarepaircorrelationsenoughtoidentifyastochastic process? 16 Exercises 17 2 Martingales,Markov,andnonstationarity 18 2.1 Statisticallyindependentincrements 18 2.2 Stationaryincrements 19 2.3 Martingales 20 2.4 Nonstationaryincrementprocesses 21 2.5 Markovprocesses 22 2.6 Driftplusnoise 22 2.7 Gaussianprocesses 23 2.8 Stationaryvs.nonstationaryprocesses 24 Exercises 26 3 Stochasticcalculus 28 3.1 TheWienerprocess 28 3.2 Ito’stheorem 29 vii viii Contents 3.3 Ito’slemma 30 3.4 Martingalesforgreenhorns 31 3.5 First-passagetimes 33 Exercises 35 4 ItoprocessesandFokker–Planckequations 37 4.1 Stochasticdifferentialequations 37 4.2 Ito’slemma 39 4.3 TheFokker–Planckpde 39 4.4 TheChapman–Kolmogorovequation 41 4.5 Calculatingaverages 42 4.6 Statisticalequilibrium 43 4.7 Anergodicstationaryprocess 45 4.8 Earlymodelsinstatisticalphysicsandfinance 45 4.9 Nonstationaryincrementsrevisited 48 Exercises 48 5 SelfsimilarItoprocesses 50 5.1 Selfsimilarstochasticprocesses 50 5.2 Scalingindiffusion 51 5.3 Superficiallynonlineardiffusion 53 5.4 Isthereanapproachtoscaling? 54 5.5 Multiaffinescaling 55 Exercises 56 6 FractionalBrownianmotion 57 6.1 Introduction 57 6.2 FractionalBrownianmotion 57 6.3 ThedistributionoffractionalBrownianmotion 60 6.4 Infinitememoryprocesses 61 6.5 Theminimaldescriptionofdynamics 62 6.6 Paircorrelationscannotscale 63 6.7 Semimartingales 64 Exercises 65 7 Kolmogorov’spdesandChapman–Kolmogorov 66 7.1 ThemeaningofKolmogorov’sfirstpde 66 7.2 Anexampleofbackward-timediffusion 68 7.3 DerivingtheChapman–Kolmogorovequationfor anItoprocess 68 Exercise 70 Contents ix 8 Non-MarkovItoprocesses 71 8.1 FinitememoryItoprocesses? 71 8.2 AGaussianItoprocesswith1-statememory 72 8.3 McKean’sexamples 74 8.4 TheChapman–Kolmogorovequation 78 8.5 Interactingsystemwithaphasetransition 79 8.6 ThemeaningoftheChapman–Kolmogorovequation 81 Exercise 82 9 Black–Scholes,martingales,andFeynman–Kac 83 9.1 Localapproximationtosdes 83 9.2 Transitiondensitiesviafunctionalintegrals 83 9.3 Black–Scholes-typepdes 84 Exercise 85 10 Stochasticcalculuswithmartingales 86 10.1 Introduction 86 10.2 Integrationbyparts 87 10.3 Anexponentialmartingale 88 10.4 Girsanov’stheorem 89 10.5 AnapplicationofGirsanov’stheorem 91 10.6 TopologicalinequivalenceofmartingaleswithWiener processes 93 10.7 SolvingdiffusivepdesbyrunninganItoprocess 96 10.8 First-passagetimes 97 10.9 Martingalesgenerallyseen 102 Exercises 105 11 Statisticalphysicsandfinance:Abriefhistoryofeach 106 11.1 Statisticalphysics 106 11.2 Financetheory 110 Exercise 115 12 Introductiontonewfinancialeconomics 117 12.1 Excessdemanddynamics 117 12.2 AdamSmith’sunreliablehand 118 12.3 Efficientmarketsandmartingales 120 12.4 Equilibriummarketsareinefficient 123 12.5 HypotheticalFXstabilityunderagoldstandard 126 12.6 Value 131