Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications

263 Pages·04.199 MB·English

by  René Carmona

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Financial Mathematics
p
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oj Carmona, René, Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial
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FM01_Carmona_FM-01-14-16.indd 2 1/14/2016 9:19:16 AM
RENÉ CARMONA
Princeton University
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ojs Princeton, New Jersey
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M li Games with Financial
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Society for Industrial and Applied Mathematics
Philadelphia
FM01_Carmona_FM-01-14-16.indd 3 1/14/2016 9:19:16 AM
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a. Copyright © 2016 by the Society for Industrial and Applied Mathematics
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rn All rights reserved. Printed in the United States of America. No part of this book may
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o be reproduced, stored, or transmitted in any manner without the written permission of
rg/j the publisher. For information, write to the Society for Industrial and Applied Mathematics,
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m. 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA.
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si Trademarked names may be used in this book without the inclusion of a trademark symbol.
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w These names are used in an editorial context only; no infringement of trademark is intended.
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p:// Royalties from the sale of this book will be donated to the SIAG/FME to be used for a prize.
htt Publisher David Marshall
ee Acquisitions Editor Elizabeth Greenspan
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ht; Developmental Editor Gina Rinelli
g Managing Editor Kelly Thomas
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y Production Editor Ann Manning Allen
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c Copy Editor Nicola Howcroft
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o Production Manager Donna Witzleben
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s Production Coordinator Cally Shrader
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c Compositor Techsetters, Inc.
M li Graphic Designer Lois Sellers
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o Library of Congress Cataloging-in-Publication Data
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bj Names: Carmona, R. (René)
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n s Title: Lectures on BSDEs, stochastic control, and stochastic differential
o games with financial applications / René Carmona, Princeton University,
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b Princeton, New Jersey.
stri Other titles: Lectures on backward stochastic differential equations,
di stochastic control, and stochastic differential games with financial
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1. applications
3 Description: Philadelphia : Society for Industrial and Applied Mathematics,
2
1. [2016] | Series: Financial mathematics ; 01 | Includes bibliographical
9.
3 references and index.
2
2. Identifiers: LCCN 2015038344 | ISBN 9781611974232
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1 Subjects: LCSH: Stochastic differential equations. | Stochastic control
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7 t theory. | Business mathematics.
1 Classification: LCC QA274.23 .C27 2016 | DDC 519.2/7--dc23
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1 LC record available at http://lccn.loc.gov/2015038344
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o Contents
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p:// Preface vii
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e ListofNotation ix
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ht; I StochasticCalculusPreliminaries 1
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p 1 StochasticDifferentialEquations 3
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r 1.1 NotationandFirstDefinitions . . . . . . . . . . . . . . . . . . . . . . 3
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e 1.2 ExistenceandUniquenessofStrongSolutions:TheLipschitzCase . . 4
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n 1.3 SDEsofMcKean–VlasovType . . . . . . . . . . . . . . . . . . . . . 11
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M li 1.4 ConditionalPropagationofChaos . . . . . . . . . . . . . . . . . . . . 18
1.5 Notes&Complements. . . . . . . . . . . . . . . . . . . . . . . . . . 26
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o 2 BackwardStochasticDifferentialEquations 27
ct t 2.1 IntroductionandFirstDefinitions . . . . . . . . . . . . . . . . . . . . 27
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bj 2.2 Mean-FieldBSDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
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s 2.3 ReflectedBackwardStochasticDifferentialEquations(RBSDEs) . . . 37
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o 2.4 Forward-BackwardStochasticDifferentialEquations(FBSDEs). . . . 40
buti 2.5 ExistenceandUniquenessofSolutions . . . . . . . . . . . . . . . . . 46
stri 2.6 TheAffineCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
di 2.7 Notes&Complements. . . . . . . . . . . . . . . . . . . . . . . . . . 63
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1. II StochasticControl 65
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2 3 ContinuousTimeStochasticOptimizationandControl 67
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3 3.1 OptimizationofStochasticDynamicalSystems. . . . . . . . . . . . . 67
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o 3.2 FirstFinancialApplications . . . . . . . . . . . . . . . . . . . . . . . 75
17 t 3.3 DynamicProgrammingandtheHJBEquation . . . . . . . . . . . . . 79
0/ 3.4 InfiniteHorizonCase . . . . . . . . . . . . . . . . . . . . . . . . . . 85
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4/ 3.5 ConstraintsandSingularControlProblems . . . . . . . . . . . . . . . 87
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d 3.6 ViscositySolutionsofHJBEquationsandQVIs . . . . . . . . . . . . 101
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a 3.7 ImpulseControlProblems . . . . . . . . . . . . . . . . . . . . . . . . 107
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nl 3.8 ErgodicControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
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4 ProbabilisticApproachestoStochasticControl 119
4.1 BSDEsandStochasticControl . . . . . . . . . . . . . . . . . . . . . 119
4.2 PontryaginStochasticMaximumPrinciple . . . . . . . . . . . . . . . 125
4.3 Linear-Quadratic(LQ)Models . . . . . . . . . . . . . . . . . . . . . 136
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4.4 OptimalControlofMcKean–VlasovDynamics . . . . . . . . . . . . 141
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g/j 5.1 IntroductionandFirstDefinitions . . . . . . . . . . . . . . . . . . . . 165
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m. 5.2 SpecificExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
a 5.3 WeakFormulationandtheCaseofUncontrolledVolatility . . . . . . . 182
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5.5 ASimpleModelforSystemicRisk . . . . . . . . . . . . . . . . . . . 198
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htt 5.7 Notes&Complements. . . . . . . . . . . . . . . . . . . . . . . . . . 216
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s 6 Mean-FieldGames 219
ht; 6.1 IntroductionandFirstDefinitions . . . . . . . . . . . . . . . . . . . . 219
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ri 6.2 AFullSolutionWithouttheCommonNoise . . . . . . . . . . . . . . 226
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op 6.3 PropagationofChaosandApproximateNashEquilibriums . . . . . . 236
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r 6.4 ApplicationsandOpenProblems . . . . . . . . . . . . . . . . . . . . 243
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e 6.5 Notes&Complements. . . . . . . . . . . . . . . . . . . . . . . . . . 250
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M lic Bibliography 253
A AuthorIndex 261
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o Preface
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w. ThisbookgrewoutofthelecturenotesIpreparedforagraduateclassItaughtatPrince-
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w tonUniversityin2011–12,andagainin2012–13. Mygoalwastointroducethestudents
p:// tostochasticanalysistools,whichplayanincreasingroleintheprobabilisticapproachto
htt optimization problems, including stochastic control and stochastic differential games. I
ee hadinvestedquiteabitofeffortintryingtounderstandthegroundbreakingworksofLasry
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ht; and Lions on mean field games, and of Caines, Huang, and Malhame´ on Nash certainty
g equivalence. These initial results were intriguing, and definitely screaming for a proba-
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y bilisticinterpretation. Whilethetoolsofoptimalcontrolofstochasticdifferentialsystems
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co are taughtin many graduateprogramsin applied mathematicsand operationsresearch, I
or was intriguedby the factthat gametheory,and especially the theoryof stochastic differ-
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s ential games, are rarely taught in these programs. In fact, I was shocked by the lack of
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ce publishedliteratureinbookform,asadstateofaffairswhichpromptedmetowritelecture
M li notesfortheclass.
A This would have been the end of the story if I hadn’tshared some of my notes with
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S a friend andco-authorof mine, Jean-PierreFouque, who decidedto convinceme to turn
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ct t these lecture notes into a book. His perseverance, together with my desire to help those
e appliedmathematicianstryingto learn the theoryofstochastic differentialgamesdespite
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u the lack of sources in textbookform, helped me to find the time to clean up my original
s
n classnotes. Still,thisshortprefaceshouldcontainacleardisclaimeremphasizingthefact
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uti that the present manuscriptis more a set of lecture notes than a polished and exhaustive
b
ri textbookonthesubjectmatter. Manyexpertsonthesubjectcouldproducebettertreatises.
dist But untilthen, I hopethese noteswill behelpfulto youngresearchersandnewcomersto
Re stochasticanalysis,eagertounderstandthesubtletiesofthemodernprobabilisticapproach
1. tostochasticcontrolandstochasticdifferentialgames.
3
2
1. Acknowledgments. Iwouldlike tothankthestudentswhosepatienceallowedmetome-
9.
3 anderthroughthemazeofconceptsandtheoriesIwasdiscoveringwhileteaching.Special
2
2. thanksaredueto DanLacker,KevinWebster andGeoffreyZhu, whoseregularfeedback
3
1 helped me keep the lectures on track. Last but not least, I want to thank Franc¸ois De-
o
7 t larue, not only for a couple of enlightening guest lectures, but for an enjoyable collabo-
0/1 ration which taught me the subtle intricacies of forward-backwardstochastic differential
1 equations—nothingcanrivallearningfromthemaster!
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d ImportantAside. Theroyaltiesgeneratedbythesales(ifany)ofthisbookwillbeused,in
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a theirentirety,tofundastudentprizeawardedeveryotheryearduringtheSIAMConference
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Rene´Carmona
Princeton,NJ
August6,2015
vii
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o List of Notation
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p:// BSDE(Ψ,ξ),27 S2,3 DPP,81
htt C([0,T];Rn),69 PW0,19
ee Cb2(Rd),7 μN,16 FBSDE,41,42
ght; s DC1,,925([0,T]×Rd),8 μLNxs,,82,2100 FPS,167
yri Mhf,8 Lt,10 HJB,67,82,119
op W(0),11,12 Lt∗,10
r c W(2),11,12 σ†,71 LQ,63,136,179
e o Fx,69 var{η},73
ns H2(A),123 H˜,71 MFG,172,216
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M lic QPX,3,313 aa==((aa1t),t·∈·[·0,,Ta],N6)8,167 MMPRST,,12677
A Ξ,244 gconv,80
SI A,68,167 v∗,102 NCE,251
ect to AB(iE,1)6,768 (vE∗,),110021 ODE,84
bj C,69 OL,68,167
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n s E,33,74 BAU,76 OLNE,170
butio LM(X(R)d,)1,384 BSDE,27 PDE,7,42
stri P,27,68,69 CL,68
di Pp(E),142 CLFFNE,170 QVI,44,87,94,109
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R δx,16,220 CLNE,170
1. (cid:2)ϕ,μ(cid:3),10 CLPS,167 RBSDE,37
3
2 Bk,3 CRRA,89,98,132,246
9.1. H0,k,3 SDE,3,16,42
23 H2,k,3 DNE,170 SPDE,222
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ix
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ht; Thisintroductorychapterisdevotedtotheanalysisofstochasticdifferentialequations
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ri ofItoˆ type. Inpreparationforthestudyofstochasticcontrolproblemsandstochastic
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p differentialgames,weproveexistenceanduniquenessforequationswhosecoefficients
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c canberandom. Next,wediscusstheMarkovpropertyandtheconnectionwithpartial
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o differentialequationswhenthecoefficientsdonotdependuponthepast. Thelasttwo
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s sectionsarenotclassicalinnatureandtheycanbeskippedinafirstreading. Theyare
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ce devotedtoMcKean–Vlasovequations,whosecoefficientsdependuponthedistribution
M li of thesolution and totheir connection withtheso-called propagation of chaos. The
A lastsectionconsidersthecasewhenthemarginaldistributionsareconditionedbythe
SI knowledgeofaseparateItoˆ diffusion.
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e 1.1 Notation and First Definitions
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n s Weassumethat(Ω,F,F,P)isastochasticbasiswherethefiltrationF = (Ft)0≤t≤T sup-
utio ports an m-dimensionalF-Brownian motion W = (Wt)0≤t≤T in Rm. For each integer
b k ≥ 1, we denoteby H0,k the collectionof all Rk-valuedprogressivelymeasurablepro-
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st cesseson[0,T]×R,andweintroducethesubspaces
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Re (cid:2) (cid:3) T (cid:4) (cid:2) (cid:4)
1. H2,k := Z ∈H0,k; E |Zs|2ds<∞ and S2 := Y ∈H0,k; E sup |Ys|2 <∞ .
3
2 0 0≤t≤T
1.
39. WealsousethenotationBk forthesubspaceofboundedprocesses,namely,
2 (cid:2) (cid:4)
2.
13 Bk := Z ∈H0,k; sup |Zt|<∞, P−a.s. .
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1 Whendealingwith severalpossiblefiltrations, weaddasubscripttospecifythefiltration
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1 withrespecttowhichtheprogressivemeasurabilityneedstobeunderstood. Inthecaseof
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0 scalarprocesses,whenk =1,weskiptheexponentkfromournotation.Weareinterested
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e instochasticdifferentialequations(SDEs)oftheform
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nl dXt =b(t,Xt)dt+σ(t,Xt)dWt, (1.1)
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D wherethecoefficientsbandσ
(b,σ): [0,T]×Ω×Rd →Rd×Rd×m
satisfythefollowingassumptions.
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4 StochasticDifferentialEquations
(A1) Foreachx ∈ Rd,theprocesses(b(t,x))0≤t≤T and(σ(t,x))0≤t≤T areinH2,d and
hp H2,dm,respectively.
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ojs (A2) ∃c>0,∀t∈[0,T],∀ω ∈Ω,∀x,x(cid:3) ∈Rd,
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nal |(b,σ)(t,ω,x)−(b,σ)(t,ω,x(cid:3))|≤c|x−x(cid:3)|.
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or We shallfollowthestandardpracticeofnotmakingthedependenceuponω ∈ Ω explicit
m. intheformulaswheneverpossible.
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ww. Definition1.1. We saythatanF-progressivelymeasurableprocessX = (Xt)0≤t≤T isa
w strongsolutionoftheSDE(1.1)if
p:// (cid:5)
e htt • 0T(|b(t,Xt)(cid:5)|+|σ(t,Xt)|2)d(cid:5)t<∞ P-almostsurely,
ht; se • Xt =X0+ 0tb(s,Xs)ds+ 0T σ(s,Xs)dWs, 0≤t≤T.
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co 1.2 Existence and Uniqueness of Strong Solutions: The
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M li Theorem1.2. LetusassumethatX0 ∈ L2 isindependentofWandthatthecoefficients
A bandσsatisfytheassumptions(A1)and(A2)above. Then,thereexistsauniquesolution
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o S of (1.1)inH2,d,andforsomec > 0dependingonlyuponT andtheLipschitzconstantof
ct t bandσ,thissolutionsatisfies
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subj E sup |Xt|2 ≤c(1+E|X0|2)ecT. (1.2)
n 0≤t≤T
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b Here andin the following,we use the letter c fora genericconstantwhose valuecan
stri changefromlinetoline.
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31. R cPersosoefs.,wFoerdeeaficnheXthe∈prHo2c,eds,stUhe(Xsp)acbeyofsquareintegrableprogressivelymeasurablepro-
2
1. (cid:3) (cid:3)
9. t t
23 U(X)t =X0+ b(s,Xs)ds+ σ(s,Xs)dWs. (1.3)
2. 0 0
3
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7 to FirstweprovethatU(X) ∈ H2,d, andthen,sinceXisasolutionoftheSDE(1.1)ifand
1 onlyif U(X) = X, we provethatU is a strictcontractionin theHilbertspaceH2,d. By
10/ definitionofthenormofH2,d,wehave
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ed (cid:9)U(X)(cid:9)2 ≤(i)+(ii)+(iii)
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nl with
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Do (i)=3TE|X0|2 <∞,
and(ii)and(iii)definedbelow.Usingthefactthat
|b(t,x)|2 ≤c(1+|b(t,0)|2+|x|2),