Loughboough Univeiy Iniuional Repoioy Appoximaing oluion of backwad doubly ochaic diffeenial equaion wih meauable coefficien uing a ime diceizaion cheme Thi iem wa ubmied o Loughboough Univeiy' Iniuional Repoioy by he/an auho. Addiional Infomaion: A Docoal Thei. Submied in paial fulfilmen of he equiemen fo he awad of Doco of Philoophy of Loughboough Univeiy. Meadaa Recod: hp://dpace.lboo.ac.uk/2134/20643 Publihe: c Cyu Yeadon Righ: Thi wok i made available accoding o he condiion of he Ceaive Common Aibuion-NonCommecial-NoDeivaive 4.0 Inenaional CC BY-NC-ND 4.0) licence. Full deail of hi licence ae available a: hp://ceaivecommon.og/licene/by-nc-nd/4.0/ Pleae cie he publihed veion.
Appoximaing oluion of backwad doubly ochaic diffeenial equaion wih meauable coefficien uing a ime diceizaion cheme by Cyu Yeadon A Docoal Thei Submied in paial fulfilmen of he equiemen fo he awad of Doco of Philoophy in Mahemaic of Loughboough Univeiy Mach 2015 Cyu Yeadon, 2015
To Lauen, my paen, Uncle Fed, ou lile daling Alexa and he wondeful aff of he lizabeh Gae Andeon Wing a Univeiy College Hopial who aved he. i
Acknowledgemen I would like o hank my upevio, Pofeo Huaizhong Zhao and D Chunong Feng and he PSRC fo he oppouniy o coninue my educaion in Mahemaic. I would alo like o hank my upevio fo genly eeing me away me fom he many abbi hole I have found o inviing ove he pa fou yea. I am alo gaeful fo he uppo ha I have eceived fom my family and Lady Vivien Cockof and would like o acknowledge he fee ofwae diibued by he nice people a Debian - you have all made he wiing of hi hei much moe pleaan. ii
Abac I ha been hown ha backwad doubly ochaic diffeenial equaion BDS- D) povide a pobabiliic epeenaion fo a ceain cla of nonlinea paabolic ochaic paial diffeenial equaion SPD). I ha alo been hown ha he oluion of a BDSD wih Lipchiz coefficien can be appoximaed by fi diceizing ime and hen calculaing a equence of condiional expecaion. Given fixed poin in ime and pace, hi appoximaion ha been hown o convege in mean quae. In hi hei, we inveigae he appoximaion of oluion of BDSD wih coefficien ha ae meauable in ime and pace uing a ime diceizaion cheme wih a view owad applicaion o SPD. To achieve hi, we equie he undelying fowad diffuion o have mooh coefficien and we conide convegence in a nom which include a weighed paial inegal. Thi combinaion of moohe fowad coefficien and weake nom allow he ue of an equivalence of nom eul which i key o ou appoach. We addiionally ake a bief look a he appoximaion of oluion of a cla of infinie hoizon BDSD wih a view owad appoximaing aionay oluion of SPD. Whil we emain agnoic wih egad o he implemenaion of ou diceizaion cheme, ou cheme hould be amenable o a Mone Calo imulaion baed appoach. If hi i he cae, we popoe han addiion o being aacive fom a pefomance pepecive in highe dimenion, uch an appoach ha a poenial advanage when conideing meauable coefficien. Specifically, ince we only diceize ime and effecively ely on imulaion of he undelying fowad diffuion o exploe pace, we ae poenially le vulneable o yemaically oveeimaing o undeeimaing he effec of coefficien wih paial diconinuiie han alenaive appoache uch iii
Abac a finie diffeence o finie elemen cheme ha do diceize pace. Anohe advanage of he BDSD appoach i ha poible o deive an uppe bound on he eo of ou mehod fo a faily boad cla of condiion in a ingle analyi. Fuhemoe, ou condiion eem moe geneal in ome epec han i ypically conideed in he SPD lieaue. iv
Conen Acknowledgemen Abac ii iii 1. Inoducion 1 2. Backgound Maeial 4 3. Review of Lieaue 19 3.1. BSD................................... 19 3.2. Appoximaion of BSD......................... 24 3.3. BDSD.................................. 31 4. Noaion and Poblem Saemen 40 4.1. Noaion.................................. 40 4.2. Poblem Saemen........................... 41 4.2.1. Finie Hoizon Poblem..................... 42 4.2.2. Infinie Hoizon Poblem..................... 44 5. Coninuou-Time Appoximaion 46 5.1. Inoducion................................ 46 5.2. BDSD wih Sep Coefficien..................... 47 5.3. BDSD wih Lipchiz Coefficien................... 65 v
Conen 5.4. BDSD wih Smooh Coefficien................... 73 5.5. Concluion................................. 76 6. Regulaiy of Y and Z 77 6.1. Inoducion................................ 77 6.2. Regulaiy of Y fo BDSD wih Lipchiz Coefficien........ 77 6.3. Regulaiy of Z fo BDSD wih Smooh Coefficien......... 85 6.4. Concluion................................. 104 7. Diceizaion Scheme 105 7.1. Inoducion................................ 105 7.2. Peliminay Reul............................ 106 7.3. Definiion of he Scheme......................... 119 7.4. o imae.............................. 122 7.5. Concluion................................. 130 8. Infinie Hoizon Cae 133 8.1. Inoducion................................ 133 8.2. Peliminay Reul............................ 134 8.3. Diceizaion Scheme.......................... 140 8.4. Concluion................................. 145 9. Dicuion 146 A. Ueful Reul 148 Bibliogaphy 151 vi
1 Inoducion A explained in he abac: I ha been hown ha backwad doubly ochaic diffeenial equaion BDSD) povide a pobabiliic epeenaion fo a ceain cla of nonlinea paabolic ochaic paial diffeenial equaion SPD). In hi hei, we inveigae he appoximaion of oluion of BDSD wih coefficien ha ae meauable in ime and pace uing a ime diceizaion cheme wih a view owad applicaion o SPD. To achieve hi, we equie he undelying fowad diffuion o have mooh coefficien and we conide convegence in a nom which include a weighed paial inegal. Thi combinaion of moohe fowad coefficien and weake nom allow he ue of an equivalence of nom eul which i key o ou appoach. We addiionally ake a bief look a he appoximaion of oluion of a cla of infinie hoizon BDSD wih a view owad appoximaing aionay oluion of SPD. The connecion beween BDSD and nonlinea paabolic SPD wa eablihed in 36 fo BDSD wih mooh coefficien and exended o he meauable coefficien cae of hi hei in 5 and 50. The appoximaion cheme we define fo BDSD wih meauable coefficien i baed upon he appoximaion cheme of 8 and 49 fo BSD wih Lipchiz coefficien. We noe han 2, an appoximaion cheme fo BDSD wih Lipchiz coefficien alo baed upon 8 and 49 i defined. Whil hee i ome ovelap beween he wok of hi hei and 2, he wo wok 1
1. Inoducion wee developed independenly of each ohe and a ou condiion ae ignificanly weake han hoe of 2, he ovelap i no ubanial. SPD of paabolic ype have a vaiey of applicaion including ee e.g. 12): chemical eacion, neuophyiology, populaion geneic, ubulence and geophyical fluid dynamic. A i he cae fo PD, he exac oluion of a SPD i ypically a difficul poblem which moivae hei appoximaion by numeical mehod. A a conequence, he developmen of new numeical mehod fo SPD wih weakened condiion will only help o boaden he applicabiliy of SPD. Thee ae a vaiey of aegie fo he numeical appoximaion of paabolic SPD of which we menion ju a few: 1 conide weak convegence of finie diffeence and finie elemen cheme fo linea SPD wih L 2 coefficien and addiive noie; 30, 18 and 19 conide finie diffeence cheme fo nonlinea SPD wih coninuou coefficien; 21 conuc ochaic Taylo expanion fo nonlinea SPD wih mooh coefficien and addiive noie; 32 conide a Mone Calo cheme baed upon he mehod of chaaceiic fo linea SPD wih mooh coefficien; 44 conide weak convegence of a finie diffeence cheme fo he ochaic hea equaion; 47 conide finie elemen cheme fo nonlinea SPD wih Lipchiz coefficien; 2 conide he appoximaion of BDSD wih Lipchiz coefficien via ime diceizaion which a i obeved in he abac above) implicily povide an appoximaion o a cla of nonlinea paabolic SPD. Wih he excepion of 32 and 2, he equaion conideed in he above ae eenially he ochaic hea equaion wih addiional em. I eem an advanage of pobabiliic cheme uch a he Mone Calo cheme of 32 and he BDSD appoach leveaged in hi hei and in 2 ha eaie o conide moe geneal dif and diffuion em. We noe, howeve, ha boh 32 and 2 equie ignificanly moe egulaiy on he coefficien of hei epecive SPD han we do. The ucue of hi hei i a follow. In Chape 2 we povide ome backgound maeial on andad noaion and ochaic analyi. In Chape 3 we povide a eview of he lieaue on BSD, BDSD and hei numeical appoximaion. In Chape 4 we inoduce he noaion ued in hi hei and define he poblem ha we wih o olve. The main wok of hi hei commence in Chape 5. The appoach aken in hi hei i o fi appoximae he BDSD ha we wih o olve wih BDSD wih moe egula coefficien. We hen define a diceizaion cheme fo hee BDSD 2
1. Inoducion wih moe egula coefficen. To hi end, in Chape 5 we peen he appoximaion of BDSD wih meauable coefficien wih BDSD wih Lipchiz coefficen and BDSD wih mooh coefficen. In Chape 6 we hen deive ome eul on he egulaiy of he oluion of BDSD wih Lipchiz coefficien and BDSD wih mooh coefficien. In Chape 7 we define a diceizaion cheme fo BDSD wih Lipchiz coefficen and deemine an uppe bound fo he eo of he cheme uing he egulaiy eul of Chape 6. In Chape 8 we conide he poblem of defining a diceizaion cheme fo infinie hoizon BDSD wih conacive coefficien. Chape 9 i a dicuion of hi hei and poenial fuue poblem o conide and finally Chape A in he appendix i a collecion of ueful eul. 3
2 Backgound Maeial Thi chape conain backgound maeial on noaion, funcion pace and ochaic analyi. Pleae poceed o Chape 3 if hi maeial i aleady familia. The maeial fo hi chape i aken fom 4, 16, 22, 23, 26, 33, 35, 37, 40, 41, 43 and 45. Fuhe deail on he majoiy of he maeial can be found in 23, 37 and 40. Geneal Noaion Le d be a poiive inege. Fo any veco x R d, x will denoe he andad uclidean nom of x. Fo any d d maix A, A := TAA T. Le p and q be poiive inege. C 0 R p, R q ) denoe he pace of coninuou funcion f : R p R q. Fo k 1, C k R p, R q ) coni of all funcion in C 0 R p, R q ) whoe deivaive of ode le han o equal o k ae coninuou. Fo k 1, Cb krp, R q ) coni of all funcion in C k R p, R q ) whoe deivaive of ode le han o equal o k ae bounded. Noe ha hi doe nomply ha he funcion ielf i bounded. Fo k 1, C0 k R p, R q ) coni of all funcion in C k R p, R q ) whoe uppo a compac ube of R p. Le Ω, F, µ) be a meaue pace, p be a eal numbe wih p 1 and d a poiive inege. Then L p Ω, F, µ); R d ) coni of all R d valued Boel-meauable funcion uch ha Ω f p dµ <. If d 1 and d 2 ae poiive inege hen L p R d 1 ; R d 2 ) L p R d 1, BR d 1 ), l); R d 2 ) whee l denoe he Lebegue meaue. Fo a non-negaive funcion ρ L 1 R d 1 ; R), he ρ-weighed pace L p ρr d 1 ; R d 2 ) coni of all R d 2 -valued Boel-meauable funcion uch ha R d 1 fx) p ρx)dx <. 4
2. Backgound Maeial Filaion, Maingale and Bownian Moion A ochaic poce i a mahemaical model fo he occuence, a each momen afe he iniial ime, of a andom phenomenon. 23 Fo he emainde of hi chape we will aume a given a complee pobabiliy pace Ω, F, P ). Definiion. A family of σ-algeba {F ; 0} uch ha F F F fo 0, ) i aid o be a filaion of F. Definiion. A ochaic poce X on Ω, F, P ) i a collecion of R d -valued wih d 1) andom vaiable {X ; 0} on Ω, F, P ). If Ω, F, P ) i equipped wih a filaion {F ; 0} and X an F -meauable andom vaiable fo each, hen he poce X i aid o be adaped o {F } and we wie {X, F ; 0}. Definiion. A eal-valued, adaped poce {M, F ; 0} i called a maingale epecively upemaingale, ubmaingale) wih epec o he filaion {F } if fo evey 0 <, 1. M L 1 Ω, F, P ); R). 2. M F = M a.. epecively M, M ). Definiion. Le X and Y be ochaic pocee defined on Ω, F, P ). They ae aid o be veion o modificaion of each ohe if X = Y a.. fo each 0. They ae aid o be indiinguihable if a.. i hold ha X = Y fo all 0. Definiion. A ochaic poce X i aid o be coninuou epecively lef-coninuou, igh-coninuou, cadlag) if i a.. ha ample pah which ae coninuou epecively lef-coninuou, igh-coninuou, cadlag). Definiion. A filaion {F ; 0} i aid o aify he uual condiion if 1. F 0 conain he P -null e of F. 2. F = u> F u fo all 0. 5
2. Backgound Maeial Remak. I eay o find an example of why he fi of he uual condiion i deiable. Fo example, if {X, F ; 0} i an adaped poce, Y i a modificaion of X and F 0 conain he P -null e of F hen Y i alo adaped o {F }. The following heoem povide an example of he benefi of addiionally auming he econd of he uual condiion. Theoem. Suppoe ha {F ; 0} aifie he uual condiion and le {M, F ; 0} be a maingale. Then hee exi a unique modificaion of M which i cadlag. Definiion. Le X be a ochaic poce. The naual filaion of X, denoed {F X ; 0}, i defined fo each 0 by F X := σx ; 0, ). Obviouly, X i adaped o {F X }. If he P -null e of F ae denoed by N hen he augmened filaion of X i defined fo each 0 by σf X N ). Definiion. An adaped, coninuou poce {W, F ; 0} aking value in R d wih d 1) i called a d-dimenional {F } andad Wiene poce o andad Bownian moion if 1. W 0 = 0 a.. 2. Fo 0, ), W W i independen of F. 3. Fo 0, ), W W i a Gauian andom vaiable wih mean zeo and covaiance maix )I d, whee I d denoe he d d ideniy maix. Remak. If {F ; 0} i aken o be he augmened filaion of he Wiene poce W, hen W i omeime called a Wiene poce wihou pecifying he filaion. The following heoem how ha no difficul o aain he uual condiion of a filaion. Theoem. The augmened filaion of he andad Wiene poce aifie he uual condiion. Definiion. A andom vaiable T : Ω 0, i aid o be a andom ime. If in addiion hee i a filaion {F ; 0} uch ha he even {T } F fo all 0, hen T i aid o be a opping ime of {F }. Definiion. Le X be a ochaic poce and T be a andom ime. The andom vaiable X T i defined on he even {T < } by X T ω) := X T ω) ω). 6
2. Backgound Maeial Definiion. Fo each a 0 le S a denoe he opping ime T of {F ; 0} uch ha T a a.. and le X be a igh-coninuou ochaic poce. Then X i aid o be of cla DL if he family X T ) T Sa i unifomly inegable fo evey 0 < a <. Definiion. An adaped poce A i called inceaing if 1. A 0 = 0 a.. 2. A a non-deceaing, igh-coninuou funcion a.. 3. A < fo evey 0. Theoem Doob-Meye Decompoiion). Le {F ; 0} aify he uual condiion. If X i a igh-coninuou {F }-ubmaingale of cla DL, hen hee exi a ighconinuou {F }-maingale M and an inceaing poce A adaped o {F } uch ha X = M + A fo each 0. Definiion. Le X be a igh-coninuou maingale. M i aid o be quae-inegable if X 2 < fo evey 0. Definiion. Le M be a quae-inegable maingale wih M 0 = 0 a.. The quadaic vaiaion of M i defined o be he poce M := A, whee A i he inceaing poce in he Doob-Meye decompoiion of M 2. Remak. Le W be a one-dimenional andad Bownian moion. quae-inegable maingale wih W =. Then W i a Definiion. Le X and Y be quae-inegable maingale wih X 0 = 0, Y 0 = 0 a.. The co-vaiaion poce of X and Y i defined by X, Y := 1 4 X + Y X Y ), 0. Definiion. Le X be a ochaic poce, fix > 0 and le Π = { 0, 1,..., n } wih 0 = 0 < 1 <... < n = be a paiion of 0,. The p-h vaiaion fo p > 0 of X ove Π i defined o be V p) Π) := n X k X k1 p. k=1 7
2. Backgound Maeial Remak. Define he meh of Π a Π := max 1 k n k k1. If V 2) Π) convege a Π 0 in ome ene), hen he limi could alo be called he quadaic vaiaion of X on 0,. The following heoem how ha hee wo definiion of quadaic vaiaion ae conien. Theoem. Le M be a quae-inegable maingale wih M 0 = 0 a.. and le Π 1, Π 2,... be a equence of paiion of 0, uch ha lim n Π n = 0. V 2) Π n ) M n pobabiliy a n. Then Definiion. The ochaic poce X T defined fo all 0 by X T := X T i aid o be he poce opped a T. Definiion. A ochaic poce {M, F ; 0} i called a local maingale if hee exi a non-deceaing equence of opping ime of {F }, {T n ; n 1}, uch ha he opped poce M Tn i a maingale fo each n 1 and lim n T n = a.. Sochaic Inegaion A conequence of he non-zeo quadaic vaiaion of Bownian moion and he coninuiy of i ample pah i ha ha ample pah of unbounded vaiaion on any ineval a.. A a eul, i no poible o define Riemann-Sielje inegal of geneal coninuou ochaic pocee wih epec o Bownian moion. In hi ubecion, he heoy of ochaic inegaion iniiaed by Iô ha avoid he poblem inheenn a Riemann-Sielje appoach i eviewed. The appoach i o define ochaic inegaion wih epec o maingale inegao fo inegand ha ae adaped o he ame filaion a he maingale. Thoughou hi ubecion, le {F ; 0} be a filaion aifying he uual condiion and le {M, F ; 0} be a coninuou quae-inegable maingale. Definiion. Le X be an {F ; 0}-adaped poce. Denoe by X 2 T := X 2 d M and X := 2 n 1 + X n ). n=1 0 8
2. Backgound Maeial Definiion. A ochaic poce X i aid o be pogeively meauable wih epec o he filaion {F ; 0} if fo each 0 and A BR d ), {, ω); 0,, ω Ω, X ω) A} B0, ) F. Remak. The following heoem how ha he condiion of pogeive meauabiliy i no difficul o aain. Indeed, Bownian moion ha a pogeively meauable modificaion. Theoem. Le {X, F ; 0} be an adaped poce. If evey ample pah of X i igh-coninuou o evey ample pah of X i lef-coninuou hen X i pogeively meauable wih epec o {F }. Definiion. Le L denoe he e of equivalence clae of pogeively meauable pocee aifying X T < fo all T > 0. Definiion. A poce X i called imple if hee exi a icly inceaing equence of eal numbe { n } n=0 wih 0 = 0 and lim n n = uch ha: 1. Thee exi a equence of andom vaiable {ξ n } n=0 and a conan C uch ha up n 0 ξ n ω) C fo evey ω Ω. 2. ξ n i F n -meauable fo evey n 0. 3. X i defined fo all 0 and ω Ω by X ω) := ξ 0 ω)i {0} ) + ξ i ω)i i,+1 ). i=0 The cla of imple pocee i denoed by L 0. Definiion. Fo X L 0, he ochaic inegal of X wih epec o M, I X), i defined fo 0 by I X) := ξ i M i+1 M i ). i=0 Remak. The exenion of he definiion of he ochaic inegal fom he e L 0 o he e L in a well-defined i.e. unique) way i baed upon he obevaion ha fo 9
2. Backgound Maeial X L 0, I X)) 2 = X 2 d M and he following eul. 0 Theoem. L 0 i dene in L wih epec o he meic dx, Y ) := X Y fo X, Y L. Definiion. Fo X L, he ochaic inegal of X wih epec o M i he unique quae-inegable maingale IX) = {I X), F ; 0} which aifie lim n k=1 IX n) ) IX) 2 1 2 k = 0 fo evey equence {X n) } n=1 L 0 wih lim n X n) X = 0. Denoe fo 0 I X) = 0 X dm. Theoem. Fo X L, and I X)) 2 = X 2 d M IX) = 0 0 X 2 d M. Theoem Iô Fomula). Le {M := M 1) coninuou local maingale wih M 0 = 0 a.. and {A := A 1),..., M d) ), F ; 0} be a veco of,..., A d) ), F ; 0} a veco of adaped pocee of bounded vaiaion wih A 0 = 0. Se X = X 0 +M +A fo 0 whee X 0 i an F 0 -meauable andom veco in R d and le f C 1,2 R + R d ; R). Then a.. i hold ha fo all 0, f, X ) = f0, X 0 ) + 0 f, X )d + d i=1 0 f, X )da i) x i 10
+ + d i=1 d i=1 0 d 2. Backgound Maeial f, X )dm i) x i 0 2 x i x j f, X )d M i), M j). Remak. We noe ha Iô Fomula i a fundamenal ool in ochaic analyi and make i eay o conide funcion of emi-maingale. Theoem Bukholde-Davi-Gundy Inequaliy). Le M be a coninuou local maingale uch ha M 0 = 0 a.. Then fo evey m > 0 hee exi poiive conan k m and K m uch ha fo evey opping ime T k m M m T up M 2m K m M m T. 0,T Remak. A we will ee, he Bukholde-Davi-Gundy Inequaliy i a vey ueful ool and one which we hall make fequen ue of. coninuou local maingale M will be a ochaic inegal. In ou ue of he inequaliy, he Theoem Maingale Repeenaion Thoeem). Le {W, F ; 0} be a d-dimenional andad Bownian moion whee {F } i he augmened filaion of W. Then fo any quae-inegable maingale {M, F ; 0} wih M 0 = 0 a.. and cadlag pah a.., hee exi quae-inegable pogeively meauable pocee {Y j), F ; 0} uch ha fo 0 M = d 0 Y j) dw j). Remak. A we will ee, he Maingale Repeenaion Thoeem i fundamenal o he heoy of backwad ochaic diffeenial equaion. Now le {W, F ; 0} be a d-dimenional andad Bownian moion and ecall ha {F } aifie he uual condiion. Le {X, F ; 0} be a d-dimenional veco of adaped pocee aifying fo 1 i d and any 0, 0 ) X i) 2 d < a.. 11
2. Backgound Maeial Define hen Z X) := exp { d i=1 0 X i) dw i) 1 2 0 X i) 2 d } Z X) = 1 + d i=1 0 Z X)X i) dw i). I follow ha ZX) i a coninuou local maingale wih Z 0 X) = 1. If ZX) i a maingale, define fo each T 0 he pobabiliy meaue Q T on F T by Q T A) := I A Z T X) fo each A F T. Theoem Novikov Condiion). If fo all T 0 { 1 T } exp X 2 d < 2 0 hen ZX) i a maingale. Theoem Baye Rule). Fix T 0 and aume ha ZX) i a maingale. 0 T and Y i an F -meauable andom vaiable wih QT Y < hen If QT Y F = 1 Z X) Y Z X) F, P -a.. and Q T -a.. Theoem Gianov Theoem). Aume ha ZX) i a maingale and define he d-dimenional poce { W, F ; 0} by W i) := W i) 0 X i) d; 1 i d, 0. Then fo each fixed T 0, { W, F ; 0, T } i a d-dimenional andad Bownian moion on Ω, F T, Q T ). Sochaic Diffeenial quaion Le b i, x), σ ij, x); 1 i d, 1 j, be Boel-meauable funcion fom R + R d ino R. Define he veco b, x) := {b i, x); 1 i d} and he maix 12
2. Backgound Maeial σ, x) := {σ ij, x); 1 i d, 1 j }. Fuhemoe, le W = {W ; 0} be an -dimenional andad Bownian moion and ake {F ; 0} o be he augmened filaion of W. Conide he ochaic diffeenial equaion dx X = x. = b, X )d + σ, X )dw,, T 2.1) Definiion. A ong oluion of he ochaic diffeenial equaion 2.1) i an adaped poce {X, F ; 0} wih coninuou ample pah uch ha 1. Fo evey 1 i d, 1 j and 0 0 { bi, X ) + σ 2 ij, X ) } d < a.. 2. a.. i hold fo all ha X = x + b, X )d + σ, X )dw. Theoem. If hee exi a conan K uch ha fo evey 0 and x, y R d b, x) b, y) + σ, x) σ, y) K hen he ochaic diffeenial equaion 2.1) ha a unique up o indiinguihabiliy ong oluion. Theoem Feynman-Kac). Suppoe ha he condiion of he peviou heoem hold and define he diffeenial opeao Lu := d i=1 b i, x) x i + 1 2 d 2 a ij, x), a ij, x)) := σσ T, x). x i x j i, Fix T > 0 and aume ha hee exi conan L > 0 and λ 2 uch ha he coninuou funcion f : R d R, g : 0, T R d R and k : 0, T R d R + aify 1. fx) L1 + x λ ) ; x R d. 13
2. Backgound Maeial 2. g, x) L1 + x λ ) ; 0, T, x R d. Suppoe fuhe ha u C 1,2 0, T ) R d ; R) aifie he PD u, x) + k, x)u, x) = Lu, x) + g, x); 0, T ), x R d, 2.2) ut, x) = fx); x R d. If in addiion hee exi conan M > 0 and µ 0 uch ha max u, x) M1 + 0,T x 2µ ); x R d, hen fo 0, T and x R d, u, x) = + fx T ) exp { g, X ) exp { k, X } )d k, X )d } d. Remak. Heuiically, hi ay ha we can olve he PD 2.2) by leing he ochaic poce X exploe pace unil ime T and hen calculae a funcional of i pah. The poce X exploe pace in ju he igh way o geneae he diffeenial opeao L. The heme of pobabiliic epeenaion of he oluion of a PD i fundamenal o he opic of hi hei. A we will ee in ou eview of lieaue in Chape 3, backwad doubly ochaic diffeenial equaion BDSD) povide pobabiliic epeenaion of a cla of ochaic PD and o by appoximaing oluion of ceain BDSD, we ae in fac able o appoximae oluion of ochaic PD. Remak. We noe ha he concep of a pobabiliic epeenaion i implicin he following elemenay eul on he hea equaion: he oluion o he iniial value poblem u, x) = 1 2 u, x), u0, x) = hx) 2 x 2 i given by u, x) = 1 2π e xy)2 /2 hy)dy hx + W ). 14
2. Backgound Maeial Backwad Sochaic Diffeenial quaion Backwad ochaic diffeenial equaion BSD) have a imila fom o SD wih andom coefficien in he ene ha hey conain a Lebegue inegal em and a ochaic inegal em. Fuhemoe, a in he cae of he oluion of an SD, he oluion of a BSD i adaped o he filaion of he diving noie. The backwad nomenclaue efe o he poviion of a eminal condiion a pa of he pecificaion of a BSD a oppoed o a aing condiion. A a eul of hi, he oluion of a BSD i no longe a ingle poce bun fac a pai of pocee. BSD have applicaion o ochaic conol and alo povide pobabiliic epeenaion fo a cla of emilinea PD ee fo example 40 o Chape 3). A we will ee in ou eview of lieaue in Chape 3, he advanage of he BSD epeenaion of PD i ha allow moe non-lineaiy in he coefficien of he PD. To inoduce BSD, le {W, F ; 0} be a d-dimenional Bownian moion wih {F } he augmened filaion of W and fix T > 0. Le ξ L 2 Ω, F T, P ); R) and f : Ω 0, T R R d R. fω,, y, z) i wien a f, y, z) i.e. he dependence on ω i implici) and i aumed ha 1. Fo any fixed y R and z R d, he andom funcion f., y, z) i pogeively meauable. T 2. f, 0, 0 0) 2 d <. 3. Thee exi a conan C uch ha fo all y 1, y 2 R, z 1, z 2 R d almo evey 0, T and a.. f, y 1, z 1 ) f, y 2, z 2 ) C y 1 y 2 + z 1 z 2 ). Conide he BSD dy = f, Y, Z )d Z, dw, Y T = ξ. 2.3) Definiion. A oluion o he BSD 2.3) i a pogeively meauable pai Y, Z) aifying fo all 0, T Y = ξ + f, Y, Z )d Z, dw 15
2. Backgound Maeial uch ha up Y 2 + 0,T 0 Z 2 d <. Theoem. Given a pai ξ, f) aifying he condiion above, hee exi a unique oluion o he BSD 2.3). Remak. Given ha he andom eminal condiion i pecified in he definiion of BSD 2.3), i no clea how o conuc he oluion Y, Z) o ha hey ae boh adaped o {F }. To give he main idea of how hi i done, we epoduce pa of he poof aken fom 40) below wih eveal echnical poin omied. Poof. The poof i baed upon he fixed poin mehod. U, V ) Y, Z) defined by Conide he mapping Y = ξ + f, U, V )d Z, dw. The pai Y, Z) i conuced a follow: fi define he maingale M := ξ + 0 f, U, V )d F. Then, by he Maingale Repeenaion Theoem, hee exi an adaped poce Z uch ha M = M 0 + Define he poce Y by Y = 0 ξ + Z, dw. I follow ha ince Y T = ξ, f, U, V )d F. Y = M = M 0 0 0 f, U, V )d f, U, V )d + 0 Z, dw 16
= ξ + 2. Backgound Maeial f, U, V )d Z, dw a equied. The emainde of he poof how ha hi mapping i a conacion and i omied. Backwad Maingale In hi ecion we inoduce backwad maingale which will be helpful in undeanding he heoy of backwad doubly ochaic diffeenial equaion. To hi end, le T > 0 be ome fixed finie ime. Definiion. A family of σ-algeba {F,T ; 0, T } uch ha F,T F,T F if 0 T i aid o be a backwad filaion of F. Remak. Fo he emainde of hi ecion we will aume a backwad filaion, {F,T ; 0, T }, a given. Definiion. A ochaic poce X = {X ; 0, T } i aid o be adaped o {F,T } if X F,T -meauable fo evey 0, T. Definiion. A eal-valued poce {M ; 0, T } i called a {F,T } backwad maingale if 1. M L 1 Ω, F, P ); R) fo evey 0, T. 2. M i adaped o {F,T }. 3. M F,T = M a.. fo evey 0 T. Definiion. A coninuou poce { W ; 0, T } aking value in R d wih d 1) i called a d-dimenional {F,T } backwad Wiene poce if 1. W i adaped o {F,T }. 2. W T = 0 a.. 3. Fo 0 T, W W independen of F,T. 4. Fo 0 T, W W a Gauian andom vaiable wih mean zeo and covaiance maix )I d, whee I d denoe he d d ideniy maix. 17
2. Backgound Maeial Remak. Le W be a andad one-dimenional Wiene poce and define he backwad filaion F,T := σ{w W T ;, T }. Then he poce W := W W T i a one-dimenional {F,T } backwad Wiene poce. Definiion. A poce X i called imple wih epec o {F,T } if hee exi a icly inceaing equence of eal numbe { k } n k=0, wih 0 = 0 and n = T uch ha: 1. Thee exi a equence of andom vaiable {ξ k } n k=1 max k=1,...,n ξ k ω) C fo evey ω Ω. and a conan C uch ha 2. ξ k i F k,t -meauable fo evey k 1. 3. X i defined fo all 0, T and ω Ω by X ω) := n ξ k ω)i k1, k )) + ξ n ω)i {T } ). k=1 Definiion. Le {M ; 0, T } be a coninuou quae-inegable {F,T } backwad maingale and X be a imple poce wih epec o {F,T }. Then he backwad ochaic inegal of X wih epec o M, I X), i defined fo 0, T by I X) := n ξ k M k M k1 ). k=1 Remak. Ju a he fowad) ochaic inegal i defined a he limin pobabiliy of he ochaic inegal of a equence of imple pocee wih epec o a fowad) filaion, he backwad ochaic inegal i defined a he limin pobabiliy of he backwad ochaic inegal of imple pocee wih epec o a backwad filaion. Indeed, a emaked in 37, if W i an {F,T } backwad Wiene poce and X i a coninuou poce adaped o {F,T } hen he backwad ochaic inegal of X wih epec o W can be defined a X dw := lim Π 0 n X k W k W k1 ) k=1 in pobabiliy whee Π := max 1 k n k k1 and i implici ha n a Π 0). 18
3 Review of Lieaue The eview of lieaue in hi chape i epaaed ino hee ecion. Secion 3.1 cove he geneal heoy and applicaion of backwad ochaic diffeenial equaion BSD), Secion 3.2 cove he appoximaion of BSD and finally Secion 3.3 cove backwad doubly ochaic diffeenial equaion BDSD). The wok in hi hei elie heavily upon hee of he efeence eviewed in hi chape: he pape 49 Zhang, 2004) on he appoximaion of BSD develop a geneal aegy baed upon deiving a egulaiy eul fo Z which we adap o he BDSD eing; he pape 36 Padoux and Peng, 1994) and 50 Zhang and Zhao 2007) on he geneal heoy of BDSD conain eul which we make epeaed ue of. A a conequence, hee efeence ae coveed in addiional deail in hee Key Refeence ubecion. 3.1. BSD Moivaed by ochaic conol heoy, BSD wee inoduced by Padoux and Peng in 1990 in he pape 34. They conide BSD of he fom Y = ξ + f, Y, Z )d + g, Y, Z )dw 3.1) fo 0, T whee {W ; 0} i a k-dimenional andad Wiene poce on a complee pobabiliy pace Ω, F, P ) wih {F ; 0} he augmened naual filaion of W ; ξ L 2 Ω, F T, P ); R d ). f : Ω 0, T R d R d k R d and g : Ω 0, T R d R d k R d k ae andom funcion uch ha 19
3. Review of Lieaue 1. f i P BR d R d k )/BR d )-meauable and g i P BR d R d k )/BR d k )- meauable whee P denoe he σ-algeba of {F }-pogeively meauable ube of Ω 0, T. T 2. { f, 0, 0 0) 2 + g, 0, 0) 2 } d <. 3. Fo a.e., ω), f and g ae globally Lipchiz in y and z. 4. Thee exi a conan α > 0 uch ha fo evey y and a.e., ω), g, y, z 1 ) g, y, z 2 ) α y 1 y 2. In hi conex, hey how ha hee exi a unique {F }-pogeively meauable T pai Y, Z) aifying { Y 0 2 + Z 2 } d < which olve 3.1). In 39 1991), Peng eablihed he connecion beween BSD and quailinea PD. Thi wa achieved by inoducing looely-coupled BSD of he fom X = x + Y = hx T ) + b, X )d + f, X σ, X )dw,, Y, Z )d Z dw 3.2) fo, T. We noe ha he eminology looely-coupled efe o he fac ha he backwad equaion fo Y and Z depend upon he fowad equaion fo X bu he fowad equaion i independen of he backwad equaion. Given diffeeniabiliy condiion on he coefficien and non-degeneacy of σ, Peng howed haf u olve he paabolic PD whee u, x) = Lu, x) + f, x, u, x), u, x)σ, x)), 3.3) ut, x) = hx) Lu := d i=1 b i, x) x i + 1 2 d 2 a ij, x), a ij, x)) := σσ T, x) x i x j i, hen u, x) = Y. Peng alo eablihed imila elaion fo paabolic and ellipic PD defined on bounded domain of R d. 20
3. Review of Lieaue In 35 1992) Padoux and Peng made ignificannoad in exending he heoy of looely-coupled BSDS. Given diffeeniabiliy condiion on he coefficien of BSD 3.2), Padoux and Peng howed ha u, x) := Y olve he PD 3.3). An inemediae ep o hi eul wa he deivaion of he key elaionhip Z = Y X ) 1 σx ). They alo howed haf f and h ae ju Lipchiz coninuou in y and z hen u, x) := Y a vicoiy oluion of he PD 3.3). Depie hi being a key pape in he heoy of BSD and hei connecion o PD, we do no povide fuhe deail hee. Thi ha been done o avoid epeiion when in he nex ecion we povide deail of hei pape 36 on BDSD which i vey much in he ame vein a 35. Remak. A we will ee, in 49 Zhang ue he epeenaion Z = Y X ) 1 σx ) deived in 35 o conuc hi key eul on he egulaiy of Z. In 36 Padoux and Peng deive he ame epeenaion of Z fo he BDSD cae and we will make ue of hi epeenaion o pove ou eul on he egulaiy of Z. In 6 1997) Bale, Buckdahn and Padoux conide looely-coupled BSD and incopoae a Poion andom meaue ino he diving noie of boh he fowad and backwad equaion. In hi eing, he oluion of he BSD i no longe a pai Y, Z) bu now a iple Y, Z, U). Wih condiion imila o 35, hey how ha he BSD ha a unique oluion. They hen again in a imila vein o 35) connec he oluion of he BSD o he vicoiy oluion of a yem of paabolic inegal-paial diffeenial equaion. In 25 2000) Kobylanki conide BSD of he fom Y = ξ + f, Y, Z )d Z dw fo 0, T whee he eminal condiion ξ i bounded and f i coninuou and ha quadaic gowh in Z. She hen pecialize hee condiion o he looely coupled BSD eing and connec he oluion of he looely coupled BSD o he vicoiy 21
3. Review of Lieaue oluion of he coeponding PD. In 3 1993) Anonelli conide adaped oluion o fully coupled fowad-backwad SD FBSD) of he fom U = J + f, U, V )dx 0 V = g, U, V )dz + Y F V T = Y,, 0, T, whee Y i an F T -meauable andom vaiable, f and g ae unifomly Lipchiz in u and v, X and Z ae emimaingale and J a cadlag poce. We noe ha hee, a boh U and V appea in boh he fowad and backwad equaion, we efe o he equaion a fully coupled. We alo noe ha whil he equaion ae in ome ene moe geneal fo example he fully coupling and he genealizaion o emimaingale noie), hey ae le geneal in he ene ha he dive g of he backwad equaion only depend upon U and V. We noe ha hi i a ignifican depaue fom he looely coupled cae a in he looely coupled cae he backwad equaion i allowed o depend upon he oluion of he fowad equaion bu he fowad equaion may no depend upon he oluion of he backwad equaion. In 29 1994) Ma, Poe and Yong conide fully coupled FBSD of he fom X = x + Y b, X = hx T ) +, Y, Z )d + f, X, Y, Z )d + σ, X, Y )dw, g, X, Y, Z )dw whee W i a d-dimenional Bownian moion and b, σ, f and g ae all mooh funcion. Unde hee ong condiion, hey how ha hi vey geneal fom of FBSD ha a unique adaped oluion iple X, Y, Z) : 0, T Ω R n R m R m d. They achieve hi by following wha hey call he Fou Sep Scheme : Sep 1 Find a mooh mapping z : 0, T R n R m R m n R m d aifying fo all, x, y, p) 0, T R n R m R m n pσ, x, y) + g, x, y, z, x, y, p)) = 0. 22
3. Review of Lieaue Sep 2 Uing z, olve he following PD fo u, x): u k + 1 2 u xxσ, x, u)σ, x, u) T )) + b, x, u, z, x, u, u x )), u k x f k, x, u, z, x, u, u x )) = 0, k = 1,..., m,, x) 0, T ) R n, ut, x) = hx), x R n. Sep 3 Uing u and z, olve he SD: X = x + b, X )d + 0 0 σ, X )dw whee b, x) := b, x, u, x), z, x, u, x), u x, x))) and σ, x) := σ, x, u, x)). Sep 4 Se Y := u, X ), Z := z, X, u, X ), u x, X )). I ineeing o noe he ue of he PD connecion in he Fou Sep Scheme. In 13 2002) Delaue conide fully coupled FBSD of he fom X = ξ + 0 Y = hx T ) + b, X, Y, Z )d + f, X, Y, Z )d 0 σ, X, Y )dw, Z dw unde condiion weake han hoe of 29. Delaue deive exience and uniquene of he above FBSD unde faily andad Lipchiz condiion on he coefficien along wih a non-degeneacy condiion on σ. We noe ha a in 29, he PD connecion play a ole in he poof; namely Delaue pove he exience and uniquene of an adaped iple X, Y, Z) ha olve he FBSD ove a mall enough) ime ineval and ue he PD connecion o exend he eul o an abiay bu fixed) ime ineval. We noe ha he PD in hi cae dio 29) i ignificanly moe geneal han he cae of 35 ince he coefficien b i now allowed o depend upon Y and Z and σ i allowed o depend upon Y. 23
3. Review of Lieaue A we have een, one applicaion of BSD i o povide a pobabiliic epeenaion fo a cla of PD. The PD connecion i no, howeve, he only aea of applicaion fo BSD. Fo example, in 15 1997) l Kaoui e al conide efleced BSD wih oluion iple Y, Z, K) of he fom Y = ξ + Y S, 0, T 0 f, Y, Z )d + K T K Z dw, 0, T, 3.4) whee S i a given coninuou, pogeively meauable poce known a he obacle. In addiion, he new componen o he oluion, K, mu be coninuou and inceaing and Y and K mu aify 0 {Y S T } dk = 0. They how haf f i Lipchiz in y and z hen hee exi a unique oluion o he efleced BSD. They hen elae he oluion componen Y o he value funcion of opimal opping poblem and, by looely coupling he BSD 3.4) o he oluion of an SD, he vicoiy oluion of PD obacle poblem. Fuhemoe, in 20 2007) Hamadene and Jeanblanc apply efleced BSD o eal opion poblem uch a deemining he opimal aegy fo eleciciy poducion by a powe aion. In 11 2014) Cohen conide BSD whee he noie i geneaed by a coninuou ime Makov chain and he eminal value i pecibed by a opping ime. Wih hi fomulaion, he find applicaion o deemining he opimal policy fo each meage ending node) fo anmiing meage ove a finie newok and he opimal conol on he peed of individual edge aveal) fo aveing a dieced gaph. 3.2. Appoximaion of BSD Thee have been eveal appoache o olving BSD numeically o a lea via diceizaion cheme upon which a numeical cheme could poenially be baed). Mo of hee appoache fall quie comfoably ino one of hee camp. The fi camp make ue of he PD connecion o BSD and olve he aociaed PD numeically. Indeed, one could imply appoximae looley coupled BSD 24
3. Review of Lieaue of he fom X = x + Y = hx T ) + b, X )d + f, X σ, X )dw,, Y, Z )d by appoximaing he elaed PD uing andad echnique. Z dw 3.5) A econd camp appoximae he diving Bownian moion wih a imple ochaic poce, uch a a andom walk and how ha he oluion o he imple equaion convege o he oluion of he oiginal BSD in ome ene. Whil hi appoach i diec and appealing fom a mahemaical poin of view, o dae i ha ended o equie ong aumpion on he coefficien of he BSD. A final camp aack he BSD poblem diecly by imply diceizing he BSD ielf. Fo example, a imple ule-like diceizaion cheme of he BSD given by 3.5) would ake he fom Y Y u + fu, X u, Yu, Zu )u ) Z, W u W fo u. A poblem wih hi appoximaion, howeve, i ha fail o enue ha Y i F -meauable. To fix hi, we can ake condiional expecaion o give he explici appoximaion Y Y u + fu, X u, Yu, Zu )u ) F. We noe ha hi i effecively he appoach aken by Zhang in 49 and he appoach aken in hi hei fo he BDSD cae. We begin ou eview on he appoximaion of BSD wih a pape fom he PD camp. In 14 1996) Dougla, Ma and Poe conide fully coupled FBSD of he fom conideed in 29: X = x + Y b, X = hx T ) +, Y, Z )d + f, X, Y, Z )d + σ, X, Y )dw, g, X, Y, Z )dw whee W i a d-dimenional Bownian moion and b, σ, f and g ae all mooh 25
3. Review of Lieaue funcion. To appoximae he above equaion, hey follow he Fou Sep Scheme of 29. They ue andad echnique o appoximae he PD of Sep 2 and ue he ule cheme o appoximae he SD of Sep 3. We noe han 31 2006) Milein and Teyakov efine hi appoach o obain a moe efficien cheme by uiliing moe advanced echnique o appoximae he equaion in ep 2 and 3 of he Fou Sep Scheme of 29 The fi cheme o diecly appoximae looely coupled BSD wih condiion imila o hoe equied fo exience and uniquene a deived by Padoux and Peng in 35 wa ha of Zhang in 49 of which we give a deailed oveview in hi ecion. Pio o hi, we menion hee ealie aemp a diec appoximaion. Fily in 10 1997) Chevance conide looely coupled BSD of he fom X = ξ + 0 Y = hx T ) + b, X )d + 0 f, X, Y )d σ, X )dw, Z dw whee he coefficien b, σ, f and h aify ong moohne condiion. To decibe hi cheme, le = 0, 1,..., n = T be a unifom paiion of 0, T, h := T, U n be a n dicee-ime appoximaion of W and {Fj n } he naual filaion of U n. Chevance cheme appoximae X and Y wih X and Ŷ epecively whee X i given by X 0 := ξ, X j := X j1 + hb j1, X j1 ) + hσ j1, X j1 )Uj n, 1 j n and Ŷ i given by Ŷ n := h X n ), Ŷ j := Ŷj+1 + hf j+1, X j+1, Ŷj+1) Fj n, 1 j n. In compaion o he eul deived in 49, Chevance cheme equie much onge condiion on he coefficien and doe no allow f o depend upon z. In 9 2001) Biand, Delyon and Mémin do conuc a cheme ha allow f o 26
3. Review of Lieaue depend upon z. They conide equaion of he fom Y = ξ + fy, Z )d Z dw whee ξ L 2 Ω, F T, P ), R) and f i Lipchiz in y and z. Thei cheme i baed upon appoximaing he andad Bownian moion W wih a caled andom walk. Moe pecifically, hey unifomly diceize 0, T ino n ubineval, define h := T n and ake {ɛ k } 1 k n o be an i.i.d. Benoulli ymmeic equence. They define hei appoximaion cheme by y n = ξ n, y k = y k+1 + hfy k, z k ) hz k ɛ k+1, k = n 1,..., 0 z k = h 1/2 y k+1 ɛ k+1 G k. They hen how haf W n := /h h ɛ n k, 0, T, k=1 aifie up 0,T W n W 0 in pobabiliy hen up Y n Y 2 + Z n Z 2 d 0 0,T 0 in pobabiliy whee Y n := y n /h, Zn := z n /h. We noe han 28 2002) Ma e al alo conuc an appoximaion cheme baed upon appoximaing andad Bownian moion W wih a caled andom walk. They, howeve, conide equaion of he fom Y = ξ + f, Y )d Z dw whee ξ L 2 Ω, F T, P ), R) and f i Lipchiz in and y. Whil ome of hei condiion ae weake han hoe of 9, hei dive f canno depend upon z - a lea non a nonlinea way. 27
3. Review of Lieaue Key Refeence: 49 - Zhang, 2004 To decibe he appoach of Zhang in 49, le T > 0 be a fixed eminal ime, Ω, F, P ) be a complee pobabiliy pace on which i defined a andad Bownian moion W and ake {F ; 0} o be he augmened filaion of W. Denoe by D he pace of all eal-valued cadlag funcion defined on 0, T. Le b,σ : 0, T R R and f : 0, T R R R be deeminiic funcion and Φ : D d R a deeminiic funcional. Zhang hen conide he following looely-coupled BSD whee X, Y and Z ae all eal-valued ochaic pocee: X = x + Y = ΦX) + b, X )d + 0 0 f, X, Y, Z )d σ, X )dw, 3.6) Z dw. Zhang aume ha: 1. b, σ and f ae unifomly 1 Hölde coninuou in and unifomly Lipchiz 2 coninuou in hei emaining vaiable. 2. Thee exi a conan K uch ha fo all x 1, x 2 D, Φx 1 ) Φx 2 ) K up x 1 ) x 2 ). 0 T 3. Thee exi a conan K uch ha up { b, 0) + σ, 0) + f, 0, 0, 0) } + Φ0) K. 0 T Remak. Fom he eul of 35 and unde he above condiion, he SD and BSD given by 3.6) have a unique oluion. Zhang define Π : 0 = 0 <... < n = T o be a paiion of 0, T and define := 1 and Π := max i. Befoe inoducing hi diceizaion cheme, he deive he following eul on he egulaiy of Z. Remak. The following eul eally he key eul of he pape and i wha make Zhang appoach wok. A peviouly noed, he eul hinge on he epeenaion of Z deived in 35. 28
3. Review of Lieaue Theoem. Suppoe ha he above condiion hold, ha Z i cadlag and le Π be any paiion of 0, T. Thee exi a conan C > 0 depending only upon T and K uch ha n i=1 i { Z Z i1 2 + Z Z i 2} d C1 + x 2 ) Π. 3.7) 1 Zhang hen define hi diceizaion cheme a follow. Define π) := 1 fo 1, ) and le X π be he oluion o he SD X π = x + Now define Y π n = ξ π, Z π,1 n = 0 0 bπ), X π) )d + and fo 1, ), i = n, n 1,..., 1 0 σπ), X π) )dw. Y π = Y π i + f, Θ π,1 ) i Z π dw whee ξ π L 2 Ω, F T, P ); R), Θ π,1 := X π, Y π, Z π,1 ) and Z π,1 := 1 i+1 +1 Z π d F. He hen pove he following heoem which povide a bound on he mean quae eo of he cheme. Theoem. Suppoe ha he above condiion hold, ha Z i cadlag and ha hee i a conan K 2 > 0 uch ha paiion Π aifie Π K 2 fo i = 1,..., n. Then max Y i Y π 0 i n i 2 + Z Z π 2 d 0 C 1 + x 2 ) Π + ΦX) ξ π 2) 29
3. Review of Lieaue whee C depend only upon T, K and K 2. In 8 2004) Bouchad and Touzi conide looely coupled BSD of he fom X = x + Y = hx T ) + b, X )d + f, X σ, X )dw,, Y, Z )d Z dw. They define he following implici diceizaion cheme which we noe i imple han he one defined by Zhang in 49. Ŷ n := h X n ), Ŷ 1 := Ŷ F i1 + f, Ẑ n := 0, Ẑ 1 := 1 Ŷ W i F i1. X 1, Ŷ 1, Ẑ 1 ), Fom hee, hey conuc a coninuou ime cheme via he Maingale Repeenaion Theoem. Taking advanage of he key eul on he egulaiy of Z deived in 49, hey how a imila fom of L 2 convegance a in 49. They hen go on o conide ome imulaion baed implemenaion of hei diceizaion cheme. In 17 2005) Gobe, Lemo and Wain alo conide looely coupled BSD of he fom X = x + Y = hx T ) + b, X )d + f, X σ, X )dw,, Y, Z )d Z dw. They develop a lea quae Mone Calo numeical cheme baed upon he implici cheme of 8. The bai of he cheme i o geneae a numbe of fowad pah inance of X, he numeical appoximaion of X) and fo each pah, ω, o calculae hx T ω)). The cheme hen poceed by epping backwad in ime calculaing condiional expecaion eenially wih epec o F i ) by egeing upon he value of X. Remak. Lea quae Mone Calo cheme of hi kind wee made popula by 27 fo picing Ameican opion. In ohe diecion, we menion han 7 2008) Bouchad and lie exend he appoach of 49 o looely coupled FBSD wih jump a conideed in 6) and 30
3. Review of Lieaue in 42 2011), Richou exend he appoach of 49 o looely coupled FBSD wih bounded eminal condiion and dive f wih quadaic gowh in Z a inoduced in 25. 3.3. BDSD Backwad doubly ochaic diffeenial equaion BDSD) wee inoduced in 1994 by Padoux and Peng in hei pape 36. In 36, hey exend he eing of BSD in 35 by inoducing a econd noie - hence he em doubly ochaic. They hen genealie he connecion beween BSD and PD o one beween BDSD and ochaic PD SPD). A hi i maeial i fundamenal o he hei, we now povide a deailed eview of 36. Key Refeence: 36 - Padoux and Peng, 1994 The eing fo 36 i a follow. On a pobabiliy pace Ω, F, P ) le {W ; 0} and {B ; 0} be muually independen andad Bownian moion aking value in R d and R m epecively. Le N denoe he P -null e of F and fix T > 0. Fo each 0, T, define F := F W F,T B whee fo any poce {φ }, F, φ := σ{φ u φ ; u, } N and F φ := F0,. φ Fo any n N, le M 2 0, T ; R n ) denoe he e of n-dimenional {F }-adaped T pocee {φ ; 0, T } ha aify φ 0 2 d <. Similaly, le S 2 0, T ; R n ) denoe he e of n-dimenional {F }-adaped pocee {φ ; 0, T } ha aify up 0 T φ 2 <. They define he coefficien of he BDSD a follow. Le f : Ω 0, T R k R k d R k g : Ω 0, T R k R k d R k m be joinly meauable and uch ha fo any y, z) R k R k d, f., y, z) M 2 0, T ; R k ), g., y, z) M 2 0, T ; R k m ) and: PP94.1 Thee exi conan C > 0 and α 0, 1) uch ha fo any ω, ) 31
3. Review of Lieaue Ω 0, T and y 1, z 1 ), y 2, z 2 ) R k R k m he following inequaliie hold: f, y 1, z 1 ) f, y 2, z 2 ) 2 C y 1 y 2 2 + z 1 z 2 2 ) g, y 1, z 1 ) g, y 2, z 2 ) 2 C y 1 y 2 2 + α z 1 z 2 2. Given ξ L 2 Ω, F T, P ); R k ), hey conide he BDSD Y = ξ + f, Y, Z )d + g, Y, Z ) db Z dw 3.8) fo 0, T whee db denoe he backwad Iô inegal wih epec o B and pove he following exience and uniquene eul. Theoem. Suppoe ha condiion PP94.1 hold. unique oluion Y, Z) S 2 0, T ; R k ) M 2 0, T ; R k m ). To make he connecion o SPD, hey poceed a follow. Then he BDSD 3.8) ha a Le funcion b C 3 b Rd, R d ) and σ C 3 b Rd, R d d ) and fo each 0, T and x R d denoe by {X ; 0, T } he unique ong oluion of he SD X = x, dx = bx )d + σx )dw,, T. The funcion f and g now ake he le geneal fom whee f, y, z) := f, X, y, z), g, y, z) := g, X, y, z) f : 0, T R d R k R k d R k, g : 0, T R d R k R k d R k m and hey inoduce he funcion h : R d R k o give he BDSD Y = hx T ) + f, X, Y, Z )d 32
+ 3. Review of Lieaue g, X, Y, Z ) db Z dw. 3.9) Fuhemoe, hey aume ha fo any 0, T, x, y, z) fx, y, z), gx, y, z)) i of cla C 3, all deivaive ae bounded on 0, T R d R k R k d and h i of cla C 2. They fi deive he following epeenaion of Z. Theoem. The andom field {Z ; 0 T, x R d } ha an a.. coninuou veion which i given by Z = Y X ) 1 σx ). Remak. Ju a he equivalen epeenaion of Z deived in 35 fo he BSD cae i ued in 49 o deive he egulaiy eul fo Z, we ue hi elaion o deive ou eul on he egulaiy of Z in he BDSD cae. Finally, in he following wo heoem hey elae he BDSD 3.9) o he following yem of quailinea backwad SPD: u, x) = hx) + + Lu, x) + f, x, u, x), u, x)σx)) d g, x, u, x), u, x)σx)) db 3.10) whee u : Ω R + R d R k, Lu = Lu 1,..., Lu k ) T and Lu = d i=1 b i x) x i + 1 2 d 2 a ij x), a ij x)) = σσ T x). x i x j i, Theoem. Suppoe ha he above condiion hold and le {u, x); 0, T, x R d } be a andom field uch ha u, x) i F,T B -meauable fo each, x), u C0,2 0, T R d ; R k ) a., and u aifie equaion 3.10). Then u, x) = Y, whee {Y, Z ); 0 T, x R d } i he unique oluion of he BDSD 3.9). Theoem. Suppoe ha he above condiion hold. Then {u, x) := Y ; 0 T, x R d } i he unique claical oluion of he yem of backwad SPD 3.10). In 5 2001) Bally and Maoui anlae he looely coupled BDSD of 36 o a weak fomulaion eing. The benefi of hi appoach i ha allow ignifican weakening on he condiion on he coefficien of he BDSDS. To ave epeiion, howeve, we do no povide an oveview of hi pape a he eul of 5 ae exended 33
3. Review of Lieaue wih ignifican ovelap by he pape 50 which we cove nex. We do noe, howeve, ha he key ool ha he weak fomulaion eing enable and which make he weakening of he condiion on he coefficien poible i an equivalence of nom eul fo he fowad diffuion X Reul A.3 on page 148). I hi equivalence of nom eul o one vey imila) ha play a key ole in 5, 50 and hi hei. We alo noe ha wa he pape 5 ha fi conideed weak oluion of BDSD. Key Refeence: 50 - Zhang and Zhao, 2007 In 50, Zhang and Zhao exend he weak oluion fomulaion of 5 o he infinie dimenional noie and infinie hoizon cae. Indeed, i hown in 50 ha he oluion of an infinie hoizon BDSD aniial ime coepond o he aionay oluion of he coeponding SPD. To decibe he eul of 50 we poceed a follow. On a pobabiliy pace Ω, F, P ) le {W ; 0} and {B ; 0} be muually independen ochaic pocee wih W a andad Bownian moion valued in R d and B a Q-Wiene poce valued on a epaable Hilbe pace U wih counable bae {e j } wih Qe j = λ j e j and λ j <. B ha he expanion ee 12) B = λj β j )e j whee β j, j = 1, 2,... ae muually independen eal-valued Bownian moion on Ω, F, P ). Le N denoe he P -null e of F and le u fix T > 0. Define F,T := F W F B,T, 0, T ; F := F W F B,, 0. Hee fo any poce {φ ; 0}, F φ, := σ{φ φ ;, } N, F φ := F φ 0, and F φ, := T 0 F φ,t. Le he weigh funcion ρ : Rd R be defined by ρx) := K ρ e υ x fo conan υ < 0 and K ρ > 0 uch ha R d ρx)dx = 1. oluion of he SD X = x + bx )d + σx )dw Fo, le X be he 34
3. Review of Lieaue whee b C 2 b Rd ; R d ) and σ C 3 b Rd ; R d R d ). Zhang and Zhao conide he following BDSD wih infinie-dimenional noie fo 0, T : Y = hx T ) + g, X f, X, Y, Z )d 3.11), Y, Z ) db Z, dw whee db denoe he backwad ochaic inegal wih epec o B. Hee h : Ω R d R, f : 0, T R d R R d R and g : 0, T R d R R d L 2 U 0 R) whee U 0 = Q 1/2 U). Seing g j := g λ j e j : 0, T R d R R d R hen equaion 3.11) i equivalen o Y = hx T ) + f, X g j, X, Y, Z )d, Y, Z ) db j Z, dw. They pove weak exience and uniquene fo he following aumpion and definiion: ZZ07.1 h i F B T, BRd )/BR) meauable and hx) 2 ρx)dx <. R d ZZ07.2 Funcion f and g ae Boel-meauable and hee exi conan C, C j, α j 0 wih C j < and α j < 1 uch ha fo any 0, T, and x R d f, x, y 1, z 1 ) f, x, y 2, z 2 ) 2 C y 1 y 2 2 + z 1 z 2 2 ) g j, x, y 1, z 1 ) g j, x, y 2, z 2 ) 2 C j y 1 y 2 2 + α j z 1 z 2 2. and ZZ07.3 0 R d f, x, 0, 0) 2 ρx)dxd < 35
3. Review of Lieaue and 0 g, x, 0, 0) 2 L 2 R d U R) ρx)dxd <. 0 Definiion. Le S be a Hilbe pace wih nom S and Boel σ-field S. Fo K R +, denoe by M 2,0, T ; S) he e of B, T ) F/S meauable andom pocee {φ);, T } wih value on S aifying: 1. φ) : Ω S i F,T F B T, meauable fo, T. 2. φ) 2 S d <. Alo denoe by S 2,0, T ; S) he e of B, T ) F/S meauable andom pocee {ψ);, T } wih value on S aifying: 1. ψ) : Ω S i F,T F B T, a.. meauable fo, T and ψ, ω) i coninuou 2. up,t ψ) 2 S <. Definiion. A pai of pocee Y,, Z, ) S 2,0 0, T ; L 2 ρr d ; R)) M 2,0 0, T ; L 2 ρr d ; R d )) i called a oluion of equaion 3.11) if fo any φ C 0 c R d ; R), R d Y φx)dx = hx T R d )φx)dx + g j, X R d R d Z φx)dx, dw f, X R d, Y, Z )φx)dx, Y, Z )φx)dxd Theoem 3.1. Unde condiion ZZ07.1)-ZZ07.4), equaion 3.11) ha a unique oluion. They hen conide he following SPD and connec weak oluion defined a.. db j 36
3. Review of Lieaue below) o he peviouly defined weak) oluion o BDSD 3.11). u, x) = hx) + + { Lu, x) + f, x, u, x), σ T x) u, x)) } d g, x, u, x), σ T x) u, x)) db 3.12) whee u : Ω R + R d R, Lu = Lu 1,..., Lu k ) T wih Lu = d i=1 b i x) x i + 1 2 d 2 a ij x), a ij x)) = σσ T x). x i x j i, Definiion. A poce u i called a weak oluion oluion in L 2 ρr d ; R)) of equaion 3.12) if u, σ T u) M 2,0 0, T ; L 2 ρr d ; R)) M 2,0 0, T ; L 2 ρr d ; R d )) whee σ T u), x) i inepeed a σ T x) u, x)) and fo an abiay ψ C 1, c 0, T R d ; R), u, x) ψ, x)dxd + u, x)ψ, x)dx hx)ψt, x)dx R d R d R d 1 σ T x) u, x))σ T x) ψ, x))dxd 2 R d T u, x)divb Ã)ψ), x)dxd R d T = f, x, u, x), σ T x) u, x))ψ, x)dxd R d T g j, x, u, x), σ T x) u, x))ψ, x)dx db j a.. R d Hee à j := 1 2 d i=1 a ij x) x i and à = Ã1,..., Ãd) T. Theoem 3.2. Aume condiion ZZ07.1)-ZZ07.4) hold and define u, x) = Y whee Y, Z ) i he oluion of equaion 3.11). Then u, x) i he unique weak oluion of 3.12). They hen, fo ome K > 0, conide he following BDSD wih infinie-dimenional, 37