Stochastic Optimal Control in Infinite Dimensions: Dynamic Programming and HJB Equations

Transcription

1 Sochaic Opimal Conrol in Infinie Dimenion: Dynamic Programming and HJB Equaion G. Fabbri 1 F. Gozzi 2 and A. Świe ch3 wih Chaper 6 by M. Fuhrman 4 and G. Teiore 5 1 Aix-Mareille Univeriy (Aix-Mareille School of Economic), CNRS and EHESS. e- mail:giorgio.fabbri@univ-amu.fr 2 Diparimeno di Economia e Finanza, Univerià LUISS - Guido Carli Roma, fgozzi@lui.i 3 School of Mahemaic, Georgia Iniue of Technology, Alana, wiech@mah.gaech.edu 4 Diparimeno di Maemaica, Poliecnico di Milano, marco.fuhrman@polimi.i 5 Diparimeno di Maemaica, Univerià di Milano Bicocca, gianmario.eiore@unimib.i

2 Conen Preface 11 Chaper 1. Preliminarie on ochaic calculu in infinie dimenion Baic probabiliy Probabiliy pace, σ-field Random variable Bochner inegral Expecaion, covariance and correlaion Condiional expecaion and condiional probabiliy Gauian meaure on Hilber pace and Fourier ranform Sochaic procee and Brownian moion Sochaic procee Maringale Sopping ime Q-Wiener proce Simple and elemenary procee Sochaic inegral Definiion of ochaic inegral Baic properie and eimae Sochaic differenial equaion Mild and rong oluion Exience and uniquene of oluion Properie of oluion of SDE Uniquene in law Furher exience and uniquene reul in pecial cae SDE coming from boundary conrol problem Semilinear SDE wih addiive noie Semilinear SDE wih muliplicaive noie Traniion emigroup Iô and Dynkin formulae Bibliographical noe 81 Chaper 2. Opimal conrol problem and example Sochaic opimal conrol problem: general formulaion Srong formulaion Weak formulaion Dynamic Programming Principle: eup and aumpion The eup 87 3

3 4 CONTENTS The general aumpion The aumpion in he cae of conrol problem for mild oluion Dynamic Programming Principle: aemen and proof Pullback o he canonical reference probabiliy pace Independence of reference probabiliy pace The proof of he abrac principle of opimaliy Infinie horizon problem HJB equaion and opimal ynhei in he mooh cae Finie horizon problem: Parabolic HJB equaion Infinie horizon problem: Ellipic HJB equaion Some moivaing example Sochaic conrolled hea equaion: diribued conrol Seing of he problem The infinie dimenional eing and he HJB equaion The infinie horizon cae Sochaic conrolled hea equaion: boundary conrol Seing of he problem: Dirichle cae Seing of he problem: Neumann cae The infinie dimenional eing and he HJB equaion Sochaic conrolled hea equaion: boundary conrol and boundary noie Seing of he problem The infinie dimenional eing The HJB equaion Opimal conrol of he ochaic Burger equaion Seing of he problem The infinie dimenional eing and he HJB equaion Opimal conrol of he ochaic Navier-Soke equaion Seing of he problem The infinie dimenional eing and he HJB equaion Opimal conrol of he Duncan-Morenen-Zakai equaion An opimal conrol problem wih parial obervaion The eparaed problem Super-hedging of forward rae Opimal conrol of ochaic delay equaion Delay in he ae variable only Delay in he ae and conrol Bibliographical noe 138 Chaper 3. Vicoiy oluion Preliminary reul B-coninuiy and weak and rong B-condiion Eimae for oluion of ochaic differenial equaion Perurbed opimizaion Maximum principle Vicoiy oluion Bounded equaion 165

4 CONTENTS Coniency of vicoiy oluion Comparion heorem Degenerae parabolic equaion Degenerae ellipic equaion Exience of oluion: Value funcion Finie horizon problem Improved verion of dynamic programming principle Infinie horizon problem Exience of oluion: Finie dimenional approximaion Singular perurbaion Perron mehod and half-relaxed limi Infinie dimenional Black-Schole-Barenbla equaion HJB equaion for conrol of he Duncan-Morenen-Zakai equaion Variaional oluion Weighed Sobolev pace Opimal conrol of he Duncan-Morenen-Zakai equaion Eimae for he DMZ equaion Vicoiy oluion Value funcion and exience of oluion HJB equaion for a boundary conrol problem Definiion of vicoiy oluion Comparion and exience heorem Sochaic conrol problem HJB equaion for conrol of ochaic Navier-Soke equaion Eimae for conrolled SNS equaion Value funcion Vicoiy oluion and comparion heorem Exience of vicoiy oluion Bibliographical noe 281 Chaper 4. Mild oluion in pace of coninuou funcion The eing and an inroducion o he mehod The mehod in he parabolic cae The mehod in he ellipic cae Preliminarie G-derivaive Weighed pace Smoohing properie of raniion emigroup The cae of Ornein-Uhlenbeck emigroup The equaion The emigroup and he aociaed Kolmogorov equaion Smoohing of R :aumpionandnullconrollabiliy Smoohing of R :reulandeimae Smoohing of R :example Join pace-ime regulariy of R The cae of perurbed Ornein-Uhlenbeck emigroup 32

5 6 CONTENTS The cae of inverible diffuion coefficien Mild oluion of HJB equaion The parabolic cae Formal derivaion of he mild form Definiion of mild oluion and aumpion Exience and uniquene of mild oluion The ellipic cae for big dicoun facor Formal derivaion of he mild form Definiion of mild oluion and aumpion Exience and uniquene of mild oluion Approximaion of mild oluion: rong oluion The parabolic cae The ellipic cae HJB equaion of Ornein-Uhlenbeck ype: Lipchiz Hamilonian The parabolic cae C 2 regulariy of he mild oluion The ellipic cae C 2 regulariy of he mild oluion Exience of mild oluion for all λ> HJB equaion of Ornein-Uhlenbeck ype: Locally Lipchiz Hamilonian The parabolic cae Local exience and uniquene of mild oluion Apriori eimae Global exience of mild and rong oluion Sochaic Conrol: Verificaion Theorem and Opimal Feedback The finie horizon cae The ae equaion The opimal conrol problem and he HJB equaion The verificaion heorem Opimal feedback The infinie horizon cae The ae equaion The opimal conrol problem and he HJB equaion The verificaion heorem Opimal feedback Example Diagonal cae Inverible diffuion coefficien Mild oluion of HJB for wo pecial problem Conrol of Sochaic Burger and Navier-Soke equaion The cae of Burger equaion Two-dimenional Navier-Soke equaion Three-dimenional Navier-Soke equaion Conrol of reacion-diffuion equaion The finie horizon problem The infinie horizon problem Regular oluion hrough explici repreenaion Quadraic Hamilonian 415

6 CONTENTS Explici oluion in he homogeneou cae Bibliographical noe 416 Chaper 5. Mild oluion in L 2 pace Inroducion o he mehod Preliminarie and he linear problem Noaion The reference meaure m and he main aumpion on he linear par The operaor N The gradien operaor D Q and he pace W 1,2 Q (H, m) The operaor R Two key lemma 435 Chaper The HJB equaion Approximaion of mild oluion Applicaion o ochaic opimal conrol The ae equaion The opimal conrol problem and he HJB equaion The verificaion heorem Opimal feedback Coninuiy of he value funcion and nondegeneracy of he invarian meaure Example Opimal conrol of delay equaion Conrol of ochaic PDE of fir order Second order SPDE in he whole pace Reul in pecial cae Parabolic HJB equaion Applicaion o finie horizon opimal conrol problem Ellipic HJB equaion Bibliographical noe 466 HJB Equaion hrough Backward Sochaic Differenial Equaion Complemen on forward equaion wih muliplicaive noie Noaion on vecor pace and ochaic procee The cla G The forward equaion: exience, uniquene and regulariy Regular dependence on daa Differeniabiliy Differeniabiliy in he ene of Malliavin Backward Sochaic Differenial Equaion in Hilber pace Well-poedne Regular dependence on daa Forward-backward yem BSDE and mild oluion o HJB and oher Kolmogorov non linear equaion 52

7 8 CONTENTS 6.5. Applicaion o opimal conrol problem Ellipic HJB equaion wih arbirarily growing Hamilonian The aociaed forward-backward yem Differeniabiliy of he BSDE and a-priori eimae on he gradien Mild Soluion of he ellipic PDE Applicaion o opimal conrol in infinie horizon Ellipic cae wih non conan diffuion: aemen only 524 Appendix A. Noaion and Funcion Space 531 A.1. Baic Noaion 531 A.2. Funcion pace 531 Appendix B. Linear operaor and C -emigroup 537 B.1. Linear operaor 537 B.2. Diipaive operaor 539 B.3. Trace cla and Hilber-Schmid operaor 54 B.4. C -emigroup and relaed reul 543 B.4.1. Baic definiion 543 B.4.2. Hille-Yoida Theorem and Yoida approximaion 544 B.4.3. Analyic emigroup and fracional power of generaor 546 B.5. π-convergence, K-convergence, π- andk-coninuou emigroup 547 B.5.1. π-convergence and K-convergence 547 B.5.2. π- andk-coninuou emigroup and heir generaor 551 B The definiion 551 B The generaor 553 B Cauchy problem for π-coninuou and K-coninuou emigroup 554 B.6. Approximaion of coninuou funcion hrough K-convergence 556 B.7. Approximaion of oluion of Kolmogorov equaion hrough K-convergence 558 B.7.1. Claical and rong oluion of (B.3) 559 B.7.2. The Ornein-Uhlenbeck emigroup aociaed o (B.3), he mild oluion and heir properie 562 B.7.3. The approximaion reul 567 Appendix C. Parabolic equaion wih non-homogeneou boundary condiion573 C.1. Dirichle and Neumann map 573 C.2. Non-zero boundary condiion, he Dirichle cae 575 C.3. Non-zero boundary condiion, he Neumann cae 577 C.4. Boundary noie, Neumann cae 579 C.5. Boundary noie, Dirichle cae 579 Appendix D. Funcion, derivaive and approximaion 583 D.1. Coninuiy properie, modulu of coninuiy 583 D.2. Fréche and Gâeaux derivaive 584

8 CONTENTS 9 D.3. Inf-Sup convoluion 585 D.4. Two verion of Gronwall Lemma 589 Appendix E. Vicoiy oluion in R N 591 E.1. Second order je 591 E.2. Definiion of vicoiy oluion 593 E.3. Finie dimenional maximum principle 594 E.4. Perron mehod 595 Bibliography 597

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10 Preface The main objecive of hi book i o give an overview of he heory of Hamilon- Jacobi-Bellman (HJB) parial differenial equaion (PDE) in infinie dimenional Hilber pace and i applicaion o ochaic opimal conrol of infinie dimenional procee and relaed field. Boh area have been developing very rapidly in he la few decade. While here exi everal excellen monograph on hi ubjec in finie dimenional pace (ee e.g. [194, 195, 294, 349, 364, 382, 449]), much le ha been wrien in infinie dimenional pace. A good accoun of he infinie dimenional cae in he deerminiic conex can be found in [312]. Oher book ha ouch he ubjec are [23, 129, 364]. We aemp o fill hi gap in he lieraure. Infinie dimenional diffuion procee appear naurally and are ued o model phenomena in phyic, biology, chemiry, economic, mahemaical finance, engineering and many oher area (ee e.g. [93, 127, 13, 282, 444]. Thi book inveigae he PDE approach o heir ochaic opimal conrol, however infinie dimenional PDE can alo be ued o udy oher properie of uch procee a large deviaion, invarian meaure, ochaic viabiliy, ochaic differenial game for infinie dimenional diffuion, ec. (ee [6, 127, 129, 182, 184, 192, 361, 363, 425, 428]). To illurae he main heme of he book le u begin wih a model diribued parameer ochaic opimal conrol problem. We wan o conrol a proce (called he ae) given by an abrac ochaic differenial equaion in a real, eparable Hilber pace H dx() =(AX()+b(, X(),a()))d + σ(, X(),a())dW (), > X() =x H, where A i he generaor of a C emigroup in H, b, σ are cerain bounded funcion and W i a o called Q-Wiener proce 1 in H. The funcion a( ), calledconrol, are ochaic procee wih value in ome meric pace Λ, which aify cerain meaurabiliy properie. The above abrac ochaic differenial equaion i very general and include variou emi-linear ochaic PDE, a well a oher equaion which can be rewrien a ochaic funcional evoluion equaion, for inance ochaic differenial delay equaion. In a mo ypical opimal conrol problem we wan o find a conrol a( ), called opimal, which minimize a co funcional T J(, x; a( )) = E l(, X(),a())d + g(x(t )). (for ome T>)amongalladmiibleconrolforomefuncionl :[,T] H Λ R, g : H R. The dynamic programming approach o he above problem i baed on udying he properie of he o called value funcion V (, x) =infj(, x; a( )) a( ) 1 Q i a uiable elf-adjoin poiive operaor in H, he covariance operaor for W. 11

11 12 PREFACE and characerizing i a a oluion of a fully nonlinear PDE, he aociaed HJB equaion. Since he ae X() evolve in he infinie dimenional pace H, hi PDE i defined in [,T] H. The link beween he value funcion V and he HJB equaion i eablihed by he Bellman principle of opimaliy known a he dynamic programming principle (DPP), η V (, x) =inf E l (, X (),a()) d + V (η, X (η)), for all η [, T ]. a( ) Heuriically he DPP can be ued o define a wo parameer nonlinear evoluion yem and he aociaed HJB equaion 1 V + Ax, DV +inf a Λ 2 (σ(, Tr x, a)q 1 2 )(σ(, x, a)q 1 2 ) D 2 V + b(, x, a),dv + l(, x, a) =, V (T,x)=g(x). (.1) i i generaing equaion. Such PDE i called infinie dimenional or a PDE in infiniely many variable. We alo call i unbounded ince i ha a erm wih an unbounded operaor A which i well defined only on he domain of A. Ohererm may alo be undefined for ome value of DV and D 2 V,heFréchederivaive of V, which we may idenify wih elemen of H and wih bounded, elf-adjoin operaor in H repecively. In paricular, he erm Tr[(σQ 1 2 )(σq 1 2 ) D 2 V ] i well defined only if (σq 1 2 )(σq 1 2 ) D 2 V i of race cla. The main idea i o ue he HJB equaion o udy properie of he value funcion, find condiion for opimaliy, obain formula for ynhei of opimal feedback conrol, ec. Thi approach urned ou o be very ucceful for finie dimenional problem becaue of i clariy and impliciy and hank o he developmen of he heory of fully nonlinear ellipic and parabolic PDE, in paricular he inroducion of he noion of vicoiy oluion and advance in he regulariy heory. However even here many open queion remain, epecially if he HJB equaion are degenerae. We hope he dynamic programming approach will be equally valuable for infinie dimenional problem even hough a complee heory i no available ye. Equaion (.1) i an example of a fully nonlinear econd-order PDE of (degenerae) parabolic ype. In he book we will deal wih more general and differen verion of uch equaion and heir degenerae ellipic counerpar. If Λ i a ingleon, (.1) i ju a erminal value problem for a linear Kolmogorov equaion. If Λ i no a ingleon bu he diffuion coefficien σ i independen of he conrol parameer a, (.1) i emi-linear. The heory of linear equaion (and ome pecial emi-linear one) ha been udied by many auhor and can be found in book [23, 8, 129, 453]. The emphai of hi book i on emi-linear and fully nonlinear equaion. There are everal noion of oluion applicable o PDE in Hilber pace which are dicued in he book: claical oluion, rong oluion, mild oluion in he pace of coninuou funcion, oluion in L 2 (µ), vicoiy oluion. Claical oluion are he mo regular one. Thi noion of oluion require C 1,2 regulariy in he Fréche ene and impoe addiional condiion o ha all erm in he equaion make ene poinwie for (, x) [,T] H. Whenclaicaloluionexi we can apply he claical dynamic programming approach o obain verificaion heorem and he ynhei of opimal feedback conrol. Unforunaely in almo all inereing cae i i no poible o find uch oluion, however hey are very ueful a a heoreical ool in he heory. The noion of rong oluion, mild oluion in he pace of coninuou funcion, and oluion in L 2 (µ) are inroduced and

12 PREFACE 13 udied only for emi-linear equaion and define oluion which have a lea fir derivaive (in ome uiable ene). Verificaion heorem and ynhei of opimal feedback conrol can ill be developed wihin heir framework. The noion of vicoiy oluion i he mo general and applie o fully nonlinear equaion, however a he curren age here are no reul on verificaion heorem and ynhei of opimal feedback conrol. Infinie dimenional problem preen unique challenge, among hem are he lack of local compacne and no equivalen of Lebegue meaure. Thi mean ha andard finie dimenional ellipic and parabolic echnique which are baed on meaure heory canno be carried over o he infinie dimenional cae. Moreover, he equaion are moly degenerae and conain unbounded erm which are ingular. So he mehod o find regular oluion o PDE in infinie dimenion like our end o be global and are baed on emigroup heory, moohing properie of raniion emigroup (like he Ornein-Uhlenbeck one), fixed poin echnique, and ochaic analyi. Thee mehod are moly rericed o equaion of emi-linear ype. On he oher hand, he noion of vicoiy oluion i perfecly uied for fully nonlinear equaion. I i local and i doe no require any regulariy of oluion excep coninuiy. A in finie dimenion i i baed on maximum principle hrough he idea of differeniaion by par, i.e. replacing he non exiing derivaive of vicoiy uboluion (repecively, uperoluion) by derivaive of mooh e funcion a poin where heir graph ouch he graph of uboluion (repecively, uperoluion) from above (repecively, below). However a he reader will ee, hi idea ha o be carried ou very carefully in infinie dimenion. The book conain chaper on he mo imporan opic in HJB equaion and he DPP approach o infinie dimenional ochaic opimal conrol. Chaper 1 conain baic maerial on infinie dimenional ochaic calculu which i needed in he ubequen chaper. I i however no inended o be an inroducion o ochaic calculu and he reader are expeced o have ome familiariy wih i. Chaper 1 i included o make he book more elf-conained. Mo of he reul preened here are well known, hence we only provide reference where he reader can find he proof and find more informaion abou he concep, example, ec. We provide proof only in cae where we could no find good reference in he lieraure. In Chaper 2 we inroduce a general ochaic opimal conrol problem and prove a key reul in he heory, namely he dynamic programming principle. We formulae i in an abrac and general form o ha i can be ued in many cae wihou he need of reproving i. Soluion of ochaic PDE mu be inerpreed in variou way (rong, mild, variaional, ec.) and our formulaion of he DPP rie o capure hi phenomenon. Our proof of he DPP i baed on andard idea however we ried o avoid heavy probabiliic mehod regarding weak uniquene of oluion of ochaic differenial equaion. Our proof i hu more analyical. We alo inroduce many example of ochaic opimal conrol problem which can be udied in he framework of he approach preened in he book. They hould give he reader an idea abou he range and applicabiliy of he maerial. Chaper 3 i devoed o he heory of vicoiy oluion. The reader hould keep in mind he following principle when i come o unbounded PDE in infinie dimenion: here i no ingle definiion of vicoiy oluion ha applie o all equaion. Thi i due o he fac ha here are many differen PDE which conain differen unbounded operaor and erm which are coninuou in variou norm. Alo he oluion have o be coninuou wih repec o weaker opologie. However he main idea of he noion of vicoiy oluion i alway he ame a we decribed before. Wha change i he choice of e funcion, pace, opologie, and he

13 14 PREFACE inerpreaion of variou erm in he equaion. In hi book we focu on he noion of o called B-coninuou vicoiy oluion which wa inroduced by Crandall and P. L. Lion in [13, 14] for fir order equaion and laer adaped o econd order equaion in [422]. The key reul in he heory i he comparion principle which i very echnical. I main componen i he o called maximum principle for emiconinuou funcion. The proof of uch reul in finie dimenion wa fir obained in [28] and wa laer implified and generalized [99, 1, 11, 27]. I i heavily baed on meaure heory and i no applicable o infinie dimenion. Thu he heory ue a finie dimenional reducion echnique inroduced by P. L. Lion in [321]. I reric he cla of equaion which can be conidered, in paricular hey have o be highly degenerae in he econd order erm. We preen hree echnique o obain exience of vicoiy oluion. The fir and mo imporan for hi book i he DPP and he ochaic opimal conrol inerpreaion, howing direcly ha he value funcion i a vicoiy oluion. Thi echnique applie o HJB equaion. The oher echnique are finie dimenional approximaion and Perron mehod. Boh can be applied o more general equaion, for inance Iaac equaion aociaed o wo-player, zero-um ochaic differenial game, however hey have limiaion of heir own. Moreover we dicu oher opic of he heory of vicoiy oluion like coniency, ingular perurbaion, ec.. Several pecial equaion are alo udied in he book becaue of heir imporance and becaue hey are good example o how how he definiion of vicoiy oluion and ome echnique can be adjued o paricular cae. They are he HJB equaion for he opimal conrol of Duncan-Morenen-Zakai equaion, ochaic Navier-Soke equaion, and ochaic boundary conrol. In paricular he la one alo conain idea on how o handle HJB equaion which may be non-degenerae, for inance if Q i no of race cla. Finally we preen applicaion o infinie dimenional Black-Schole-Barenbla equaion of mahemaical finance. Chaper 4 i devoed o he heory of mild and rong oluion in pace of coninuou funcion hrough fixed poin echnique baed on he moohing properie of raniion emigroup uch a Ornein-Uhlenbeck one. Thi heory applie only o emi-linear equaion, i.e. when he coefficien σ doe no depend on he conrol parameer a, andhioricallyiwahefirapproachinroducedinhe lieraure. I wa fir inroduced by Barbu and Da Prao [23] and laer improved and developed in variou paper, ee e.g [63, 64, 79, 81, 226, 231, 232, 235]. Chaper 4 i divided ino four main par. In he fir one (Secion ), we preen he baic ool needed for he analyi: he heory of generalized gradien and he moohing of raniion emigroup. In he econd one (Secion 4.4 o 4.7), we develop he heory for a general ype of emi-linear HJB equaion (parabolic and ellipic) wihou connecion wih opimal conrol problem. The main idea behind hi approach i he following. Conider he HJB equaion (.1) in he emi-linear auonomou cae: V + AV +inf a Λ b(x, a),dv + l(, x, a) =, (.2) V (T,x)=g(x), where A i he linear operaor Aϕ = Ax, Dϕ Tr (σ(x)q )(σ(x)q 2 ) D 2 ϕ. If uch operaor generae a emigroup e A hen, by he variaion of conan formula, one can rewrie (.2) in he inegral form a V (, x) =e (T )A g(x)+ T e (T )A F (, ) (x) d

14 PREFACE 15 where F (, x) :=inf a Λ {b(x, a),dv + l(, x, a)}. The oluion of hi inegral equaion i called a mild oluion and i obained by fixed poin echnique. To define i, he oluion mu a lea have fir order paial derivaive. Thu one need uiable moohing properie of he emigroup e A (which i he Ornein- Uhlenbeck emigroup in he imple cae). Since hi emigroup i no rongly coninuou, apar from very pecial cae, one need o ue he heory of π-emigroup inroduced in [386] or he one of weakly coninuou (or K-coninuou) emigroup [75, 82, 225]. In he hird par (Secion 4.8), we develop a connecion wih ochaic opimal conrol problem. The fac ha mild oluion have fir order paial derivaive allow o give a meaning o formulae for opimal feedback. However he proof of verificaion heorem and opimal feedback formulae canno be done raighforwardly a one need o apply Io formula in infinie dimenion which require mooh funcion. For hi reaon (following [231]), one inroduce he noion of rong oluion of he HJB equaion (.2) a a uiable limi of claical oluion and prove ha any mild oluion i alo a rong oluion. The fourh and he la par of he chaper (Secion ) deal wih ome pecial equaion. In Secion 4.9 we how how he echnique developed in he previou ecion can be adaped o HJB equaion and analyi of opimal conrol problem for ochaic Burger equaion, ochaic Navier-Soke equaion and ochaic reacion diffuion equaion. In Secion 4.1 we dicu ome equaion for which explici repreenaion of he oluion can be found. Such cae are alway of inere in applicaion. Chaper 5 i devoed o a relaively new and promiing heory of mild and rong oluion in pace of L 2 funcion wih repec o a uiable meaure µ (ee [223, 3, 4, 94]). The conen of hi chaper are imilar o he previou one a he main idea behind he definiion of mild and rong oluion of HJB equaion are he ame. The difference i in he fac ha he reference pace i no he pace of coninuou funcion bu he pace of quare inegrable one wih repec o he meaure µ. The reul are imilar: exience and uniquene of oluion of HJB equaion hrough fixed poin argumen, verificaion heorem hrough approximaion, exience of opimal feedback. The advanage of hi approach i ha he reul require weaker aumpion on he daa, hu enlarging he range of poible applicaion, including for inance he conrol of delay equaion, however a a co of weaker aemen, for example he fir order paial derivaive i now defined in a Sobolev weak ene and i no in general a Fréche derivaive. The main ool ued here are he heory of invarian meaure for infinie dimenional ochaic differenial equaion and he properie of raniion emigroup in he pace of inegrable funcion wih repec o uch meaure. Chaper 6 i devoed o a differen and in many repec complemenary echnique of Backward Sochaic Differenial Equaion (BSDE). I i wrien by wo leading exper in he field. BSDE are Iô ype equaion in which he iniial condiion i replaced by a final condiion and a new unknown proce appear correponding o a uiable maringale erm. In he nonlinear, finie dimenional cae BSDE have been inroduced in [372] while heir direc connecion wih opimal ochaic conrol wa firly inveigaed in [154] and [377]. Since hen, he general heory of BSDE ha developed coniderably, ee [56, 58, 152, 289, 325, 371]. Beide ochaic conrol, applicaion were given o many field, for inance o opimal opping, ochaic differenial game, nonlinear parial differenial equaion and many opic relaed o mahemaical finance. Infinie dimenional BSDE have alo been conidered, ee for inance [97, 211, 252, 264, 373]. The inere for u i ha BSDE provide an alernaive way o repreen he value funcion of an opimal

15 16 PREFACE conrol problem and conequenly o udy he correponding HJB equaion and o olve he conrol problem. I urn ou ha he mo uiable noion of oluion for he HJB equaion i, in hi conex, he one of mild oluion on pace of coninuou funcion bu, unlike in Chaper 4, he BSDE mehod eem paricularly adaped o rea degenerae cae in which he raniion emigroup ha no moohing properie. The price o pay i ha normally we need more regular coefficien and a rucural condiion (impoing, roughly peaking, ha he conrol ac wihin he image of he noie). If hee requiremen are aified he BSDE echnique reveal o be very flexible. In paricular, in Chaper 6 we will how how hey allow o rea boh parabolic and ellipic HJB equaion (ee [55, 212, 265, 336]). Moreover we will preen exenion o he cae of locally Lipchiz Hamilonian, quadraically growing wih repec o he gradien (ee [54, 55, 25]), o he cae of HJB equaion correponding o ergodic conrol problem ([26]) and he cae of ae equaion wih noie and conrol on he boundary ([131, 338]). We will finally decribe how he regulariy requiremen of he coefficien can be parially removed inroducing auiableconcepof generalizedgradien,ee[213]. I i impoible o cover all apec of he heory of HJB equaion in infinie dimenion and i connecion o ochaic opimal conrol. In paricular he heory of inegro-pde i an emerging area which i no preened in he book. We do no dicu fir order equaion and exenion o Banach pace. Equaion in he pace of probabiliy meaure i anoher emerging opic. We have choen a elecion of opic which give a broad overview of he field and enough informaion o ha he reader can ar exploring he ubjec on heir own. There are already enough imporan applicaion o juify he inere in he ubjec. The reader hould no be rericed o he boundarie drawn by he book. We hope he book will pur he inere and reearch in he field among heoreical and applied mahemaician, and i will be ueful o all kind of cieni and reearcher working in area relaed o ochaic conrol.

16 CHAPTER 1 Preliminarie on ochaic calculu in infinie dimenion 1.1. Baic probabiliy We recall baic noion of meaure heory and give a hor inroducion o random variable and he heory of Bochner inegral Probabiliy pace, σ-field. Definiion 1.1 (π-yem, σ-field) Conider a e Ω and denoe by P(Ω) he power e of Ω. (i) A nonempy cla of ube of Ω, F P(Ω), i called a π-yem if i i cloed under finie inerecion. (ii) A cla of ube of Ω, F P(Ω) i called a σ-field in Ω if Ω F and F i cloed under complemen and counable union. (iii) A cla of ube of Ω, F P(Ω) i called a λ-yem if: Ω F ; if A, B F,A B, hen B \ A F ; if A i F,i=1, 2,...,A i A, hen A F. If G and F are wo σ-field in Ω and G F,weayhaG i a ub-σ-field of F. Given a cla C P(Ω), hemalleσ-field conaining C i called he σ-field generaed by C. I i denoed by σ(c ). A σ-field F in Ω i aid o be counably generaed if here exi a counable cla of ube C P(Ω) uch ha σ(c )=F. If C P(Ω) and A Ω we denoe C A := {B A : B C }. Wedenoeby σ A (C A) he σ-field of ube of A generaed by C A. I i eay o ee ha σ A (C A) =σ(c ) A (ee for inance [12], page 5). For A Ω we denoe i complemen by A c := Ω \ A, andfora, B Ω we denoe heir ymmeric difference by A B := (A \ B) (B \ A). We will wrie R + =[, + ), R + =[, + ) {+ }, R = R {± }. Theorem 1.2 Le G be a π-yem and F be a λ-yem in ome e Ω, uch ha G F. Then σ(g ) F. Proof. See [281], Theorem 1.1, page 2. Corollary 1.3 Le G be a π-yem and F be he malle family of ube of Ω uch ha: G F ; if A F hen A c F ; if A i F,A i A j = for i, j =1, 2,...,i= j, hen i=1 A i F. Then σ(g )=F. Proof. Since σ(g ) aifie he hree condiion for F,weobvioulyhave F σ(g ). For he oppoie incluion i remain o noice ha F i a λ-yem. (For a elf-conained proof ee alo [13], Propoiion 1.4, page 17.) 17

17 18 1. PRELIMINARIES ON STOCHASTIC CALCULUS IN INFINITE DIMENSIONS Definiion 1.4 (Meaurable pace) If Ω i a e and F i a σ-field in Ω, he pair (Ω, F ) i called a meaurable pace. Definiion 1.5 (Probabiliy meaure, probabiliy pace) Conider a meaurable pace (Ω, F ). A funcion µ : F [, + ) {+ } i called a meaure on (Ω, F ) if µ( ) =, and whenever A i F,A i A j = for i, j =1, 2,...,i= j, hen µ A i = µ(a i ). i=1 The riple (Ω, F,µ) i called a meaure pace. If µ(ω) < + we ay ha µ i a bounded meaure. If Ω= n=1 A n, where A n F,µ(A n ) < +,n =1, 2,...,we ay ha µ i a σ-finie meaure. Ifµ(Ω) = 1 we ay ha µ i a probabiliy meaure. We will ue he ymbol P o denoe probabiliy meaure. The riple (Ω, F, P) i called a probabiliy pace. Thu a probabiliy meaure i a σ-addiive funcion P : F [, 1] uch ha P(Ω) = 1. Given a meaure pace (Ω, F,µ), wedenoen := {F Ω : G F,F G, µ(g) =}. The elemen of N are called µ-null e. If N F,hemeaure pace (Ω, F,µ) i aid o be complee. The σ field F := σ(f, N ) i called he compleion of F (wih repec o µ). I i eay o ee ha σ(f, N )={A B : A F,B N}. If G F i anoher σ-field hen σ(g, N ) i called he augmenaion of G by he null e of F. The augmenaion of G may be differen from i compleion, a he laer i ju he augmenaion of G by he ube of he e of meaure zero in G.Wealohaveσ(G, N )={A Ω:A B N for ome B G }. Lemma 1.6 Le µ 1,µ 2 be wo bounded meaure on a meaurable pace (Ω, F ), and le G be a π-yem in Ω uch ha Ω G and σ(g )=F. Then µ 1 = µ 2 if and only if µ 1 (A) =µ 2 (A) for every A G. Proof. See [281], Lemma 1.17, page 9. Le Ω, T be a family of e. We will denoe he Careian produc of he family Ω by T Ω. If T i finie (T = {1,...,n}) orcounable(t = N), we will alo wrie Ω 1... Ω n,repecivelyω 1 Ω 2... If each Ω i a opological pace, we endow T Ω wih he produc opology. If each Ω ha a σ-field F, we define he produc σ-field T F in T Ω a he σ-field generaed by he one-dimenional cylinder e A ( = Ω ). If T = {1,...,n} (repecively, T = N) we will ju wrie T F = F 1... F n (repecively, T F = F 1 F 2...). If S i a opological pace, he σ-field generaed by he open e of S i called Borel σ-field. I will be denoed by B(S). If S i a meric pace, unle aed oherwie, i defaul σ-field will alway be B(S). I i no difficul o ee ha if S 1,S 2,... are eparable meric pace, hen i=1 B(S 1 S 2...) =B(S 1 ) B(S 2 )... If (S, ρ) i a meric pace, A S, andweconider(a, ρ) a a meric pace, hen B(A) =A B(S). AcompleeeparablemericpaceicalledaPolih pace. Alo B(R + )=σ(b(r + ), {+ }), B(R) =σ(b(r), { }, {+ }). Ameaurable pace(ω, F ) i called counably deermined (or F i called counably deermined) if here i a counable e F F uch ha any wo probabiliy meaure on (Ω, F ) ha agree on F mu be he ame. I follow from Lemma 1.6 ha if F i counably generaed hen F i counably deermined. If S i a Polih pace hen B(S) i counably generaed.

18 1.1. BASIC PROBABILITY 19 If (Ω i, F i,µ i ),i =1,...,n are meaure pace, heir produc meaure on (Ω 1... Ω n, F 1... F n ) i denoed by µ 1... µ 2. I i he meaure uch ha µ 1... µ 2 (A 1... A n )=µ 1 (A 1 )... µ n (A n ) for all A i Ω i,i=1,...,n. If S i a meric pace, a bounded meaure µ on (S, B(S)) i called regular if µ(a) =up{µ(c) :C A, C cloed} =inf{µ(u) :A U, U open} A B(S). Every bounded meaure on (S, B(S)) i regular (ee [374], Chaper II, Theorem 1.2). A bounded meaure µ on (S, B(S)) i called igh if for every > here exi a compac e K S uch ha µ(s \ K ) <. If S i a Polih pace hen every bounded meaure on (S, B(S)) i igh (ee [374], Chaper II, Theorem 3.2). We refer o [42, 44, 199, 281, 374] for more on he general heory of meaure and probabiliy Random variable. Definiion 1.7 (Random variable) A meaurable map X beween wo meaurable pace (Ω, F ) and ( Ω, G ) i a called a random variable. Thi mean ha X i a random variable if X 1 (A) F for every A G. We wrie i horly a X 1 (G ) F. Someime we will ju ay ha X i F /G meaurable. If Ω =R (rep. R + ) and G i he Borel σ-field B(R) (rep. B(R + )) hen X i aid o be a real random variable (rep. poiive random variable). If Ω, Ω are opological pace and F, G are he Borel σ-field hen X i aid o be Borel meaurable. Given a random variable X :(Ω, F ) ( Ω, G ) we denoe by σ(x) he malle ub-σ-field of F ha make X meaurable, i.e. σ(x) :=X 1 (G ). I i called he σ-field generaed by X. Given a e of indice I and a family of random variable X i :(Ω, F ) ( Ω, G ), i I, heσ-field σ {X i } i I generaed by {Xi } i I i he malle ub-σ-field of F ha make all he funcion X i : Ω,σ {X i } i I ( Ω, G ) meaurable, i.e. σ {X i } i I = σ X 1 i (G ):i I. Lemma 1.8 Le (Ω, F ) be a meaurable pace. Then: (i) If ( Ω, G ) i a meaurable pace, X : Ω Ω, and C G i uch ha σ(c )=G, hen X i F /G meaurable if and only if X 1 (C ) F. Moreover, σ(x) =σ(x 1 (C )). (ii) If X n :Ω R,n =1, 2,... are random variable, hen up n X n, inf n X n, lim up n X n, lim inf n X n are random variable. (iii) Le X n :Ω S, n =1, 2,... be random variable, where S i a meric pace. Then: if S i complee hen {ω : X n (ω) converge} F ; if X n X on Ω, hen X i a random variable. (iv) Le (Ω i, F i ),i = 1, 2 be meaurable pace, and X : Ω 1 Ω 2 Ω be (F 1 F 2 )/F meaurable. Then, for every ω 1 Ω 1, X ω1 ( ) =X(ω 1, ) i F 2 /F meaurable, and, for every ω 2 Ω 2, X ω2 ( ) =X(,ω 2 ) i F 1 /F meaurable. Proof. See for inance [281], Lemma 1.4, 1.9, 1.1, and [47], Theorem 7.5, page 138. Theorem 1.9 Le (Ω, F ) and ( Ω, G ) be wo meaurable pace and (S, d) a Polih pace (i.e. a complee and eparable meric pace). Le X :(Ω, F ) ( Ω, G ) and φ: (Ω, F ) (S, B(S)) be wo random variable. Then φ i meaurable a a map from (Ω,σ(X)) o (S, B(S)) if and only if here exi a meaurable map η :( Ω, G ) (S, B(S)) uch ha φ = η X.

19 2 1. PRELIMINARIES ON STOCHASTIC CALCULUS IN INFINITE DIMENSIONS Proof. See [281], Lemma 1.13, page 7, or [449] Theorem 1.7 page 5. We refer o [42, 199, 281, 47] for more on meaurabiliy and for he general heory of inegraion. Definiion 1.1 (Borel iomorphim) Le (Ω, F ) and ( Ω, G ) be wo meaurable pace. A bijecion f from Ω ono Ω i called a Borel iomorphim if f i F /G meaurable and f 1 i G /F meaurable. We hen ay ha (Ω, F ) and ( Ω, G ) are Borel iomorphic. Definiion 1.11 (Sandard meaurable pace) A meaurable pace (Ω, F ) i called andard if i i Borel iomorphic o one of he following pace: (i) (ii) (iii) ({1,..,n}, B({1,..,n})), (N, B(N)), {, 1} N, B({, 1} N ), where we have he dicree opologie in {1,..,n} and N, and he produc opology in {, 1} N. The following heorem collec reul ha can be found in [374] (Chaper I, Theorem 2.8 and 2.12). Theorem 1.12 If S i a Polih pace, hen (S, B(S)) i andard. If a Borel ube of S i uncounable, hen i i Borel iomorphic o {, 1} N. Two Borel ube of S are Borel iomorphic if and only if hey have he ame cardinaliy. If (Ω, F ) i andard and A F, hen (A, F A) i andard. In paricular we have he following reul. Theorem 1.13 If (Ω, F ) i andard, hen i i Borel iomorphic o a cloed ube of [, 1] (wih i induced Borel igma field). Definiion 1.14 (Simple random variable) Le (Ω, F ) be a meaurable pace, and (S, d) be a meric pace (endowed wih he Borel σ-field induced by he diance). A random variable X : (Ω, F ) (S, B(S)) i called imple (or imple funcion) if i ha a finie number of value. Lemma 1.15 Le f : (Ω, F ) S be a meaurable funcion beween a meaurable pace (Ω, F ) and a eparable meric pace (S, d) (endowed wih he Borel σ-field induced by he diance). Then here exi a equence f n : Ω S of imple, F /B(S) meaurable funcion, uch ha d (f(ω),f n (ω)) i monoonically decreaing o for every ω Ω. Proof. See [13], Lemma 1.3, page 16. Lemma 1.16 Le S be a Polih pace wih meric d. Le(Ω, F, P) be a complee probabiliy pace and le G 1, G 2 F be wo σ-field wih he following propery: for every A G 2 here exi B G 1 uch ha P(A B) =. Le f :(Ω, G 2 ) (S, B(S)) be a meaurable funcion. Then here exi a funcion g :(Ω, G 1 ) (S, B(S)) uch ha f = g, P a.e., and imple funcion g n :(Ω, G 1 ) (S, B(S)) uch ha d(f(ω),g n (ω)) monoonically decreae o, P-a.e.. Proof. The proof follow he line of he proof of Lemma 1.25, page 13, in [281]. Sep 1: Le u aume fir ha f = x1 A (he characeriic funcion) for ome A G 2 and x S. Byhypohei,wecanfindB G 1.. P(A B) =and hen he claim i proved if we chooe g n g = x1 B. The ame argumen hold for a imple funcion f. Sep 2: For he cae of a general f, hankolemma1.15wecanfindaequence

20 1.1. BASIC PROBABILITY 21 of imple, G 2 -meaurable funcion f n uch ha d(f(ω),f n (ω)) monoonically decreae o. BySep1,wecanfindimple,G 1 -meaurable funcion g n uch ha f n = g n, P-a.e. Thu he claim follow aking g(ω) :=limg n (ω) if he limi exi and g(ω) = (for ome S) oherwie. Lemma 1.17 Le (Ω, F ) be a meaurable pace, and V E be wo real eparable Banach pace uch ha he embedding of V ino E i coninuou. Then: (i) B(E) V B(V ) and B(V ) B(E). (ii) If X :Ω V i F /B(V ) meaurable, hen i i F /B(E) meaurable. (iii) If X :Ω E i F /B(E) meaurable, hen X 1 {X V } i F /B(V ) meaurable. (iv) X :Ω E i F /B(E) meaurable if and only if for every f E, f X i F /B(R) meaurable. Proof. The embedding of V ino E i coninuou, o B(E) V B(V ). Since he embedding i alo one-o-one, i follow from [374], Theorem 3.9, page 21, ha B(V ) B(E), which complee he proof of (i). Par (ii) and (iii) are direc conequence of (i). f(ω) i eparable becaue E i eparable, o Par (iv) i a paricular cae of he Pei heorem, ee [381] Theorem 1.1. Noaion 1.18 If E i a Banach pace we denoe by E i norm. Given wo Banach pace E and F,wedenoebyL(E,F) he Banach pace of all coninuou linear operaor from E o F. If E = F we will uually wrie L(E) inead of L(E,F). If H i a Hilber pace we denoe by, i inner produc. We will alway idenify H wih i dual. If V,H are wo real eparable Hilber pace, we denoe by L 2 (V,H) he pace of Hilber-Schmid operaor from V o H (ee Appendix B.3). The pace L 2 (V,H) i a real eparable Hilber pace wih he inner produc, 2,eePropoiionB.25. Lemma 1.19 Le (Ω, F ) be a meaurable pace and V,H be real eparable Hilber pace. Suppoe ha F :Ω L 2 (V,H) i a map uch ha for every v V, F ( )v i F /B(H) meaurable. Then F i F /B(L 2 (V,H)) meaurable. Proof. Since L 2 (V,H) i eparable, by Lemma 1.17-(iv) i i enough o how ha for every T L 2 (V,H) ω F (ω),t 2 = + k=1 F (ω)e k,te k i F /B(R) meaurable, where {e k } i any orhonormal bai of V.Buhiiclear ince for every ω F (ω),t 2 = lim F n T (ω), n + where F T n (ω) = n F (ω)e k,te k k=1 and Fn T (ω) i F /B(R) meaurable becaue i i a finie um of funcion ha are F /B(R) meaurable. Le I be an inerval in R, E, F be wo real Banach pace, and le E be eparable. If f : I E F i Borel meaurable hen for every I he funcion f(, ) :E F i Borel meaurable (by Lemma 1.8-(iv)). Aume now ha, for all I and for ome m, f(, ) B m (E,F) (he pace of Borel meaurable funcion wih polynomial growh m, ee Appendix A.2

21 22 1. PRELIMINARIES ON STOCHASTIC CALCULUS IN INFINITE DIMENSIONS for he precie definiion). I i no rue in general ha he funcion I B m (E,F), f(, ) i Borel meaurable. A a counerexample 1 one can ake he funcion [, 1] L 2 (R) L 2 (R), (, x) S x, where (S ) i he emigroup of lef ranlaion. Indeed he map [, 1] L(L 2 (R)), S i no meaurable (ee e.g. [13], Secion 1.2). Since L(L 2 (R)) B 1 (L 2 (R),L 2 (R)) and he norm in L(L 2 (R)) i equivalen o he one induced by B 1 (L 2 (R),L 2 (R)), he claim follow in a raighforward way. On he oher hand, we have he following ueful reul. Lemma 1.2 Le I and Λ be wo Polih pace. Le µ be a meaure defined on he he Borel σ-field B(I) and denoe by B(I) he compleion of B(I) wih repec o µ. Le f : I Λ R be Borel meaurable and uch ha for every I, f(, ) i bounded from below (repecively, above). Then he funcion f : I R, inf f(, a) (1.1) a Λ (repecively, f : I R, up a Λ f(, a)) if/b(r) meaurable 2. In paricular if I i an inerval in R, E, F are wo real Banach pace wih E eparable, if ρ : I E F i Borel meaurable and, for all I and for ome m, ρ(, ) B m (E,F), hen he funcion i Lebegue meaurable. ρ 1 : I R, f(, ) Bm(E,F) (1.2) Proof. The fir par i Example in Volume 2 of [44] (recall ha Polih pace are Soulin pace, ee [44], Definiion 6.6.1, and o I Λ i a Soulin pace). For he econd claim, oberve ha ince f i Borel meaurable, alo he funcion f : I E R, f(, x) := ρ(, x) F (1 + x 2 E )m/2 i Borel meaurable (ince i i he produc of a coninuou funcion wih he compoiion of a coninuou funcion and a Borel meaurable one). The reul hu follow from par one. Definiion 1.21 (Independence) Conider a probabiliy pace (Ω, F, P). LeI be a e of indice, and C i F for all i I. We ay ha he familie C i,i I, are independen if, for every finie ube J of I and every choice of A i C i, (i J), we have P A i = P(A i ). i J i J If C i F i, for all i I, aπ-yem (rep. σ-field), he definiion above give in paricular he noion of independen π-yem (rep. σ-field). Random variable are aid o be independen if hey generae independen σ-field. A random variable X i independen of ome σ-field G if σ(x) and G are independen σ-field. Lemma 1.22 Conider a probabiliy pace (Ω, F, P). LeC i F,beaπ-yem for every i I. IfC i, i I are independen, hen σ (C i ),i I, are independen Thi example ha been uggeed o u by Mauro Roeolao. 2 Oberve ha f i no alway Borel meaurable, ee [44] Volume 2, Exercice (ii), page

22 1.1. BASIC PROBABILITY 23 Proof. See [281] Lemma 2.6 page Bochner inegral. Throughou hi ecion (Ω, F, µ) i a meaure pace where µ i σ-finie, and E i a eparable Banach pace wih norm E.We endow E wih he Borel σ-field B(E). Lemma 1.23 Le X :(Ω, F ) E be a random variable. Then he real valued funcion X E i meaurable. Proof. See [13] Lemma 1.2 page 16. Le p 1. WedenoebyL p (Ω, F,µ; E) he quoien pace of he e L p (Ω, F,µ; E) := X :(Ω, F ) (E,B(E)) meaurable : X(ω) p E dµ(ω) < + wih repec o he equivalence relaion of equaliy µ-a.e. L p (Ω, F,µ; E) i a Banach pace when endowed wih he norm 1/p X L p (Ω,F,µ;E) = X(ω) p E dµ(ω) Ω (ee e.g. [138] Theorem 7.17 page 14). We will ofen denoe he norm by X L p when he conex i clear. If H i a eparable Hilber pace, hen L 2 (Ω, F,µ; H) i a Hilber pace a well equipped wih he calar produc X, Y L 2 (Ω,F,µ;H) = Ω X(ω),Y(ω) H dµ(ω). The pace L (Ω, F,µ; E) i he quoien pace of he pace of bounded F /B(E) meaurable funcion wih repec o he relaion of being equal a.e.. I i a Banach pace equipped wih he norm X L (Ω,F,µ;E) =eup X(ω) E. Ω We will denoe L p (Ω, F,µ; R) imply by L p (Ω, F,µ) or L p (Ω,µ) when he σ- algebra F i clear from he conex. In he pecial cae when Ω=I i an inerval wih endpoin a and b wih a<b(which may be ± ), F i he Borel σ-field of I, andµ i he Lebegue meaure on I, we will imply wrie L p (I; E) or L p (a, b; E) for L p (I,F,µ; E). FinallywedenoebyL p loc (I; E) he e of meaurable funcion f : I E uch ha K f() p Ed i finie for every compac ube K of I. Lemma 1.24 If F i counably generaed apar from null e hen L p (Ω, F,µ; E) i a eparable Banach pace. Proof. See [14] p. 92. Definiion 1.25 (Bochner inegral) Le X :(Ω, F,µ) E be a imple random variable X = N i=1 x i1 Ai, where x i E, A i F,µ(A i ) < +. The Bochner inegral of X i defined a N X(ω)dµ(ω) := x i µ(a i ). Ω Le X be in L 1 (Ω, F,µ; E). The Bochner inegral of X i defined a X(ω)dµ(ω) := X n (ω)dµ(ω), Ω lim n + where X n :(Ω, F,µ) E are imple random variable uch ha X(ω) X n (ω) E dµ(ω) =. (1.3) lim n + Ω i=1 Ω Ω

23 24 1. PRELIMINARIES ON STOCHASTIC CALCULUS IN INFINITE DIMENSIONS Remark 1.26 I follow eaily from Lemma 1.15, ha for X L 1 (Ω, F,µ; E), here alway exi a equence of imple random variable X n :(Ω, F,µ) E a in Definiion 1.25, aifying (1.3). Propoiion 1.27 Le X L 1 (Ω, F,µ; E). Then he Bochner inegral of X i well defined and doe no depend on he choice of he equence. Moreover X(ω)dµ(ω) X(ω) E dµ(ω). (1.4) E Ω Proof. See [13] Secion 1.1 (in paricular inequaliy (1.6) page 19 and he par below Lemma 1.5). The proof here i done for a probabiliy meaure µ, bu he general cae i idenical. Propoiion 1.28 Aume ha (Ω, F,µ) i a complee meaure pace, E and F are eparable Banach pace and A : D(A) E F i a cloed operaor (ee Definiion B.3). If X L 1 (Ω, F,µ; E) and X D(A) a.., hen AX i an F -valued random variable, and X i a D(A)-valued random variable, where D(A) i endowed wih he graph norm of A (ee Definiion B.3). If moreover Ω AX(ω) F dµ(ω) < +, hen A X(ω)dµ(ω) = AX(ω)dµ(ω). Ω Proof. The fac ha X i a D(A)-valued random variable and AX i an F -valued random variable follow from Lemma 1.17-(ii). For he la par, ee he proof of Propoiion 1.6, Chaper 1 of [13]. Corollary 1.29 Aume ha E and F are eparable Banach pace and T : E F i a coninuou linear operaor. If X L 1 (Ω, F,µ; E), hen T X(ω)dµ(ω) = TX(ω)dµ(ω). Ω Proof. I i a paricular cae of Propoiion Definiion 1.3 (Lebegue poin) Aume ha Ω i a meric pace wih diance d. LeX be in L 1 (Ω, F,µ; E). Apoin ω Ω i aid o be a Lebegue poin for X if 1 lim µ(b Ω X(ω) X( ω) E dµ(ω) = ( ω)) B Ω( ω) where B Ω ( ω) :={ω Ω : d(ω, ω) }. Theorem 1.31 Le (Ω 1, F 1 ) and (Ω 2, F 2 ) be wo meaurable pace and µ 1 (repecively µ 2 )beaσ-finie meaure on (Ω 1, F 1 ) (repecively on (Ω 2, F 2 )). Then here exi a unique meaure µ 1 µ 2 on F 1 F 2 uch ha, for every A F 1 and B F 2 wih finie meaure, The meaure µ 1 µ 2 i σ-finie. (µ 1 µ 2 )(A B) =µ 1 (A)µ 2 (B). Proof. See Theorem 8.2, page 16 in Chaper VI, Secion 8 of [36]. Theorem 1.32 (Fubini Theorem) Le (Ω 1, F 1 ) and (Ω 2, F 2 ) be wo meaurable pace and µ 1 (repecively µ 2 )beaσ-finie meaure on (Ω 1, F 1 ) (repecively on (Ω 2, F 2 )). Le E be a eparable Banach pace wih norm E. Ω Ω Ω

24 (i) 1.1. BASIC PROBABILITY 25 Le X be in L 1 (Ω 1 Ω 2, F 1 F 2,µ 1 µ 2 ; E). Then, for µ 1 -almo every ω 1 Ω 1, he funcion X(ω 1, ) i in L 1 (Ω 2, F 2,µ 2 ; E), and he funcion given by ω 1 X(ω 1,ω 2 )dµ 2 (ω 2 ) Ω 2 for µ 1 -almo all ω 1 (and defined arbirarily for oher ω 1 ) i in L 1 (Ω 1, F 1,µ 1 ; E). Moreover, we have X(ω 1,ω 2 )d(µ 1 µ 2 )(ω 1,ω 2 )= X(ω 1,ω 2 )dµ 1 (ω 1 )dµ 2 (ω 2 ). Ω 1 Ω 2 Ω 2 (ii) Ω 1 Le X :Ω 1 Ω 2 E be an F 1 F 2 -meaurable map. Aume ha, for µ 1 -almo every ω 1 Ω 1, he funcion X(ω 1, ) i in L 1 (Ω 2, F 2,µ 2 ; E) and ha he map given by ω 1 X(ω 1,ω 2 ) dµ 2 (ω 2 ) Ω 2 for µ 1 -almo all ω 1 (and defined arbirarily for oher ω 1 ) i in L 1 (Ω 1, R). Then X i in L 1 (Ω 1 Ω 2, F 1 F 2,µ 1 µ 2 ; E) and par (i) of he heorem applie. Proof. See Theorem 8.4, page 162, and Theorem 8.7, page 165 in Chaper VI, Secion 8 of [36]. Theorem 1.33 Le E be a Banach pace and µ be a bounded meaure on (E,B(E)). Then he e of uniformly coninuou and bounded funcion UC b (E) i dene in L p (E,B(E),µ) for 1 p<+. Proof. By Lemma 1.15 and he monoone convergence heorem i i enough o prove ha every characeriic funcion 1 A for ome A B(E) can be approximaed by funcion in UC b (E). Since µ i regular, for every > we can find a cloed e C, C A and an open e U, A U uch ha µ(u \ C) < p.moreover, conidering e U n = {x U :di(x: A) > 1/n} if neceary, we can aume ha di(c, U) >. Then he funcion f (x) := di(x, U) di(x, A)+di(x, U) belong o UC b (E) and 1 A f L p < Expecaion, covariance and correlaion. Le (Ω, F, P) be a probabiliy pace and E be a eparable Banach pace wih norm E. Definiion 1.34 (Expecaion) Given X in L 1 (Ω, F, P; E), we denoe by E[X] he (Bochner) inegral X(ω)dP(ω). E[X] i aid o be he expecaion (or he Ω mean) of X. To define he covariance operaor, we recall fir ha if x E, y F, where E,F are Hilber pace, he operaor x y : F E i defined by (x y)h = xy, h F. Definiion 1.35 (Covariance operaor, correlaion) Given a real, eparable Hilber pace H and X L 2 (Ω, F, P; H), he covariance operaor of X i defined by Cov(X) :=E (X E[X]) (X E[X]).