On Zero-Sum Stochastic Differential Games

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1 O Zeo-Sum Sochac Dffeeal Game Eha Bayaka, Sog Yao Abac We geealze he eul of Flemg ad Sougad 13 o zeo-um ochac dffeeal game o he cae whe he cool ae ubouded. We do h by povg a dyamc pogammg pcple ug a coveg agume ead of elyg o a dcee appoxmao whch ued alog wh a compao pcple 13. Alo, coa wh 13, we defe ou pay-off hough a doubly efleced backwad ochac dffeeal equao. he value fuco he degeeae cae of a gle coolle cloely elaed o he ecod ode doubly efleced BSDE. Keywod: Zeo-um ochac dffeeal game, Ello-Kalo aege, dyamc pogammg pcple, ably ude pag, doubly efleced backwad ochac dffeeal equao, vcoy oluo, obacle poblem fo fully o-lea PDE, hfed pocee, hfed SDE, ecod-ode doubly efleced backwad ochac dffeeal equao. Coe 1 Ioduco 1.1 Noao ad Pelmae Doubly Refleced Backwad Sochac Dffeeal Equao Doubly Refleced Zeo-Sum Sochac Dffeeal Game wh Squae-Iegable Cool 6 3 A Obacle Poblem fo Fully o-lea PDE 1 4 Shfed Pocee Cocaeao of Sample Pah Meauably of Shfed Pocee Iegably of Shfed Pocee Shfed Sochac Dffeeal Equao Pag of Cool ad Saege Opmzao Poblem wh Squae-Iegable Cool Geeal Reul Coeco o Secod-Ode Doubly Refleced BSDE Poof Poof of Seco 1 & Poof of Dyamc Pogammg Pcple Poof of Seco Poof of Seco Poof of Seco Depame of Mahemac, Uvey of Mchga, A Abo, MI 48109; emal: eha@umch.edu. E. Bayaka uppoed pa by he Naoal Scece Foudao ude appled mahemac eeach ga ad a Caee ga, DMS , DMS , ad DMS , epecvely, ad pa by he Sua M. Smh Pofeohp. Depame of Mahemac, Uvey of Pbugh, Pbugh, PA 1560; emal: ogyao@p.edu.

2 O Zeo-Sum Sochac Dffeeal Game 1 Ioduco I h pape we ue doubly efleced backwad ochac dffeeal equao DRBSDE o geeae payoff fo a zeo-um ochac dffeeal game oduced by he emal wok of Flemg ad Sougad 13. I ou eg, he wo playe compee by choog quae-egable cool. Sce Hamadèe ad Lepele 16, 17, BSDE ad doubly efleced BSDE have bee bough o he aaly of zeo-um ochac dffeeal game o olve vaou poblem uch a deemg addle po ee e.g. 16, 17, 30, k-eve cool poblem ee e.g. 11, ad mxed cool ad oppg poblem ee e.g. 18, 19, 14. All hee wok followed 13 aumg ha cool pace compac. Amog hem, Buckdah ad L 4, 6, 5 ad Hamadèe e al. 1 eemble 13 he mo by codeg coolled dffuo coeffce ad Ello-Kalo aege. Howeve, he aaly dffee ha 13 ad ou ha hey wok wh a ufom caocal pace Ω ω C0, ; R d : ω0 0 } egadle of he ag me of he game. Fo he developme of ochac dffeeal game ohe aea, ee he efeece e.g. 4. Cvać ad Kaaza 8 aed he eeach of DRBSDE ad howed ha he oluo of a DRBSDE he value of a cea Dyk game, a zeo-um ochac game of opmal oppg ee he efeece he oduco hee fo hocal ude o Dyk game. he Hamadèe ad Lepele 18 ad Hamadèe e al. 19 added cool o DRBSDE o udy mxed cool ad oppg game wh applcao valuao of Iael opo o Ameca game opo. Oe ecoue emedou echcal dffcule whe he compace aumpo of he cool pace emoved, ce he appoxmao ool of 13 alo ee Flemg ad Heádez Heádez 1 o applcable ay loge. hee ae ome excepo o h ule: Squae-egable cool wa codeed by Bowe 3 fo a pecfc zeo-um veme game bewee wo mall veo whoe cool ae fom of he pofolo. he PDE h pecal cae have mooh oluo, heefoe he poblem ca be olved by elyg o a vefcao heoem ead of he dyamc pogammg pcple. I a moe geeal eg, Chape 6 of Kylov 4 codeed quae-egable cool. Howeve, he aaly wa doe oly fo coopeave game.e. he o called up up cae. I alo woh meog ha ped by ug-of-wa a dcee-me adom u game, ee e.g. 31 ad 5, Aa ad Budhaja 1 uded a zeo-um ochac dffeeal game wh U V x R : x 1} 0, played ul he ae poce ex a gve doma. he auho howed ha he value of uch a game he uque vcoy oluo o he homogeou fy Laplace equao. A Chape 6 of 4, hey deped o a appoxmag he game wh ubouded cool wh a equece of game wh bouded cool. hey pove a dyamc pogammg pcple fo he lae cae ad pove he equcouy of he appoxmag equece o coclude ha he value fuco a vcoy oluo o he fy Laplace equao. Iead of elyg o appoxmao agume, we decly pove a dyamc pogammg pcple fo he game wh quae-egable cool. I h pape, he cool of epecve playe ake value wo epaable mec pace U ad V. We follow he pobablc eg of 13 ad ely o he exece of he egula codoal pobably dbuo. Whe he game a fom me 0,, we code he caocal pace Ω ω C, ; R d : ω 0 }, whoe coodao poce B a Bowa moo ude he Wee meaue P0. Deoe by U ep. V he e of all U valued ep. V valued quae-egable poce. If playe I chooe a µ U ad playe II elec a ν V a cool, he ae poce X,x,µ,ν ag fom x wll evolve accodg o he followg SDE: X x + b, X, µ, ν d + σ, X, µ, ν db,,, 1.1 whee he df b ad he dffuo σ ae Lpchz x ad have lea gowh µ, ν. Ad he payoff playe I wll eceve fom playe II deemed by he f compoe of he oluo Y,x,µ,ν, Z,x,µ,ν, K,x,µ,ν, K,x,µ,ν o he followg DRBSDE: Y h X,x,µ,ν + f, X,x,µ,ν l, X,x,µ,ν Y l, X,x,µ,ν,,, Y l, X,x,µ,ν dk l, X,x,µ,ν Y dk 0,, Y, Z, µ, ν d+k K K K Z db,,, 1.

3 1. Ioduco 3 wh wo epaae obacle fuco l < l afyg l, h l,. Whe l, h, l, f ae all /q Hölde couou x fo ome q 1,, Y,x,µ,ν q egable by El A e al. 10. A we ee fom 1.1 ad 1. ha he cool µ ad ν fluece he game wo apec: ehe affec 1. va he ae poce X,x,µ,ν o appea decly he geeao f of 1. a paamee. We ue Ello-Kalo aege a 13. I h game, oe playe e.g. Playe I ha poy ad chooe a cool ad oppoe e.g. Playe II wll eac by elecg a coepodg aegy. We pecfy Playe II aegy by a meauable mappg β :, Ω U V f he game a fom me. h addoal pecfcao o accommodae a pacula meauably ue, ee Remak.. Ude a lea gowh codo o he µ vaable, β duce a mappg β : U V by β µ ω β, ω, µ ω, µ U,, ω, Ω, whch exacly a Ello-Kalo aegy. he w 1, x f β B up µ U Y,x,µ,β µ epee Playe I poy value of he game ag fom me ad ae x, whee B collec all admble aege fo Playe II. Playe II poy value w, x defed mlaly. Alhough value fuco w 1, x, w, x ae ll /q Hölde couou x, hey ae o loge 1/q Hölde couou a he cae of compac cool pace. Hece we ae o able o ue he appoach of 13 o how he dyamc pogammg pcple fo w 1 ad w, ee Remak.. Iead, we ue he couy of Y,x,µ,ν cool µ, ν, popee of hfed pocee epecally hfed SDE a well a ably ude pag of cool/aege a led below o pove a dyamc pogammg pcple, ay fo w 1 : w 1, x f β B up Y,x,µ,β µ µ U τ µ,β, w 1 τµ,β, X,x,µ,β µ τ µ,β 1.3 fo ay famly τ µ,β : µ U, β B } of Q valued oppg me. he cucal gede of he poof of dyamc pogammg pcple 1.3 ae: Whe b, σ ae λ Hölde couou µ o ν ad f λ Hölde couou µ o ν fo ome λ 0, 1, applyg a a po emae 1.7 fo DRBSDE, we oba a couou depedece eul of X,x,µ,ν ad Y,x,µ,ν o cool µ o ν, ee Lemma.1 ad Lemma.. h depedece ogehe wh wo ce opologcal popee of he caocal pace Ω, amely epaably ad Lemma 6.3 ae cucal he coveg agume whch ued o couc ε opmal aege ag a ay oppg me. Le 0. Fo ay adom vaable ξ o Ω we defe a hfed adom vaable ξ,ω ω ξω ω, ω Ω a a pojeco of ξ oo Ω alog a gve pah ω of Ω, whee ω ω he cocaeao of pah ω ad ω a me, ee 4.1. I dcee-me fe-ae couepa a eco of a bomal/omal ee of ae pce o oe of bache. Smlaly, oe ca oduce hfed pocee ad hfed adom feld, pacula, hfed cool ad hfed aege. Soe e al. 37, 40 a well a ou geealzao Subeco 4. & 4.3 how ha hee hfed adom objec almo uely he meauably ad egably. I Popoo 4.7, we exed a eul of 13 o hfed fowad SDE: Fo P0 a.. ω Ω, he hfed poce of X,x,µ,ν olve 1.1 wh paamee, X,x,µ,ν ω, µ,ω, ν,ω o he pobably pace Ω, F, P 0. Smlaly, he hfed pocee of Y,x,µ,ν, Z,x,µ,ν, K,x,µ,ν, K,x,µ,ν olve 1. wh he paamee, X,x,µ,ν ω, µ,ω, ν,ω o Ω, F, P 0, ee Popoo 4.8. hee wo popoo ae alo cucal demoag 1.3. I coucg he ε opmal aege above, we ue pag of cool ad aege. Ou e of cool ad aege ae cloed ude pag, ee Popoo 4.9 & I he lae poof we how ha a addoal pah-couy o aege fo povg he dyamc pogammg pcple alo cloed ude pag. Ug he dyamc pogammg pcple, he couy of Y,x,µ,ν cool µ, ν a well a he epaably of U, V we ca deduce ha he value fuco w 1 ad w ae dcouou vcoy oluo of he coepodg obacle poblem of fully o-lea PDE, ee heoem 3.1. Whe V become a gleo, he zeo-um ochac dffeeal game degeeae o a clacal ochac cool poblem cludg oly oe playe. I pacula, whe U all ymmec d dmeoal mace}, b, x, u b, x ad σ, x, u u, he value fuco w of he opmzao poblem cocde wh ha of

4 O Zeo-Sum Sochac Dffeeal Game 4 he ecod-ode doubly efleced BSDE ad elaed o he oe Nuz 7 va a pobably afomao of og fom 5.1. Movaed by applcao facal mahemac ad pobablc umecal mehod, Chedo e al. 7 oduced ecod-ode BSDE. Lae, Soe e al. 40 efed h oo ad Soe e al. 38 elaed o G expecao of Peg 9, 8. Amog he ece developme of ecod-ode BSDE ee e.g. 3, 6, 33, Maou e al. 6 uded he efleced veo, whch ca be exeded o he doubly-efleced cae a uual. he e of he pape ogazed a follow: Afe lg he oao ued, we wll pee wo bac popee of DRBSDE Seco 1. I Seco, we e up he zeo-um ochac dffeeal game baed o DRBSDE ad pee a dyamc pogammg pcple, heoem.1, fo poy value of boh playe defed va Ello- Kalo aege. Ug he dyamc pogammg pcple, we how Seco 3 ha he poy value ae dcouou vcoy oluo of he coepodg obacle poblem of fully o-lea PDE, ee heoem 3.1. I Seco 4, we exploe he popee of hfed pocee cludg he meauably/egably, hfed SDE ad pag of cool/aege. he coe of h eco ae all echcal ecee povg ou ma eul, heoem.1 ad 3.1. I Seco 5, we wll dcu he clacal ochac cool poblem a a degeeae cae ad coec o ecod ode doubly efleced BSDE. he poof of ou eul ae gve Seco Noao ad Pelmae We le M be a geec mec pace wh mec ρ M ad deoe by BM he Boel σ feld o M. Fo ay x M ad δ > 0, O δ x x M : ρ M x, x < δ} deoe he ope ball ceeed a x wh adu δ ad cloue O δ x x M : ρ M x, x δ}. Fo ay fuco φ : M R, we defe lm φx lm x x f x O 1/ x φx ad lm x x φx lm up φx, x M. x O 1/ x Alo, we le E deoe a geec Euclda pace. Fx d N. Fo ay 0 <, we e Q,, Q } ad le Ω, ω C, ; R d : ω0 } be he caocal pace ove he peod,, whch equpped wh he ufom om ω, up ω., We le O δ ω ω Ω, : ω ω, < δ} deoe he ope ball ceeed a ω Ω, wh adu δ > 0, ad le BΩ, be he coepodgly Boel σ feld of Ω,. We deoe by B, he caocal poce o Ω,, ad by P, 0 he Wee meaue o Ω,, BΩ, ude whch B, a d dmeoal Bowa moo. Le F, F, σ B, ;, } be he flao geeaed by B, ad le C, collec all cylde e, } 1E : m N, < 1 < < m, E } m 1 BRd. I well-kow ha B BΩ, σc,, 1E σ :,, E BR } d F,. 1.4 F,,.e. C, m 1 B, Fo ay F, oppg me τ, we defe wo ochac eval, τ, ω, Ω : < τω }, τ,, ω, Ω : τω } ad e τ, A, ω, A : τω } fo ay A F, τ. he followg wo eul ae bac, ee fo poof. Lemma 1.1. Le 0 <, fo ay,, he σ feld F, C, couably geeaed by m B, 1 Oλ 1 x : m N, Q wh 1 < < m, x Q d, λ Q + }. Lemma 1.. Le 0 S <. he ucao mappg Π,S, Π,S couou ude ufom om ad F,, ω ω ω, ω Ω,,, S P, 0 Π,S, : Ω, Ω,S defed by / F,S meauable fo ay, S. Moeove, we have 1A P,S 0 A, A F,S S.

5 1. Doubly Refleced Backwad Sochac Dffeeal Equao 5 Fom ow o, we fx a me hozo 0, ad hall dop fom he above oao,.e. Ω,,,, B,, F,, P, 0, C, Ω,, B, F, P0, C. he expecao ude P 0 wll be deoed by E. Whe S we mply deoe Π,, by Π, Lemma 1.. Gve 0,, we le P deoe he e of all pobably meaue o Ω, BΩ Ω, F by 1.4. Fo ay P P, we e N P N Ω : N A fo ome A F wh P A 0} a he colleco of all P ull e. he P augmeao F P of F co of F P σ F N P,,. I pacula, we wll we F } F }, fo FP 0 F P 0. he compleo of Ω, F, P he pobably pace Ω, F P, P, wh P P. Fo coveece, we wll mply we P fo P. F Smla o Lemma.4 of 39, we have he followg eul: Lemma 1.3. Le 0, ad P P. 1 Fo ay ξ L 1 F P, P ad,, E P ξ F P EP ξ F, P a.. Coequely, a magale ep. local magale o em-magale wh epec o F, P alo a magale ep. local magale o em-magale wh epec o F P, P. Fo ay E valued, F P adaped couou poce X },, hee ex a uque ee of P evaecece E valued, F adaped couou poce X }, uch ha P X X,, 1. Fo ay E valued, F P pogevely meauable poce X },, hee ex a uque d dp a.. ee E valued, F pogevely meauable poce X }, uch ha X ω X ω fo d dp a.., ω, Ω. I boh cae, we call X he F veo of X. Fo ay p 1,, 0, ad P P, we oduce ome pace of fuco: 1 Fo ay ub σ feld F of F P, le Lp F, E, P be he pace of all E valued, F meauable adom vaable ξ uch ha ξ L p F,P E } 1/p P ξ p <. Fo ay flao F F }, o Ω, F P, PF wll deoe he F pogevely meauable σ feld of, Ω. Le C 0 F,, E, P be he pace of all E valued, F adaped pocee X }, wh P a.. couou pah. We defe he followg ubpace of C 0 F,, E, P : C p F,, E, P X C 0F,, E, P : X C pf, E,P P up, } C ±,p F,, P X C 0 F,, R, P : X± ±X 0 C pf,, P ; V F,, P X C 0 F,, R, P : X ha P a.. fe vaao} ; } 1/p } X p < ; K F,, P X C 0 F,, R, P : X 0 ad X ha P a.. ceag pah } ; K p F,, P X K F,, P : E P X p } <. 3 Le H p,loc F,, E, P be he pace of all E valued, F pogevely meauable pocee X }, wh X p d <, P0 a.. Ad fo ay p 1,, we le H p, p F,, E, P deoe he pace of all E valued, F pogevely meauable pocee X }, wh X H p, p F,,E,P E P X p d p/p } 1/ p <. Alo, we e G q F,, P C q F,, R, P H,q F,, Rd, P K q F,, P Kq F,, P. If E R ep. P P 0, we wll dop fom he above oao. Moeove, we ue he coveo f. 1. Doubly Refleced Backwad Sochac Dffeeal Equao Le 0,. A paamee e ξ, f, L, L co of a adom vaable ξ L 0 F, a fuco f :, Ω R R d R, ad wo pocee L, L C 0, uch ha f P F BR BR d /BR meauable ad F ha L ξ L, P 0 a.. I pacula, ξ, f, L, L called a, q paamee e f ξ L q F ad L C,q F,., L C +,q F,

6 O Zeo-Sum Sochac Dffeeal Game 6 Defo 1.1. Gve 0, ad a paamee e ξ, f, L, L, a quaduple Y, Z, K, K C 0 F, H,loc,, R d K F F, K F, called a oluo of he doubly efleced backwad ochac dffeeal equao o he pobably pace Ω, F, P0 wh emal codo ξ, geeao f, lowe obacle L ad uppe obacle L DRBSDE P0, ξ, f, L, L fo ho f hold P0 a.. ha Y ξ+ f, Y, Z d+k K K K Z db,,, L Y L,, ad Y L dk L Y dk 0. he la wo equale 1.5 ae kow a he fla-off codo coepodg o L ad L epecvely, ude whch he wo ceag pocee K, K keep poce Y bewee L ad L a he mmal effo:.e., Oly whe Y ed o dop below L ep. e above L, K ep. K wll geeae a upwad ep. dowwad momeum. We f have he followg compao eul ad a po emae fo DRBSDE, whch geealze hoe 5 ad 15. Popoo 1.1. Gve 0, ad wo, q paamee e ξ 1, f 1, L 1, L 1, ξ, f, L, L wh P 0 ξ 1 ξ P0 L 1 L, L 1 L,, 1, le Y, Z, K, K C q, H,q,, R d K F F F, K F,, 1, be a oluo of DRBSDE P0, ξ, f, L, L. Fo ehe 1 o, f f Lpchz couou y, z:.e. fo ome γ > 0, hold fo d dp0 a.., ω, Ω ha f, ω, y, z f, ω, y, z γ y y + z z, y, y R, z, z R d, 1.6 ad f f 1, Y 3, Z 3 f, Y 3, Z 3, d dp0 a.., he hold P0 a.. ha Y 1 Y fo ay,. Popoo 1.. Gve 0, ad wo, q paamee e ξ 1, f 1, L 1, L 1, ξ, f, L, L wh P 0 L 1 L, L1 L,, 1, le Y, Z, K, K C q, H,q,, R d K F F F, K F,, 1, be a oluo of DRBSDE P0, ξ, f, L, L. If f1 afe 1.6, he fo ay ϖ 1, q E up Y 1 Y C, ϖ ϖ ϖ, γ E ξ1 ξ ϖ +E f 1, Y, Z f, Y, Z d }. 1.7, Gve a, q paamee e ξ, f, L, L uch ha f Lpchz couou y, z. If E f, 0, 0 q d < ad f P0L < L,, 1, he we kow fom heoem 4.1 of 10 ha he DRBSDE P0, ξ, f, L, L adm a uque oluo Y, Z, K, K G q,. F 1.5 Doubly Refleced Zeo-Sum Sochac Dffeeal Game wh Squae-Iegable Cool Le U, ρ U ad V, ρ V be wo epaable mec pace, whoe Boel σ feld wll be deoed by BU ad BV epecvely. Fo ome u 0 U ad v 0 V, we defe u U ρu u, u 0, u U ad v V ρv v, v 0, v V. Fx a o-empy U 0 U ad a o-empy V 0 V. We hall udy a zeo-um ochac dffeeal game bewee wo playe, playe I ad playe II, who chooe quae-egable U valued cool ad V valued cool epecvely o compee: Defo.1. Gve 0,, a admble cool poce µ µ }, fo playe I ove peod, a U valued, F pogevely meauable poce uch ha µ U 0, d dp0 a.. ad ha E µ d <. U Admble cool pocee fo playe II ove peod, ae defed mlaly. he e of all admble cool fo playe I ep. II ove peod, deoed by U ep. V.

7 . Doubly Refleced Zeo-Sum Sochac Dffeeal Game wh Squae-Iegable Cool 7 Ou zeo-um ochac dffeeal game fomulaed va a decoupled SDE DRBSDE yem wh he followg paamee: Fx k N, γ > 0 ad q 1,. 1 Le b : 0, R k U V R k be a B0, BR k BU BV/BR k meauable fuco ad le σ : 0, R k U V R k d be a B0, BR k BU BV/BR k d meauable fuco uch ha fo ay, u, v 0, U V ad ay x, x R k b, 0, u, v + σ, 0, u, v γ 1 + u U + v V ad b, x, u, v b, x, u, v + σ, x, u, v σ, x, u, v γ x x ;. Le l, l : 0, R k R be wo jo couou fuco uch ha l, x<l, x,, x 0, R k ad ha l, x l, x l, x l, x γ x x /q, 0,, x, x R k ;.3 3 Le h: R k R be a /q Hölde couou fuco wh coeffce γ uch ha l, x hx l, x, x R k ; 4 Le f : 0, R k R R d U V R be B0, BR k BR BR d BU BV/BR meauable fuco uch ha fo ay, u, v 0, U V ad ay x, y, z, x, y, z R k R R d f, 0, 0, 0, u, v γ 1 + u /q U + v/q V ad f, x, y, z, u, v f, x, y, z, u, v γ x x /q + y y + z z..5 Fo ay λ 0, we le c λ deoe a geec coa, depedg o λ,, q, γ, l up l, 0 ad l, l, 0, whoe fom may vay fom le o le. I pacula, c0 ad fo a geec coa depedg o up,, q, γ, l ad l. Fx 0,. Aume ha whe playe I ep. II elec admble cool µ U ep. ν V, he coepodg ae poce ag fom me a po x R k dve by he SDE 1.1 o he pobably pace Ω, F, P0. Clealy, he meauably of b ad σ mple ha fo ay x R k, boh b, x, µ, ν }, ad σ, x, µ, ν } ae, F pogevely meauable pocee. Alo, we ee fom. ha fo ay, ω, Ω, boh b,, µ ω, ν ω ad σ,, µ ω, ν ω ae Lpchz wh coeffce γ. Sce E b, 0, µ, ν + σ, 0, µ, ν d c 0 + c 0 E µ + ν U V d <.6 by.1, well-kow ee e.g. heoem.5.7 of 4 ha 1.1 adm a uque oluo X,x,µ,ν }, C F,, R k. Applyg heoem.5.9 of 4 wh ξ, ξ, b 0, σ 0 x, 0, 0, 0 hu x 0 hee ad ug.1 yeld E up, up, X,x,µ,ν c x + E.1.4 µ + ν U V d <..7 Lemma.1. Le ϖ 1,, 0, ad x, µ, ν R k U V. 1 Fo ay,, E up X,x,µ,ν x c 0 1+ x +c 0 E µ +ν U V d..8, Fo ay x R k, E X,x,µ,ν X,x,µ,ν ϖ c ϖ x x ϖ..9 3 If b ad σ ae λ Hölde couou u fo ome λ 0, 1,.e., fo ay, x, v 0, R k V ad u 1, u U b, x, u1, v b, x, u, v + σ, x, u1, v σ, x, u, v γ ρ λ u1, u U,.10

8 O Zeo-Sum Sochac Dffeeal Game 8 he fo ay µ U E up X,x,µ,ν X,x,µ,ν, ϖ c ϖ E ϖ/ ρ λ µ U, µ d..11 Smlaly, f b ad σ ae λ Hölde couou u fo ome λ 0, 1,.e., fo ay, x, u 0, R k U ad v 1, v V b, x, u, v1 b, x, u, v + σ, x, u, v1 σ, x, u, v γ ρ λ v1, v V,.1 he fo ay ν V E up X,x,µ,ν X,x,µ,ν, ϖ c ϖ E ϖ/ ρ λ ν V, ν d..13 By Lemma 1.3, X,x,µ,ν adm a uque F veo X,x,µ,ν, whch clealy belog o C F,, R k ad alo afe.7,.8,.9,.11 ad.13. If µ, ν aohe pa of U V uch ha µ, ν µ, ν d dp0 a.. o, τ fo ome F oppg me τ, he boh Xτ,x,µ,ν }, ad X,x, µ, ν } τ afy he ame SDE: X x +, b τ, X d + σ τ, X db,,,.14 wh b τ, ω, x 1 <τω} b, x, µ ω, ν ω ad σ τ, ω, x 1 <τω} σ, x, µ ω, ν ω,, ω, x, Ω R k. Clealy, fo ay x R k boh b τ,, x ad σ τ,, x ae F pogevely meauable pocee, ad fo ay, ω, Ω boh b τ, ω, ad σ τ, ω, ae Lpchz couou wh coeffce γ. hu.14 ha a uque oluo C F,, R k,.e. P0 X,x,µ,ν τ Xτ,x, µ, ν,, Le Θ ad fo he quaduple, x, µ, ν. By he couy of l ad l, L Θ l, X Θ ad L Θ l, XΘ,, ae wo eal valued, F adaped couou pocee uch ha L Θ < L Θ,,. Gve a F oppg me τ, he meauably of f, X Θ, µ, ν ad.5 mply ha fτ Θ, ω, y, z 1 <τω} f, X Θ ω, y, z, µ ω, ν ω,, ω, y, z, Ω R R d a P F BR BR d /BR meauable fuco ha Lpchz couou y, z wh coeffce γ. Ad oe ca deduce fom.3,.4,.5, Hölde equaly ad.7 ha E up L Θ τ q + up L Θ τ q + f τ Θ, 0, 0 q d c 0 +c 0 E XΘ + d <..16,, up, µ U +ν V hu, fo ay Fτ meauable adom vaable ξ wh L Θ τ ξ LΘ τ, P0 a.., follow ha E ξ q <,.e. ξ L q Fτ. he heoem 4.1 of 10 how ha he DRBSDE P 0, ξ, fτ Θ, L Θ τ, Lτ Θ adm a uque oluo Y Θ τ, ξ, Z Θ τ, ξ, K Θ τ, ξ, K Θ τ, ξ G q,. Clealy, F veo Ỹ Θ τ, ξ, Z Θ τ, ξ, K Θ τ, ξ, F K Θ τ, ξ by Lemma 1.3 belog o G q F,. A F, Ω }, Ỹ Θ τ, ξ a coa. Gve aohe F oppg me ζ uch ha ζ τ, P0 a.., Θ oe ca ealy how ha Ỹζ τ, ξ, 1 <ζ} ZΘ τ, ξ, K Θ ζ τ, ξ, K } ζ τ, Θ ξ G q F, olve he DRBSDE P0, Ỹ Θ ζ τ, ξ, f ζ Θ, LΘ ζ, L ζ Θ. o w, we have, Θ Ỹ ζ, Ỹζ Θ τ, ξ, Z Θ ζ, Ỹζ Θ τ, ξ, K Θ ζ, Ỹζ Θ τ, ξ, K Θ ζ, Ỹ Θ Ỹ Θ ζ τ, ξ, 1 <ζ} ZΘ τ, ξ, K Θ ζ τ, ξ, K Θ ζ τ, ξ ζ τ, ξ,,..17

9 . Doubly Refleced Zeo-Sum Sochac Dffeeal Game wh Squae-Iegable Cool 9 he couy of fuco h mple ha h XΘ a eal valued, F meauable adom vaable uch ha L Θ l, X Θ Θ h XΘ l, XΘ L. Hece, we ca ue 1.5 o oba ha l, x l, X Θ Y Θ, h XΘ l, XΘ l, x, P 0 a.., whch lead o ha Y,x,µ,ν l, x Ỹ Θ, h XΘ l, x..18 Iped by Popoo 6.1 of 5, we have he followg a po emae abou he depedece of, h X,x,µ,ν o al ae x ad o cool µ, ν. Lemma.. Le ϖ 1, q, 0, ad x, µ, ν R k U V. 1 Fo ay x R k, E up, h X,x,µ,ν Y,x,µ,ν Le l, l ad h afy Y,x,µ,ν,, h X,x,µ,ν ϖ c ϖ x x ϖ q..19 l, x l, x l, x l, x hx hx γ ψ x x, 0,, x, x R k.0 fo a ceag C R + fuco ψ uch ha fo ome 0 < R 1 < 1 < R ψa 1 a f a 0, R 1, ψa a /q f a R 1, R, ad ψa a /q f a > R. Clealy, ψa a /q fo ay a 0. So.0 mple.3 ad he /q Hölde couy of h. Le λ 0, 1. If b, σ ae λ Hölde couou u ee.10 ad f f λ Hölde couou u,.e., fo ay, x, y, z, v 0, R k R R d V ad u 1, u U f, x, y, z, u1, v f, x, y, z, u, v γ ρ λ u U 1, u,.1 he fo ay µ U E up Y,x,µ,ν,, h X,x,µ,ν c ϖ κ ϖ ψ Y,x,µ,ν E, h X,x,µ,ν ϖ ρ λ U µ, µ d ϖ + E } ϖ ρ λ U µ, µ d,. whee κ ψ + R1 1 up 1 ψ a + up ψ a R q. a R 1,R a R 1,R Smlaly, f b, σ ae λ Hölde couou v ee.1 ad f f addoally λ Hölde couou v,.e., fo ay, x, y, z, u 0, R k R R d U ad v 1, v V f, x, y, z, u, v1 f, x, y, z, u, v γ ρ λ v V 1, v,.3 he fo ay ν V E up Y,x,µ,ν,, h X,x,µ,ν c ϖ κ ϖ ψ Y,x,µ,ν E, h X,x,µ,ν ϖ ρ λ V ν, ν d ϖ + E } ϖ ρ λ ν V, ν d..4 Now, we ae eady o oduce value of he zeo-um ochac dffeeal game va he followg oo of admble aege.

10 O Zeo-Sum Sochac Dffeeal Game 10 Defo.. Gve 0,, a admble aegy α fo playe I ove peod, a U valued fuco α o, Ω V ha P F BV / BU meauable ad afe: α, V 0 U 0, d dp0 a.. Fo a κ > 0 ad a o-egave meauable poce Ψ o Ω, F wh E Ψ d <, hold d dp0 a.. ha α, ω, v U Ψ ω + κv V, v V..5 Admble aege β :, Ω U V fo playe II ove peod, ae defed mlaly. he e of all admble aege fo playe I ep. II o, deoed by A ep. B. Gve 0,, a admble aegy α A duce a mappg α : V U by α ν ω α, ω, ν ω, ν V,, ω, Ω. o ee h, le ν V. Clealy, α ν a U valued, F pogevely meauable poce. Sce, ω } }, Ω : ν ω V 0, ω, Ω : α, ω, V 0 U 0, ω, Ω : α ν ω U 0}, hold d dp0 a.. ha α ν U 0. O he ohe had, oe ca deduce ha E α ν U d E Ψ d + κ E ν d <. V hu, α ν U. If ν 1 V equal o ν V, d dp 0 a.. o, τ fo ay F oppg me τ, he α ν 1 α ν, d dp 0 a.. o, τ. So α exacly a Ello Kalo aegy codeed e.g. 13. Smlaly, ay β B gve e o a mappg β : U V. Defo.3. Gve 0,, a A aegy α ad o be of  f fo ay ε > 0, hee ex a δ > 0 ad a cloed ube F of Ω wh P 0 F >1 ε uch ha fo ay ω, ω F wh ω ω < δ We defe B B mlaly. w 1, x f β B up µ U Ỹ,x,µ,β µ up, v V Fo ay, x 0, R k, we defe, h up ρ U α, ω, v, α, ω, v < ε..6 X,x,µ,β µ ad ŵ 1, x f β B up µ U Ỹ,x,µ,β µ, h X,x,µ,β µ a playe I poy value ad c poy value of he zeo-um ochac dffeeal game ha a fom me ad ae x. Coepodgly, we defe w, x up f, h X,x,α ν,ν ad ŵ, x up f Ỹ,x,α ν,ν, h X,x,α ν,ν α A ν V Ỹ,x,α ν,ν ν V α  a playe II poy value ad c poy value of he zeo-um ochac dffeeal game ha a fom me ad ae x. By.18, oe ha l, x w 1, x ŵ 1, x l, x ad l, x ŵ, x w, x l, x..7 he wo obacle fuco l, l a well a he DRBSDE ucue peve he value fuco. he value w 1, x ad ŵ 1, x, ohewe, mgh blow up ule we mpoe addoal egably codo o U ad B aalogou o e.g. Aumpo 5.7 of 40. Remak.1. Gve 0,, we ca egad µ U a a membe of A ce ν V µ U α µ, ω, v µ ω,, ω, v, Ω V clealy a P F BV BU meauable fuco uch ha α µ, V 0 µ U 0, d dp0 a.. ad ha.5 hold fo Ψ α µ U ad ay κ α > 0. Smlaly, V ca be embedded o B. he follow ha w 1, x f up Ỹ,x,µ,ν, h X,x,µ,ν ad w, x up f Ỹ,x,µ,ν, h X,x,µ,ν. Howeve, he fac ha up f µ U ν V mply ha w, x w 1, x. Ỹ,x,µ,ν, h X,x,µ,ν f up ν V µ U µ U ν V Ỹ,x,µ,ν, h X,x,µ,ν doe o ecealy

11 . Doubly Refleced Zeo-Sum Sochac Dffeeal Game wh Squae-Iegable Cool 11 Le 0, ad x 1, x R k. Fo ay β B ad µ U,.19 how ha E up Ỹ,x1,µ,β µ, h X,x 1,µ,β µ Ỹ,x,µ,β µ, h X,x,µ,β µ q c 0 x 1 x., I he follow ha Ỹ,x,µ,β µ, h X,x,µ,β µ c0 x 1 x /q Ỹ,x1,µ,β µ, h X,x 1,µ,β µ Ỹ,x,µ,β µ, h X,x,µ,β µ akg upemum ove µ U ad he akg fmum ove β B yeld ha +c0 x 1 x /q..8 w 1, x c 0 x 1 x /q w 1, x 1 w 1, x + c 0 x 1 x /q. hu w 1, x 1 w 1, x c 0 x 1 x /q, ad oe ca deduce he mla equale fo ŵ 1, w ad ŵ : Popoo.1. Fo ay 0, ad x 1, x R k, we have w1, x 1 w 1, x + ŵ1, x 1 ŵ 1, x + w, x 1 w, x + ŵ, x 1 ŵ, x c0 x 1 x /q. Howeve, hee value fuco ae geeally o 1/q Hölde couou a he cae of compac cool pace. Remak.. Whe yg o decly pove he dyamc pogammg pcple, 13 ecoueed a eou meauably poblem, ee page 99 hee. o ovecome h echcal dffculy, hey f poved ha he value fuco ae uque vcoy oluo o he aocaed Bellma-Iaac equao by a me-dcezao appoach aumg ha he lmg Iaac equao ha a compao pcple, whch ele o he followg egulay of he appoxmag value v π v π, x v π, x C 1/ + x x, x,, x 0, R k wh a ufom coeffce C > 0 fo all pao π of 0,. Sce ou value fuco ae o 1/ Hölde couou gve q, h mehod doe o wok geeal ude ou aumpo. Iead, we pecfy Ello Kalo aege a meauable adom feld fom oe cool pace o aohe ode o avod mla meauably ue whe pag aege ee Popoo h a cucal gede he poof of he upeoluo de of he dyamc pogammg pcple heoem.1. Gve 1,, ce w, couou fo ay 0,, oe ca deduce ha fo ay F oppg me τ wh couably may value } N,, ad ay R k valued, F τ meauable adom vaable ξ w τ, ξ N 1 τ}w, ξ F τ meauable..9 Smlaly, ŵ τ, ξ alo F τ meauable. he we have he followg dyamc pogammg pcple fo value fuco. heoem.1. Le, x 0, R k. Fo ay famly τ µ,β : µ U, β B } of Q, valued, F oppg me,x,µ,β µ τ µ,β, w 1 τµ,β, X w 1, x f β B ad ŵ 1, x f β B he evee equaly of.31 hold f up µ U up µ U Ỹ,x,µ,β µ Ỹ,x,µ,β µ τ µ,β.30,x,µ,β µ τ µ,β, ŵ 1 τµ,β, X τ µ,β ;.31 V λ l, l ad h afy.0; b, σ ae λ Hölde couou v ee.1 ; ad f λ Hölde couou v ee.3 fo ome λ 0, 1.

12 O Zeo-Sum Sochac Dffeeal Game 1 O he ohe had, fo ay famly τ ν,α : ν V, α A } of Q, valued, F oppg me,x,α ν,ν w, x up f τ ν,α, w τν,α, X ad ŵ, x up α A ν V f ν V α Â Ỹ,x,α ν,ν Ỹ,x,α ν,ν τ ν,α.3,x,α ν,ν τ ν,α, ŵ τν,α, X τ ν,α ;.33 he evee equaly of.33 hold f l, l ad h afy.0; b, σ ae λ Hölde couou u ee.10 ; ad U λ f λ Hölde couou u ee.1 fo ome λ 0, 1. Noe ha each Ỹ,x,µ,β µ,x,µ,β µ τ µ,β, w 1 τµ,β, X,x,µ,β µ τ µ,β.30 well-poed ce w 1 τ µ,β, X τ µ,β Fτ µ,β meauable by.9 ad ce,x,µ,β µ,x,µ,β µ,x,µ,β µ l τ µ,β, X τ µ,β w 1 τ µ,β, X τ µ,β l τ µ,β, X τ µ,β L,x,µ,β µ τ µ,β L,x,µ,β µ τ µ,β by.7. he poof of heoem.1 ee Subeco 6. ele o.,.4, popee of hfed pocee epecally hfed SDE a well a ably ude pag of cool/aege, he lae wo of whch wll be dcued Seco 4. 3 A Obacle Poblem fo Fully o-lea PDE I h eco, we how ha he c poy value ae dcouou vcoy oluo of he followg obacle poblem of a PDE wh a fully o-lea Hamloa H: m w l, x, max w, x H, x, w, x, D x w, x, Dxw, x }}, w l, x 0,, x 0, R k. 3.1 Defo 3.1. Le H : 0, R k R R k S k, be a uppe ep. lowe emcouou fuco wh S k deog he e of all R k k valued ymmec mace. A uppe ep. lowe emcouou fuco w : 0, R k R called a vcoy uboluo ep. upeoluo of 3.1 f w, x ep. hx, x R k, ad f fo ay 0, x 0, ϕ 0, R k C 1, 0, R k uch ha w 0, x 0 ϕ 0, x 0 ad ha w ϕ aa a c local maxmum ep. c local mmum a 0, x 0, we have m ϕ l 0, x 0, max ϕ 0, x 0 H 0, x 0, ϕ 0, x 0, D x ϕ 0, x 0, Dxϕ 0, x 0 }}, ϕ l 0, x 0 ep. 0. Alhough he fuco H Defo 3.1 may ake ± value, he lef-had-de of he equaly above bewee w l 0, x 0 ad w l 0, x 0 ad hu fe. Fo ay, x, y, z, Γ, u, v 0, R k R R d S k U 0 V 0, we e H, x, y, z, Γ, u, v 1 ace σσ, x, u, v Γ + z b, x, u, v + f, x, y, z σ, x, u, v, u, v ad code he followg Hamloa fuco: H 1 Ξ up u U 0 lm Ξ Ξ f HΞ, u, v, v V 0 H 1 Ξ lm up u U 0 f v O u lm U 0 u u up HΞ, u, v, Ξ O 1/ Ξ ad H Ξ f v V 0 lm Ξ Ξ up u U 0 HΞ, u, v, H Ξ lm f up v V 0 u Ov lm f U 0 u u Ξ O 1/ Ξ HΞ, u, v, whee Ξ, x, y, z, Γ, Ou } v V 0 : v V + u U ad O } v u U0 : u U + v V. Fo ay, x 0, R k, Popoo.1 mple ha w1, x lm w 1, x lm w 1, x ad w, x lm,x,x w, x lm w, x.,x,x

13 4. Shfed Pocee 13 I fac, w 1 he malle uppe emcouou fuco above w 1 alo kow a he uppe emcouou evelope of w 1, whle w he lage lowe emcouou fuco below w alo kow a he lowe emcouou evelope of w. Smlaly, fo 1,, w, x lm ŵ, x ad w, x lm, x,, x 0, R k ŵ ae he lowe ad uppe emcouou evelope of ŵ epecvely. Gve x R k, hough w, x ŵ, x hx, pobably ehe of w x, w x, w x equal o hx a he value fuco w, ŵ may o be couou. heoem If U 0 ep. V 0 a couable uo of cloed ube of U ep. V, he w 1 ad w 1 ep. w ad w ae wo vcoy uboluo ep. upeoluo of 3.1 wh he fully olea Hamloa H 1 ep. H. O he ohe had, f V λ ep. U λ hold fo ome λ 0, 1, he w 1 ep. w a vcoy upeoluo ep. uboluo of 3.1 wh he fully olea Hamloa H 1 ep. H. 4 Shfed Pocee I h eco, we fx 0 ad wll exploe popee of hf pocee fom Ω o Ω, whch ae eceay fo Seco ad Seco Cocaeao of Sample Pah We cocaeae a ω Ω ad a ω Ω a me by: ω ω ω 1,} + ω + ω 1, },,, 4.1 whch ll of Ω. Clealy, h cocaeao a aocave opeao:.e., fo ay, ad ω Ω ω ω ω ω ω ω. Gve ω Ω. we e ω ad ω à ω ω : ω Ã} fo ay o-empy à Ω. he ex eul how ha A F co of all bache ω Ω wh ω A. Lemma 4.1. Le A F. If ω A, he ω Ω A.e. A,ω Ω. Ohewe, f ω / A, he ω Ω A c.e. A,ω. Alo, fo ay A Ω we e A,ω ω Ω : ω ω A} a he pojeco of A o Ω alog ω. I pacula,,ω. Fo ay A Ã Ω ad ay colleco A } I of ube of Ω, Oe ca deduce ha A c,ω ω Ω : ω ω A c } Ω ω Ω : ω ω A} Ω A,ω A,ω c, 4. ad A,ω ω Ω : ω ω A} ω Ω : ω ω Ã} Ã,ω, 4.3,ω } A ω Ω : ω ω A ω Ω } : ω ω A A,ω. 4.4 I I I I Lemma 4.. Le ω Ω. Fo ay ope ep. cloed ube A of Ω, A,ω a ope ep. cloed ube of Ω. Moeove, gve,. we have A,ω F fo ay A F ad ω à F fo ay à F. Fo ay D, Ω, we accodgly e D,ω, ω, Ω :, ω ω D }. Smla o , fo ay D D, Ω ad ay colleco D } I of ube of, Ω, oe ha, Ω D,ω, Ω D,ω D,ω c, D,ω D,ω ad,ω D I I D,ω. 4.5

14 O Zeo-Sum Sochac Dffeeal Game Meauably of Shfed Pocee Fo ay M valued adom vaable ξ o Ω, we defe a hfed adom vaable ξ,ω o Ω by ξ,ω ω ξω ω, ω Ω. Ad fo ay M valued poce X X },, coepodg hfed poce wh epec o ad ω co of X,ω X,ω,,. I lgh of Lemma 4., hfed adom vaable ad hfed pocee he meauably he followg way: Popoo 4.1. If ξ F meauable fo ome,, he ξ,ω F meauable. Moeove, fo ay M valued, F adaped poce X },, he hfed poce } X,ω, F adaped. Popoo 4.. Gve 0,, D,ω B, 0 F 0 fo ay D B, 0 F 0. Coequely, f X }, a M valued, meauable poce o Ω, F ep. a M valued, F pogevely meauable poce, he he hfed poce } X,ω a meauable poce o Ω, F, ep. a F pogevely meauable poce. Moeove, we have D,ω P F fo ay D P F. Fo ay J, Ω M, we e J,ω, ω, x, Ω M :, ω ω, x J }. Coollay 4.1. Fo ay J P F BM, J,ω P F BM. Le M be aohe geec mec pace. If a fuco f :, Ω M M P F BM/B M meauable, he he fuco f,ω, ω, x f, ω ω, x,, ω, x, Ω M P F BM/B M meauable. Whe τω fo ome F oppg me τ, we hall mplfy he above oao by: ω τ ω ω τω ω, A τ,ω A τω, ω, D τ,ω D τω, ω, ξ τ,ω ξ τω, ω ad X τ,ω X τω, ω. he followg lemma how ha gve a F oppg me τ, a F τ meauable adom vaable oly deped o wha happe befoe τ: Lemma 4.3. Fo ay F oppg me τ ad ξ F τ, ξ τ,ω ξω. I pacula, τ ω τ Ω τω τω. 4.3 Iegably of Shfed Pocee I h ubeco, le τ be a F oppg me wh couably may value. I vue of egula codoal pobably dbuo by 41, we have he followg heace of egably fo hfed adom vaable: Popoo 4.3. Fo ay ξ L 1 F, hold fo P 0 a.. ω Ω ha ξ τ,ω L 1 F τω, P τω 0 ad E τω ξ τ,ω E ξ F τ ω R, 4.6 whee E τω ad fo E τω P. Coequely, fo ay p 1, ad ξ L p F, hold fo P 0 a.. ω Ω ha 0 ξ τ,ω L p F τω, P τω 0. Coollay 4.. Fo ay P0 ull e N, hold fo P0 a.. ω Ω ha N τ,ω a P τω 0 ull e. Coequely, fo ay wo eal-valued adom vaable ξ 1 ad ξ, f ξ 1 ξ, P0 a.., he hold fo P0 a.. ω Ω ha ξ τ,ω 1 ξ τ,ω, P τω 0 a.. Nex, le u exed Popoo 4.3 o E valued meauable pocee. Popoo 4.4. Le X }, be a E valued meauable poce o Ω, F fo ome p, p 1,. I hold fo P0 a.. ω Ω ha X τ,ω wh E τω p d p/p <. τω X τ,ω uch ha E X p p/p τ d < } τω, a meauable poce o Ω τω, F τω Coollay 4.3. Gve p, p 1,, f X }, H p, p F,, E ep. C p F,, E, he hold fo P0 a.. ω Ω, X τ,ω } τω, H p, p τω F τω,, E, P τω 0 ep. C p F τω τω τω,, E, P 0.

15 4.4 Shfed Sochac Dffeeal Equao 15 Smla o Coollay 4., a hfed d dp 0 ull e ll ha zeo poduc meaue: Popoo 4.5. Fo ay D B, F wh d dp 0 D τ, 0, hold fo P0 a.. ω Ω ha D τ,ω B τω, F τω wh d dp τω 0 D τ,ω 0. Cool ad aege ca alo be hfed. Popoo Fo ay µ U ep. ν V, hold fo P0 a.. ω Ω ha µ τ,ω µ τ,ω } τω, U τω ep. ν τ,ω ν τ,ω } τω, V τω. Fo ay α A ep. α Â, β B ad β B, hold fo P 0 a.. ω Ω ha α,ω A ep. α,ω Â, β,ω B ad β,ω B. 4.4 Shfed Sochac Dffeeal Equao I h ubeco, we ll code a F oppg me τ wh couably may value. Fx x R k, µ U, ν V ad e Θ, x, µ, ν. Fo P0 a.. ω Ω, Popoo how ha µ τ,ω, ν τ,ω U τω V τω, ad hu we kow fom Seco ha he followg SDE o he pobably pace Ω τω, F τω, P τω 0 : X X Θ τω ω + τω adm a uque oluo below, he F τω veo of X Θω τ b, X, µ τ,ω, ν τ,ω d + τω } X Θω τ } τω, τω, σ, X, µ τ,ω, ν τ,ω db τω, τω, 4.7 C τω,, R k wh Θ ω F τω τ τω, X Θτω ω, µτ,ω, ν τ,ω. A how exacly he hfed poce τ,ω} XΘ. τω, Popoo 4.7. I hold fo P 0 a.. ω Ω ha X Θω τ XΘ τ,ω, τω,. h eul ha appeaed 13 fo cae of compac cool pace ee he paagaph below 1.16 hee ad appeaed Lemma 3.3 of 7 whee oly oe ubouded cool codeed. he poof of Popoo 4.7 deped o he followg eul abou he covegece of hfed adom vaable pobably. Lemma 4.4. Fo ay ξ } N L 1 F ha covege o 0 pobably P 0, we ca fd a ubequece ξ } N of uch ha fo P0 a.. ω Ω, ξ τ,ω τω covege o 0 pobably P0. } N Fo ay F meauable adom vaable ξ wh LΘ ξ LΘ, P0 a.., Popoo 4.1, Coollay 4. ad Popoo 4.7 mply ha fo P0 a.. ω Ω, ξ τ,ω F τω ad L Θω τ l, X Θω τ l, τ,ω XΘ L Θ τ,ω ξ τ,ω L Θ τ,ω l, τ,ω XΘ l, X Θω τ L Θω τ. he Seco alo how ha fo P0 a.. ω Ω, he DRBSDE pace Ω τω, F τω, P τω 0 adm a uque oluo P τω, Θ LΘω τ, L ω τ o he pobably 0, ξ τ,ω, f Θω τ Y Θω τ, ξ τ,ω, Z Θω τ, ξ τ,ω, K Θω τ, ξ τ,ω, K Θω τ, ξ τ,ω G q τω,. Smla o Popoo 4.7, he F τω veo of Y Θω F τω τ, ξ τ,ω cocde wh he hfed poce Ỹ Θ, ξ } τ,ω. τω, Popoo 4.8. I hold fo P0 a.. ω Ω ha Ỹ Θω τ, ξ τ,ω Ỹτ Θ, ξ ω fo P0 a.. ω Ω. Ỹ Θω τ τω, ξ τ,ω Ỹ Θ, ξ τ,ω, τω,. I pacula, Popoo 4.8 ca alo be how by Pcad eao, ee 4.15 of 37 fo a BSDE veo.

16 O Zeo-Sum Sochac Dffeeal Game Pag of Cool ad Saege We defe Π,, ω, Π, ω,, ω, Ω. Aalogou o Lemma 1., oe ha Lemma 4.5. Le,. Fo ay D B, F 1, Π, D B, F ad d dp0 Π 1, D d dp 0 D. Coequely, he mappg Π, :, Ω, Ω P F /PF meauable, whee P F D PF : D, Ω } a σ feld of, Ω. Now, we ae eady o dcu pag of cool ad aege. Popoo 4.9. Le µ U fo ome 0, ad le τ be a F oppg me akg value a couable ube } N of,. Gve N N, le A }l 1 F be djo ube of τ } fo 1,, N ad e A 0 Ω N l A. he fo ay µ }l 1 1 U, 1,, N 1 µ ω µ Π, ω, f, ω τ, A, A fo 1, N ad 1,, l, 4.8 µ ω, f, ω, τ τ, A0 defe a U cool uch ha fo ay, ω τ, µ µ τ,ω, f ω A fo 1, N ad 1,, l, µ τ,ω, f ω A 0. We ca pae ν }l 1, 1,, N o a ν V wh epec o A V }l 1, 1,, N he ame mae. Popoo Le α A ep. Â fo ome 0, ad le τ be a F oppg me akg value a couable ube } N of Q,. Gve N N, le A }l 1 F be djo ube of τ } fo 1,, N ad e A 0 Ω N l A. he fo ay α }l 1 1 ep. Â A, 1,, N 1 α, ω, v α, Π, ω, v, f, ω τ, A, A fo 1, N ad 1,, l, α, ω, v, f, ω, τ τ, A0, v V 4.9 a A aegy ep. Â aegy uch ha gve ν V, hold fo P 0 a.. ω Ω ha fo ay τω, α ν τ,ω α ν,ω, f ω A fo 1, N ad 1,, l, α ν τ,ω, f ω A We ca pae β }l 1 B ep. B, 1,, N o a β B ep. B wh epec o A }l 1, 1,, N he ame mae. 5 Opmzao Poblem wh Squae-Iegable Cool I h eco, we wll emove v cool o ake V V 0 v 0 } o ha he zeo-um ochac dffeeal game dcued above degeeae a a oe-cool opmzao poblem fo playe I. 5.1 Geeal Reul We wll follow he eg of Seco excep ha we ake away he v cool fom all oao ad defo. I pacula, V, B ad B dappea o become gleo whle A equvale o U. he fo ay, x 0, R k, w 1, x, ŵ 1, x, w, x cocde a w, x up µ U Ỹ,x,µ, h X,x,µ. 5.1 A V λ vally hold fo ay λ 0, 1, he oe-cool veo of heoem.1 ead a:

17 5. Coeco o Secod-Ode Doubly Refleced BSDE 17 Popoo 5.1. Le, x 0, R k. Fo ay famly τ µ : µ U } of Q, valued, F oppg me up u U 0 w, x up µ U Ỹ,x,µ τ µ, w,x,µ τ µ, X τ µ. 5. Moeove, fo Ξ, x, y, z, Γ 0, R k R R d S k, HΞ ad HΞ mplfy epecvely a HΞ lm HΞ, u ad Ξ Ξ HΞ lm up u U 0 lm lm U 0 u u up Ξ O 1/ Ξ up HΞ, u lm up Ξ O 1/ Ξ u U 0 up HΞ, u lm u U 0 Ξ Ξ up u U 0 HΞ, u, up HΞ, u Ξ O 1/ Ξ whee we ued he fac ha up lm up he ecod equaly. he we have he followg oe-cool u U 0 U 0 u u u U 0 veo of heoem 3.1: Popoo 5.. he lowe emcouou evelope of w: w, x lm w, x,, x 0, R k a vcoy upeoluo of 3.1 wh he fully olea Hamloa H. O he ohe had, f U 0 a couable uo of cloed ube of U, he he uppe emcouou evelope of w: w, x lm w, x,, x 0, R k a vcoy uboluo of 3.1 wh he fully olea Hamloa H. Remak 5.1. Smla o 40, we oly eed o aume he meauably ep. lowe em-couy of he emal fuco h fo he ep. equaly of 5. ad hu fo he vcoy uboluo ep. upeoluo pa of Popoo Coeco o Secod-Ode Doubly Refleced BSDE Now le u fuhe ake k d, U S d, u 0 0 ad U 0 S >0 d Γ S d : deγ > 0}. Lemma 5.1. S d a epaable omed veco pace o whch he deema de couou. I follow ha U 0 S >0 d co of cloed ube of U: F u U : deu 1/}, N. We alo pecfy: b, x, u b, x ad σ, x, u u,, x, u 0, R d U. 5.3 fo a fuco b, x : 0, R d R d ha B0, BR d /BR d meauable ad Lpchz couou x wh coeffce γ > 0. Va he afomao 5.1, we wll how ha he value fuco w defed 5.1 he value fuco of ecod-ode doubly efleced BSDE. Gve 0,, we ay ha a P P a em-magale meaue f B a couou em-magale wh epec o F, P. Le Q be he colleco of all em-magale meaue o Ω, F. Lemma 5.. Le 0,. Fo, j 1,, d}, hee ex a R } valued, F pogevely meauable poce â,,j uch ha fo ay P Q, hold P a.. ha â,,j â,j, lm m B,, B,j P B,, B,j, P,, 5.4 m 1/m + whee B,, B,j P deoe he P co vaace bewee he h ad j h compoe of B. Smla o 39, we le Q W collec all P Q uch ha P a.. B P aboluely couou ad â S >0 d fo a.e.,. 5.5 I geeal, wo dffee pobable P 1, P of Q W ae muually gula, ee Example.1 of 39.

18 O Zeo-Sum Sochac Dffeeal Game 18 Lemma 5.3. Fo ay 0,, hee ex a uque S >0 d valued, F pogevely meauable poce ˆq uch ha fo ay P Q W, hold P a.. ha ˆq ˆq ˆq â fo a.e.,. Fo ay P Q W, we defe IP P 1dB ˆq,,,, whch a couou em-magale wh epec o F P, P. Sce he f pa of 5.5 ad 5.4 mply ha P a.., B P oe ca deduce fom Lemma 5.3 ha P a.. I P P â d,,, 5.6 ˆq 1 ˆq 1d B P ˆq 1 ˆq 1 â d I d d,,. I lgh of Lévy chaacezao, he magale pa W P of I P a Bowa moo ude P. Le G P G P }, deoe he P augmeed flao geeaed by he P Bowa moo W P,.e. G P σ σ W P,, N P,,. Le, x 0, R d ad µ U. Accodg o ou pecfcao 5.3, X,x,µ }, C,, R k ad F fo he uque oluo of he followg SDE o he pobably pace Ω, F, P 0 : Smla o.7, we have E X x + up, X,x,µ b, X d + c x + E µ db,,. 5.7 µ d <. 5.8 By Lemma 1.3, X,x,µ adm a uque F veo X,x,µ C F,, R k whch alo afe 5.8. A ha couou pah excep o ome Nµ,x F wh P 0 N,x µ 0, we ca vew X,x,µ 1 X, x, µ c x Nµ,x X,x,µ 5.9 a a mappg fom Ω o Ω. We clam ha X,x,µ acually a meauable mappg fom Ω, F o Ω, F : o ee h, we pck up a abay pa, E, BR d. he F adape of X,x,µ mple ha X,x,µ 1 B 1E ω Ω : X,x,µ ω B 1E } ω Ω : X,x,µ N,x µ N,x µ c ω Ω : ω E }, x, µ } X ω E x F, f 0 E, Nµ,x c ω Ω, x, µ } 5.10 : X ω E x F, f 0 / E, whee E x x + x : x E} BR d. hu 1E B Λ A Ω : X,x,µ } 1 A F. Clealy, Λ a σ feld of Ω. I follow ha B F 1E; σ,, E BR d Λ, 5.11 povg he meauably of he mappg X,x,µ. Coequely, we ca duce a pobably meaue P,x,µ P 0 X,x,µ o Ω, F,.e. P,x,µ P. Smla o 39, we e Q,x S P,x,µ } µ U.

19 5. Coeco o Secod-Ode Doubly Refleced BSDE 19 Lemma 5.4. Gve, x 0, R d ad µ U, le X,x,µ : Ω Ω be he mappg defed 5.9. I hold fo ay, ha X,x,µ 1 F P,x,µ F. Moeove, we have P,x,µ P 0 X,x,µ 1 Popoo 5.3. Fo ay, x 0, R d, we have Q,x S Q W. he followg eul abou Q,x S ped by Lemma 8.1 of 39. o F P,x,µ Popoo 5.4. Le, x 0, R d. Fo ay P Q,x S, FP cocde wh G P, he P augmeed flao geeaed by he P Bowa moo W P. Fx, x 0, R d ad e B,x x + B. By he couy of l ad l, L,x l, B,x,x ad L l, B,x,, ae wo eal-valued, F adaped couou pocee uch ha L,x < L,x,,. he meauably of f, B,x, he meauably of ˆq by Lemma 5.3 a well a he Lpchz couy of f y, z mply ha ˆf,x, ω, y, z f, B,x ω, y, z, ˆq ω,, ω, y, z, Ω R R d a P F BR BR d /BR meauable fuco ha alo Lpchz couou y, z. Gve µ U, oe ca deduce fom.3, he veo of.4 ad.5 whou ν cool, Hölde equaly, ad 5.8 ha E P,x,µ L,x q,x + up L q +,x ˆf τ, 0, 0 q d c 0 +c 0 E P,x,µ B,x + ˆq d, up, c 0 +c 0 E up, B,x X,x,µ + ˆq X,x,µ d c 0 +c 0 E up, up, X,x,µ + µ d <. A L,x hb,x L,x, follow ha E P,x,µ hb,x q <,.e. ξ L q Fτ, P,x,µ. he Popoo 5.4 ad heoem 4.1 of 10 how ha he followg Doubly efleced BSDE o he pobably pace Ω, F P,x,µ, P,x,µ Y hb,x + L,x ˆf,x, Y, Z d+k K K K P,x,µ Y L,x,,, ad Y L,x, dk Z dw P,x,µ,,, L,x Y dk 0 adm a uque oluo Y,x,P,x,µ, hb,x, Z,x,P,x,µ, hb,x, K,x,P,x,µ, hb,x, K,x,P,x,µ, hb,x G q F,, P,x,µ. P,x,µ Popoo 5.5. Fo ay, x 0, R d ad µ U P0 Y,x,µ,x,P, hb,x X,x,µ Y,x,µ, h X,x,µ,, 1. Le Ỹ,x,P,x,µ, hb,x be he F veo of Y,x,µ,x,P, hb,x F, oe ca deduce fom 5.13 ha 1 P0 Y,x,µ,x,P, hb,x X,x,µ y,x,µ} P,x,µ Y I follow ha y,x,µ Ỹ,x,P,x,µ, hb,x. Hece,, h w, x up Ỹ,x,µ µ U X,x,µ up P Q,x S. Fo he coa y,x,µ Ỹ,x,µ Ỹ,x,P,x,µ,x,P, hb,x y,x,µ}., h B,x, h, 5.14 whch exeded he value fuco of 40 ee 5.9 hee o he cae of doubly efleced BSDE baed o moe geeal fowad SDE. hu ou value fuco w cloely elaed o he ecod-ode doubly efleced BSDE. O he ohe had, whe he geeao f 0, he gh-had-de of 5.14 a doubly efleced veo of he value fuco codeed 7.,x,µ X

20 O Zeo-Sum Sochac Dffeeal Game 0 6 Poof 6.1 Poof of Seco 1 & Poof of Lemma 1.1: Fo ay,, clea ha σ C, B σ, 1E :,, E BR } d F,. o ee he evee, we fx,. Fo ay x Q d ad λ Q +, le j } j N Q, wh lm j j. Sce Ω, he e of R d valued couou fuco o, ag fom 0, we ca deduce ha whch mple ha B, 1 Oλ x λ m N B, 1 j m j Oλ 1 x σ C,. O O λ x : x Q d }, λ Q + Λ E R d : B, 1E } σ C,. Clealy, O geeae BR d ad Λ a σ feld of R d. hu, oe ha BR d Λ. he follow ha F, B, σ 1E :,, E BR } d σ C,. Poof of Lemma 1.: Fo mplcy, le u deoe Π,S, up,s by Π. We f how he couy of Π. Le A be a ope ube of Ω,S. Gve ω Π 1 A, ce Πω A, hee ex a δ > 0 uch ha O δ Πω ω Ω,S : } ω Πω < δ A. Fo ay ω O δ/ ω, oe ca deduce ha up Πω Πω ω ω + up ω ω up ω ω < δ,,s,s, whch how ha Πω O δ Πω A o ω Π 1 A. Hece, Π 1 A a ope ube of Ω,. Now, le, S. Fo ay, ad E BR d, oe ca deduce ha Π 1 B,S 1E ω Ω, : B,S } Πω E hu all he geeag e of F,S I follow ha F,S Λ,.e., Π 1 A F, ω Ω, : ω ω E } B, B, 1 E F,. 6.1 belog o Λ } A Ω,S : Π 1 A F,, whch clealy a σ feld of Ω,S. fo ay A F,S. Nex, we how ha he he duced pobably P P, 0 Π 1 equal o P,S 0 o F,S S : Sce he Wee meaue o Ω,S, BΩ,S uque ee e.g. Popoo I.3.3 of 35, uffce o how ha he caocal poce B,S a Bowa moo o Ω,S ude P : Le S. Fo ay E BR d, mla o 6.1, oe ca deduce ha hu, P B,S B,S Π 1 B,S 1E P, 0 B,S Π 1 B,S 1E B, B, B,S 1E P, 0 he dbuo of B,S B,S ude P he ame a ha of B, dbuo wh mea 0 ad vaace max I d d. O he ohe had, fo ay A F,S 0 ad 6. yeld ha fo ay E BR d P, P A B,S P, 0 B,S Π 1 A 1E P, P, 0, ce Π 1 A belog o F, 0 Π 1 B,S Π 1 A Π 1 B,S B,S 1 E. 6. B, B, 1 E, whch how ha B, ude P, 0 a d dmeoal omal, depedece fom B, B, 1E B,S 1E P A P B,S 1E B,S. Hece, B,S B,S depede of F,S ude P. ude

21 6.1 Poof of Seco 1 & 1 Poof of Lemma 1.3: 1 F, le ξ L 1 F P, P ad,. Fo ay A F P, hee ex a à F uch ha A à N P ee e.g. Poblem.7.3 of 3. hu we have ha A ξdp à ξdp à E P ξf dp A E P ξf dp, whch mple ha E P ξ F P EP ξ F, P a he ealy follow ha ay magale X wh epec o F, P alo a magale wh epec o F P, P. Nex, le X X }, be a local magale wh epec o F, P. hee ex a ceag equece τ } N of F oppg me wh P lm τ 1 uch ha Bτ, N ae all magale wh epec o F, P. Fo ay m N, ν m f, : X > m} defe a F oppg me. I lgh of he Opoal Samplg heoem, X τ ν m a magale wh epec o F, P. hu, fo ay <, oe ha E P X τ ν m E P X τ ν m X τ ν m, P a F P Sce P lm τ 1, whe 6.4, he bouded covegece heoem mple ha E P X νm F P X νm, P a.. Namely, X νm a bouded magale wh epec o F P, P. Clealy, ν m } k N ae F P oppg me wh lm ν m. Hece, X a local magale wh epec o F P, P, Moe geeal, k ay em-magale wh epec o F, P alo a em-magale wh epec o F P, P. he uquee obvou ad uffce o how he exece fo cae E R: Le X }, be a eal-valued, F P adaped couou poce. Fo each Q,, we ee fom 6.3 ha F X EP X F EP X F P X, Se N ω Ω : he pah X ω o couou } 1+ 1 X P a.. X X } N P. Q, Sce X,+ },, a eal-valued, F pogevely meauable poce fo ay N, We ee ha X lm X 1 lm X < } alo defe a eal-valued, F pogevely meauable poce. Le ω N c ad,. Fo ay N, ce, + 1 wh +, oe ha X ω X ω X ω. Clealy, lm, A, he couy of X how ha lm X ω lm X ω X ω, whch mple ha N c ω Ω : X ω X ω,, }. heefoe, X P dguhable fom X, ad follow ha X alo ha P a.. couou pah. Nex, le X }, be a eal-valued, F P pogevely meauable poce ha bouded. Sce X X d,, defe a eal-valued, F P adaped couou poce, we kow fom pa 1 ha X ha a uque F veo X. Fo ay N, X X X 1/ clealy a eal-valued, F adaped couou poce ad hu a F pogevely meauable poce. I follow ha X lm X 1 lm X < } aga defe a eal-valued, F pogevely meauable poce. Se N ω Ω : X ω X ω fo ome, } N P. Fo ay ω N c, oe ca deduce ha lm X ω lm X X 1/ lm X d X, fo a.e.,, 1/ whch mple ha X ω X ω fo d dp a.., ω, Ω. Moeove, fo geeal eal-valued, F P pogevely meauable poce X },, le X m be he F veo of X m m X m }, fo ay m N. he X lm X m 1 m lm X m m < } defe a eal-valued, F pogevely meauable poce. Le D, ω, Ω : X m ω X m ω }. Clealy, d dp D m N 0 ad hold fo ay, ω, Ω \D ha X ω X ω.

22 O Zeo-Sum Sochac Dffeeal Game Lemma 6.1. Gve 0, ad wo, q paamee e ξ 1, f 1, L 1, L 1, ξ, f, L, L wh P 0 L 1 L, L1 L,, 1, le Y, Z, K, K C q, H,q,, R d K F F F, K F,, 1, be a oluo of DRBSDE P0, ξ, f, L, L. Fo ehe 1 o, f f afe 1.6, he fo ay ϖ 1, q E up Y 1 + ϖ C, ϖ, γ E ξ1 ξ + ϖ +E f1, Y 3, Z 3 f, Y 3, Z 3 ϖ } + d., Y Poof: Whou lo of geealy, le f 1 afy 1.6. Fx ϖ 1, q. We aume ha E ohewe, he eul hold auomacally. f1, Y, Z f, Y, Z + d ϖ <, 6.5 Fo X ξ, Y, Z, we e X X 1 X. Applyg aaka fomula o poce Y + yeld ha Y + ξ Y>0} f1, Y 1, Z 1 f, Y, Z d Y>0} dk 1 dk dk 1 + dk dl 1 Y>0} Z db,, whee L a eal-valued, F adaped, ceag ad couou poce kow a local me. he we ca deduce fom Lemma.1 of 10 ha Y + ϖ Y + ϖ ϖϖ 1 + ϖ 0 1 Y>0} ϖ Y + ϖ 1 f1, Y 1, Z 1 f, Y 1 Y>0} Y + ϖ 1 Z db ϖ 1 Y>0} Y + ϖ Z d, Z d+ϖ 1 Y>0} Y + ϖ 1 dk 1 dk dk 1 +dk 1 Y>0} Y + ϖ 1 dl,. 6.6 By he lowe fla-off codo of DRBSDE P 0, ξ 1, f 1, L 1, L 1, hold P 0 a.. ha 1 Y>0} Y + ϖ 1 dk 1 1 L 1 Y 1 >Y } L 1 Y + ϖ 1 dk 1 Smlaly, he uppe fla-off codo of DRBSDE P 0, ξ, f, L, L mple ha P 0 a Y>0} Y + ϖ 1 dk 1 Y 1 >Y L } Y 1 L + ϖ 1 dk 1 L 1 >L } L 1 Y + ϖ 1 dk L 1 >L } Y 1 L + ϖ 1 dk 0. Pug hee wo equaly back o 6.6 ad ug Lpchz couy of f 1 y, z, we oba Y + ϖ ϖϖ 1 + Y + ϖ +ϖ 1 Y>0} Y + ϖ Z d 1 Y>0} Y + ϖ 1 γ Y +γ Z + f + d Z db,, 6.7 whee f + f1, Y, Z f, Y, Z +. Sce C q, C ϖ, ad H,q,, R d H,ϖ,, R d by Jee equaly, he Bukholde- F F F F Dav-Gudy equaly ad Hölde equaly mply ha fo ome c > 0 E up 1 Y>0} Y + ϖ 1 Z db ce Y + ϖ 1/ Z d, ce up, Y ϖ 1 Z d 1/ c Y ϖ 1 C ϖ F, Z H,ϖ F,,Rd <,

23 6.1 Poof of Seco 1 & 3 whch how ha 1 Y >0} Y + ϖ 1 } Z db, a ufomly egable magale wh epec o F, P he, leg, ad akg he expecao E 6.7, we ca deduce fom Hölde equaly, Youg equaly ad 6.5 ha ϖϖ 1 E E ξ ϖ +ϖe 1 Y>0} Y + ϖ Z d up, Y ϖ 1 γ up Y + γ, ϖ 1 + γ Y ϖ C ϖ F, +γϖ ϖ/ Z ϖ H,ϖ F,,Rd +E Hece, we ca defe a ceag equece of F oppg me τ f, : 1 Y>0} uch ha lm τ, P 0 a.. Fx N. Sce 1 Y>0} Y + ϖ 1 Z γ ϖ 1 leg τ ad τ 6.7 yeld ha + Y τ ϖ ϖϖ 1 + τ 4 τ + ϖγ + ϖγ τ ϖ 1 1/ Z d + f + d ϖ f + d <. Y + ϖ } Z d >, N Y + ϖ + ϖ 1 4γ Y + ϖ 1 Y>0} Z d Y τ + ϖ +ϖ τ Y + ϖ τ d ϖ τ 1 Y >0} Y + ϖ Z d,,, τ τ Y + ϖ 1 f + d 1 Y>0} Y + ϖ 1 Z db,,. 6.9 akg he expecao E, we ca deduce fom Fub heoem, 6.8 ad Opoal Samplg heoem ha + E Yτ ϖ + ϖϖ 1 τ E 1 Y>0} Y + ϖ Z d 4 τ E η + ϖγ + ϖγ + E Yτ ϖ d,,, 6.10 ϖ 1 whee η Yτ + ϖ + ϖ τ Y + ϖ 1 f + d. Le C, ϖ, γ deoe a geec coa, depedg o, ϖ, γ, whoe fom may vay fom le o le. A applcao of Gowall equaly o 6.10 yeld ha + E Yτ ϖ + ϖϖ 1 τ E 4 τ whch ogehe wh Fub heoem how ha 1 Y>0} Y + ϖ Z d C, ϖ, γe η,, τ E Y + ϖ d E + Y τ ϖ + d E Yτ ϖ d C, ϖ, γe η. he we ca deduce fom 6.9 ha E up,τ Y + ϖ C, ϖ, γe η + ϖe up, τ τ 1 Y>0} Y + ϖ 1 Z db.

24 O Zeo-Sum Sochac Dffeeal Game 4 he Bukholde-Dav-Gudy equaly aga mple ha fo ome c > 0 E up, ce + ϖc E 1 τ}1 Y>0} Y + ϖ 1 Z db ce 1 τ}1 Y>0} Y + ϖ 1/ Z d up,τ τ Y + ϖ/ τ 1 Y>0} Y + ϖ 1/ Z d 1 1 Y>0} Y + ϖ Z 1 d ϖ E up,τ whee we ued 6.11 wh he la equaly. Sce E follow fom Youg equaly ha E up,τ Hece, we have ϖ E up,τ Y + ϖ Y + ϖ +C, ϖ, γe η, up,τ Y + ϖ Y + ϖ + C, ϖ, γ E η C, ϖ, γ E Yτ ϖ +E E,τ C, ϖ, γe Yτ + ϖ + 1 E up,τ Y + ϖ up,τ Y ϖ <, C ϖ F, Y + ϖ 1 τ } f + d τ ϖ +C, ϖ, γe f + d. up Y + ϖ + ϖ C, ϖ, γ E Yτ ϖ + E f + d }. 6.1 Sce Y C ϖ, ad ce lm τ F, P0 a.., he Domaed Covegece heoem mple ha + lm Yτ ϖ E ξ + ϖ. Leg 6.1 ad applyg he Moooe Covegece heoem E o lef-had-de lead o ha E up, Y 1 Y + ϖ C, ϖ, γ E ξ1 ξ + ϖ +E f1, Y, Z f, Y, Z + d ϖ } Poof of Popoo 1.1: Fo ehe 1 o, f f afe 1.6, applyg Lemma 6.1 wh ϖ q yeld ha E up Y 1 + q 0. Hece, hold P0 a.. ha Y + 0, o Y 1 Y fo ay,., Y Poof of Popoo 1.: Fo ay ϖ 1, q, follow fom Lemma 6.1 ha E up, Y 1 Y + ϖ C, ϖ, γ E ξ1 ξ + ϖ +E f1, Y, Z f, Y, Z + d ϖ }.6.14 Exchagg he ode of Y 1, Y ad applyg Lemma 6.1 aga gve ha E up Y Y 1 + ϖ C, ϖ, γ E ξ ξ 1 + ϖ +E f, Y, Z f 1, Y, Z ϖ } + d,, whch ogehe wh 6.14 mple 1.7. Poof of Lemma.1: 1 Se Θ, x, µ, ν ad fx,. Fo ay,, oe ca deduce fom 1.1,. ad.1 ha up X Θ x γ, 1+ x + up X Θ x +µ U +ν V d + up,, σ, X Θ, µ, ν db, P 0 a..

25 6.1 Poof of Seco 1 & 5 he Hölde equaly, Doob magale equaly,.,.1 ad Fub heoem mply ha E up X Θ x c 0 E 1+ x + up X Θ x +µ +ν, U d + c V 0 E σ, X Θ, µ, ν d, c 0 1+ x + c 0 E up, X Θ x d+ c 0 E µ U +ν V A applcao of Gowall equaly yeld ha E up X Θ x c 0 e c0 1+ x + c 0 e c0 E µ +ν, U V d,,. akg gve.8. d,,. Gve x R k, we e X X,x,µ,ν X,x,µ,ν,,. By., up X x x +γ X d+ up σ, X,x,µ,ν, µ, ν σ, X,x,µ,ν, µ, ν db,,.,, he he Bukholde-Dav-Gudy equaly ad. mply ha E up X ϖ c ϖ x x ϖ + c ϖ E, ϖ ϖ/ X d + X d ϖ c ϖ x x ϖ + c ϖ E X d + up X ϖ/ c ϖ x x ϖ + c ϖ E, ϖ X d + 1 E ϖ/ X d up X, ϖ,., A E up X ϖ 1 + E up X,x,µ,ν + up X,x,µ,ν < by.7, follow fom Hölde,,, equaly ad Fub heoem ha E up X ϖ c ϖ x x ϖ +c ϖ E X ϖ d cϖ x x ϖ +c ϖ E up X d, ϖ,. 6.15, A applcao of Gowall equaly yeld.9., 3 Nex, Le u aume.10 fo ome λ 0, 1. Gve µ U, we e X X,x,µ,ν X,x,µ,ν,,. By. ad.10, up X γ, X +ρ λ U µ, µ d+ up, σ, X,x,µ,ν, µ, ν σ, X,x,µ,ν, µ, ν db,,. he oe ca deduce fom he Bukholde-Dav-Gudy equaly,.,.10 ad Hölde equaly ha ϖ ϖ E up X ϖ c ϖ E X d + ρ λ U µ, µ d + X +ρ λ U µ, µ ϖ/ d, c ϖ E c ϖ E ϖ X d + ϖ X d + ρ λ µ, µ ϖ/ U d + up, ρ λ µ, µ ϖ/ U d + 1 E X ϖ/ ϖ/ X d up X, ϖ,., Smla o 6.15, follow fom Hölde equaly ad Fub heoem ha E up X ϖ c ϖ E up X ϖ d+c ϖ E ρ λ µ, µ ϖ/ U d,,.,,

26 O Zeo-Sum Sochac Dffeeal Game 6 he a applcao of Gowall equaly yeld.11. Smlaly, wh.1 we ca deduce.13 fo each ν V. Lemma 6.. Le M be a epaable mec pace wh mec ρ M. Fo ay wo M valued, F adaped ep. F pogevely meauable pocee Y, Z, he oegave-valued poce ρ M Y, Z alo F adaped ep. F pogevely meauable. Poof: Le x } N be he couable dee ube of M ad deoe by BM he Boel σ feld of M. We f clam ha fo ay y, z M ad λ > 0, ρ M y, z<λ f ad oly f hee ex N ad Q 0, λ uch ha ρ M y, x < ad ρ M x, z<λ : h deco obvou due o he agle equaly. : If ρ M y, z < λ, we le be a pove aoal umbe ha le ha 1 λ ρm y, z. hee ex a N, uch ha ρ M y, x <. By he agle equaly, ρ M x, z ρ M x, y + ρ M y, z < + ρ M y, z < λ. So we poved he clam Now, gve wo M valued, F adaped pocee Y ad Z, fo ay, ad λ>0, 6.16 mple ha ω Ω : ρ Y M ω, Z ω <λ } ω Ω : Y ω O x } ω Ω : Z ω O λ x } F, N Q 0,λ whch how ρ M Y, Z alo F adaped. If Y ad Z ae fuhe F pogevely meauable, he fo ay, ad λ>0, we ee fom 6.16 ha, ω, Ω : ρ M Y ω, Z ω <λ } N Q 0,λ, ω, Ω : Y ω O x }, ω, Ω : Z ω O λ x } B, F. Namely, ρ M Y, Z F pogevely meauable a well. Poof of Lemma.: We e Θ, x, µ, ν. 1 Fo ay x R k, le Θ, x, µ, ν ad X X Θ X Θ. he meauably of f Θ, X ad.5 how ha f ±, ω, y, z f Θ, ω, y, z ± γ X ω /q,, ω, y, z, Ω R R d defe wo P F BR BR d /BR meauable fuco ha ae Lpchz couou y, z wh coeffce γ. We ee fom Hölde equaly,.16 ad.7 ha E f±, 0, 0 q d c 0 E f Θ, 0, 0 q d + up X <. 6.17, Fx ε > 0. he fuco φx x + ε 1/q, x R k ha he followg devave: fo ay, j 1,, k} φx q φ1 q x x ad j φx q φ1 q x δ j + 4 q 1 qφ1 q x x x j. I eay o emae ha x /q φx x /q + ε 1/q, Dφx q φ1 q x x q x q 1, x R k Fo ay z R k d, ce ace D φxzz q φ1 q x z + 4 q 1 qφ 1 q x z x, we alo have q x q z 4 q 1 q x q z ace D φxzz q x q z, x R k. 6.19