1. Introduction. Markovian stochastic control problems on a given horizon of time T can typically be written as. β(xu α,α u,e)ñ(du, de).

Transcription

1 SIAM J. CONROL OPIM. Vol. 54, No. 4, pp c 2016 Sociey for Idurial ad Applied Mahemaic A WEAK DYNAMIC PROGRAMMING PRINCIPLE FOR COMBINED OPIMAL SOPPING/SOCHASIC CONROL WIH E f -EXPECAIONS ROXANA DUMIRESCU, MARIE-CLAIRE QUENEZ, AND AGNÈS SULEM Abrac. We udy a combied opimal corol/oppig problem uder a oliear expecaio E f iduced by a BSDE wih jump, i a Markovia framework. he ermial reward fucio i oly uppoed o be Borelia. he value fucio u aociaed wih hi problem i geerally irregular. We fir eablih a ub- rep., uper- opimaliy priciple of dyamic programmig ivolvig i upper- rep., lower- emicoiuou evelope u rep., u. hi reul, called he weak dyamic programmig priciple DPP, exed ha obaied i [Bouchard ad ouzi, SIAM J. Corol Opim., , pp i he cae of a claical expecaio o he cae of a E f -expecaio ad Borelia ermial reward fucio. Uig hi weak DPP, we he prove ha u rep., u iavicoiy ub- rep., uper- oluio of a oliear Hamilo Jacobi Bellma variaioal iequaliy. Key word. Markovia ochaic corol, mixed opimal corol/oppig, oliear expecaio, backward ochaic differeial equaio, weak dyamic programmig priciple, Hamilo Jacobi Bellma variaioal iequaliy, vicoiy oluio, E f -expecaio AMS ubjec claificaio. 93E20, 60J60, 47N10 DOI /15M Iroducio. Markovia ochaic corol problem o a give horizo of ime ca ypically be wrie a [ 1.1 u0,x=upe α A fα,x α d + gx α, 0 where A i a e of admiible corol procee α,adx α i a corolled proce of he form X α = x + bxu α,α u du + σxu α,α u dw u βxu α,α u,eñdu, de. R he radom variable gx α may repree a ermial reward ad fα,x αai- aaeou reward proce. Formally, for all iiial ime i [0, ad iiial ae y, he aociaed value fucio i defied by [ 1.2 u, y =upe α A fα,x α d + gx α Xα = y. Received by he edior Jue 19, 2015; acceped for publicaio i revied form May 17, 2016; publihed elecroically Augu 24, hp:// Fudig: he reearch of he fir auhor leadig o hee reul ha received fudig from he Regio Ile-de-Frace. Iiu für Mahemaik, Humbold-Uiveriä zu Berli, Uer de Lide 6, Berli, Germay roxaa@ceremade.dauphie.fr. LPMA, Uiverié Pari 7 Dei Didero, Boie courrier 7012, Pari cedex 05, Frace queez@mah.uiv-pari-didero.fr. INRIA Pari, 3 rue Simoe Iff, CS 42112, Pari Cedex 12, Frace, ad Uiverié Pari-E, Pari, Frace age.ulem@iria.fr. 2090

2 WEAK DPP WIH E f -EXPECAIONS 2091 he dyamic programmig priciple DPP ca formally be aed a [ 1.3 u0,x=upe fα,x α d + u, Xα for i [0,. α A 0 hi priciple i claically eablihed uder aumpio which eure ha he value fucio u aifie ome regulariy properie. From hi priciple, i ca be derived ha he value fucio i a vicoiy oluio of he aociaed Hamilo Jacobi Bellma HJB equaio. Similar reul are obaied for opimal oppig ad mixed opimal oppig/corol problem. he cae of a dicoiuou value fucio ad i lik wih vicoiy oluio ha bee udied for deermiiic corol i he eighie. Barle ad Perhame udy i [3 a deermiiic opimal oppig problem wih a reward map g oly uppoed o be Borelia. o hi purpoe, hey iroduce a oio of vicoiy oluio which exed he claical oe o he dicoiuou cae: a fucio v i aid o be a weak vicoiy oluio of he HJB equaio if i upper emicoiuou u..c. evelope, deoed by v,iavicoiy uboluio of hi PDE, ad if i he lower emicoiuou l..c. evelope, deoed by v,iavicoiy uperoluio of hi equaio. he, by he claical DPP provided i he previou lieraure, hey ge ha he u..c. evelope u of he value fucio aifie a ubopimaliy priciple i he ee of Lio ad Sougaidi i [19. Uig hi ubopimaliy priciple, hey he how ha u i a vicoiy uboluio of he HJB equaio. Moreover, uig he fac ha he l..c. evelop u of u i he value fucio of a relaxed problem, hey how ha u i a vicoiy uperoluio ad hu ge ha u i a weak vicoiy oluio of he HJB equaio. hey re ha i geeral, he weak vicoiy oluio of hi PDE i o uique. However, uder a regulariy aumpio o he reward g, by uig he corol formulae, hey obai ha he u..c. evelope u of he value fucio i he uique u..c. vicoiy oluio of he HJB equaio. See Remark 23 for addiioal referece ad comme. More recely, i a ochaic framework, Bouchard ad ouzi [8 have proved a weak DPP whe he ermial map g i irregular: hey prove ha he value fucio u aifie a ubopimaliy priciple of dyamic programmig ivolvig i u..c. evelope u, ad uder a addiioal regulariy lower emicoiuiy aumpio of he reward g, hey obai a uperopimaliy priciple ivolvig he l..c. evelope u. he, uig he ubopimaliy priciple, hey derive ha u i a vicoiy uboluio of he aociaed HJB equaio. Moreover, whe g i l..c., uig he uperopimaliy priciple, hey how ha u i a vicoiy uperoluio ad hu ge ha u i a weak vicoiy oluio of hi PDE i he ame ee a above or [3. A weak DPP ha bee furher eablihed, whe g i l..c. for problem wih ae corai by Bouchard ad Nuz i [7, ad, whe g i coiuou, for zero-um ochaic game by Bayrakar ad Yao i [4. I hi paper we are iereed i geeralizig hee reul o he cae whe g i oly Borelia ad whe he liear expecaio E i replaced by a oliear expecaio iduced by a backward ochaic differeial equaio BSDE wih jump. ypically, uch problem i he Markovia cae ca be formulaed a 1.4 up E0, α [gx α, α A where E α i he oliear expecaio aociaed wih a BSDE wih jump wih corolled driver fα,x α,y,z,k. Noe ha problem 1.1 i a paricular cae of

3 2092 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM 1.4 whe he driver f doe o deped o he oluio of he BSDE, ha i, whe fα,x α,y,z,k fα,x α. We fir provide a weak DPP ivolvig he u..c. ad l..c. evelope of he value fucio. For hi purpoe, we prove ome prelimiary reul, i paricular ome meaurabiliy ad pliig properie. No regulariy codiio o g i required o obai he ub- ad uper-opimaliy priciple, which i o he cae i he previou lieraure i he ochaic cae, eve wih a claical expecaio ee [8, [7, ad [4. Uig hi weak DPP, we he how ha he value fucio, which i geerally eiher u..c. or l..c., i a weak vicoiy oluio i he ee of [3 of a aociaed oliear HJB equaio. Moreover, i hi paper, we coider he combied problem whe here i a addiioal corol i he form of a oppig ime. We hu coider mixed geeralized opimal corol/oppig problem of he form 1.5 up up E0,τ α [ hτ,x τ α, α A τ where deoe he e of oppig ime wih value i [0,, ad h i a irregular reward fucio. Noe ha i he lieraure o BSDE, ome paper ee, e.g., Peg [20, Li ad Peg [18, Buckdah ad Li [9, ad Buckdah ad Nie [10 udy ochaic corol problem wih oliear E-expecaio i he coiuou cae wihou opimal oppig. heir approach i differe from our ad relie o he coiuiy aumpio of he reward fucio. he paper i orgaized a follow: i ecio 2, we formulae our geeralized mixed corol-opimal oppig problem. Uig reul o refleced BSDE RBSDE, we expre hi problem a a opimal corol problem for RBSDE. I ecio 3, we prove a weak DPP for our mixed problem wih E f -expecaio. hi require ome pecific echique of ochaic aalyi ad BSDE o hadle meaurabiliy ad oher iue due o he olieariy of he expecaio ad he lack of regulariy of he ermial reward. Uig he DPP ad properie of RBSDE, we prove i ecio 4 ha he value fucio of our mixed problem i a weak vicoiy oluio of a oliear HJB variaioal iequaliy. I he appedix, we give everal fie meaurabiliy properie which are ued i he paper. 2. Formulaio of he mixed oppig/corol problem. We coider he produc pace Ω := Ω W Ω N,whereΩ W := C[0, i he Wieer pace, ha i, he e of coiuou fucio ω 1 from [0,ioR p uch ha ω 1 0 = 0, ad Ω N := D[0, i he Skorohod pace of righ-coiuou wih lef limi RCLL fucio ω 2 from [0,ioR d, uch ha ω 2 0 = 0. Recall ha Ω i a Polih pace for he Skorohod meric. Here p, d 1, bu, for oaioal impliciy, we hall coider oly R-valued fucio, ha i, he cae p = d =1. Le B =B 1,B 2 be he caoical proce defied for each [0,adeach ω =ω 1,ω 2 bybω i =Bω i i :=ω i for i =1, 2. Le u deoe he fir coordiae proce B 1 by W.LeP W be he probabiliy meaure o Ω W, BΩ W uch ha W i a Browia moio. Here BΩ W deoe he Borelia σ-algebra o Ω W. Se E := R \{0} equipped wih i Borelia σ-algebra BE, where 1. We defie he jump radom meaure N a follow: for each >0adeachB BE, 2.1 N., [0, B := 1 {ΔB 2 B}. 0<

4 WEAK DPP WIH E f -EXPECAIONS 2093 he meaurable e E, BE i equipped wih a σ-fiie poiive meaure ν uch ha E 1 e νde <. Le P N be he probabiliy meaure o Ω N, BΩ N uch ha N i a Poio radom meaure wih compeaor νded ad uch ha B 2 = 0< ΔB2 a.. Noe ha he um of jump i well defied up o a P N -ull e. We e Ñdr, de :=Ndr, de νded. he pace Ω i equipped wih he σ-algebra BΩ ad he probabiliy meaure P := P W P N. Le F := F 0 be he filraio geeraed by W ad N compleedwihrepecobω ad P, defied a follow ee [17, p. 3 or [13, IV: le F be he compleio σ-algebra of BΩ wih repec o P 1 For each [0,, F i he σ-algebra geeraed by W,N, ad he P -ull e. Noe ha F = F ad F 0 i he σ-algebra geeraed by he P -ull e. Le P be he predicable σ-algebra o Ω [0, aociaed wih he filraio F. Le >0befixed. LeH 2 deoed alo by H2 be he e of real-valued predicable procee Z uch ha E 0 Z2 d < ad le S 2 be he e of real-valued RCLL adaped procee ϕ wih E[up 0 ϕ 2 <. Le L 2 ν be he e of meaurable fucio l :E, BE R, BR uch ha l 2 ν := E l2 eνde <. he e L 2 ν i a Hilber pace equipped wih he calar produc l, l ν := E lel eνde for all l, l L 2 ν L2 ν. Le H2 ν deoe he e of predicable real-valued procee k wih E 0 k 2 L d <. 2 ν Le A be he e of corol, defied a he e of predicable procee α valued i a compac ube A of R p,wherep N.Foreachα Aad each iiial codiio x i R, le X α,x 0 be he uique R-valued oluio i S 2 of he ochaic differeial equaio SDE: 2.2 X α,x = x + bxr α,x,α r dr + σxr α,x,α r dw r + βx α,x r,α r,eñdr, de, 0 0 where b, σ : R A R, are Lipchiz coiuou wih repec o x ad α, ad β : R A E R i a bouded meaurable fucio uch ha for ome coa C 0, ad for all e E βx, α, e C Ψe, x R,α A, where Ψ L 2 ν, βx, α, e βx,α,e C x x + α α Ψe, x,x R,α,α A. he crierio of our mixed corol problem, depedig o α, i defied via a BSDE wih driver fucio f aifyig he followig hypohei. Aumpio 1. f : A [0, R 3 L 2 ν R, BR i BA B[0, BR3 BL 2 ν -meaurable ad aifie i fα,, x, 0, 0, 0 C1 + x p for all α A, [0,,x R, wherep N, ii fα,, x, y, z, k fα,,x,y,z,k C α α + x x + y y + z z + k k L 2 ν for all [0,, x, x,y,y,z,z R, k, k L 2 ν,α,α A, iii fα,, x, y, z, k 2 fα,, x, y, z, k 1 γα,, x, y, z, k 1,k 2,k 2 k 1 ν for all, x, y, z, k 1,k 2,α, where γ : A [0, R 3 L 2 ν 2 L 2 ν, BL 2 ν i BA B[0, BR 3 BL 2 ν 2 - meaurable, aifyig γ.e 1ad γ.e Ψe, dνe-a.., where Ψ L 2 ν. For all x R ad all corol α A,lef α,x be he driver defied by f α,x r, ω, y, z, k :=fα r ω,r,xr α,x ω,y,z,k. 1 For he defiiio of he compleio of a σ-algebra ad he oe of P -ull e, ee, e.g., Lemma E

5 2094 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM We iroduce he oliear expecaio E f α,x deoed more imply by E α,x aociaed wih f α,x, defied for each oppig ime τ ad for each η L 2 F τ a Er,τ α,x [η :=Xr α,x, 0 r τ, where Xr α,x i he oluio i S 2 of he BSDE aociaed wih driver f α,x,ermial ime τ, ad ermial codiio η, ha i aifyig dx α,x r = fα r,r,x α,x r Zr α,x dw r, Xr α,x E,Z α,x r,kr α,x dr Kr α,x α,x eñdr, de; XS = η, ad Z α,x, K α,x are he aociaed procee, which belog repecively o H 2 ad H 2 ν. Codiio iii eure he odecreaig propery of he Efα,x -expecaio ee [21. For all x R ad all corol α A, we defie he reward by h, X α,x for 0 < ad gx α,x for =,where g : R R i Borelia, h :[0, R R i a fucio which i Lipchiz coiuou wih repec o x uiformly i, ad coiuou wih repec o o [0,, h, x + gx C1 + x p for all [0,,x R, wih p N. Le be he e of oppig ime wih value i [0,. Suppoe he iiial ime i equal o 0. Noe ha E α,,x 0,τ [ hτ,xτ α,,x ca be ake a coa. 2 For each iiial codiio x R, we coider he mixed opimal corol/oppig problem: 2.3 u0,x:=upup α A τ E α,x α,x 0,τ [ hτ,xτ, where h, x :=h, x1 < + gx1 =. Noe ha h i Borelia bu o ecearily regular i, x. We ow make he problem dyamic. We defie for [0,adeachω Ωhe -ralaed pah ω =ω := ω ω. Noe ha ω 1, := ω 1 ω 1 correpod o he realizaio of he ralaed Browia moio W := W W ad ha he ralaed Poio radom meaure N := N,,. ca be expreed i erm of ω 2, := ω 2 ω 2 imilarly o 2.1. Le F =F be he filraio geeraed by W ad N compleed wih repec o BΩ ad P.Noeha for each [,, F i he σ-algebra geeraed by W r, N r, r, adf 0. Recall alo ha we have a marigale repreeaio heorem for F -marigale a ochaic iegral wih repec o W ad Ñ. Le u deoe by he e of oppig ime wih repec o F wih value i [,. Le P be he predicable σ-algebra o Ω [, equipped wih he filraio F. We ow iroduce he followig pace of procee. Le [0,. Le H 2 be he P -meaurable procee Z o Ω [, uch ha Z H 2 := E[ Zudu 2 <. We defie H 2,ν a he e of P -meaurable procee K o Ω [, uch ha K H 2,ν := E[ K u 2 ν du <. We deoe by S2 he e of real-valued RCLL procee ϕ o Ω [,, F -adaped, wih E[up ϕ 2 <. 2 Ideed, he oluio of a BSDE wih a Lipchiz driver i uique up o a P -ull e. I iiial value may hu be ake coa for all ω, modulo a chage of i value o a P -ull e, becaue F 0 i he σ-algebra geeraed by he P -ull e.

6 WEAK DPP WIH E f -EXPECAIONS 2095 Le A be he e of corol α : Ω [, A, which are P -meaurable. We coider he oluio deoed by X α,,x i S 2 of he followig SDE drive by he ralaed Browia moio W ad he ralaed Poio radom meaure N wih filraio F : 2.4 X α,,x = x + + E bx α,,x r,α r dr + βx α,,x r,α r,eñ dr, de. σxr α,,x,α r dwr For all, x [0, R ad all corol α A,lef α,,x be he driver defied by f α,,x r, ω, y, z, k :=fα r ω,r,xr α,,x ω,y,z,k. Le E.,τ α,,x [ hτ,xτ α,,x deoed alo by X α,,x be he oluio i S 2 of he BSDE wih driver f α,,x α,,x, ermial ime τ, ad ermial codiio hτ,xτ, drive by W ad N, which i olved o [, Ω wih repec o he filraio F : dxr α,,x = fα r,r,xr α,,x, Xr α,,x,zr α,,x,kr α,,x dr 2.5 Zr α,,x dwr E Kα,,x r eñ dr, de Xτ α,,x α,,x = hτ,xτ, where Z α,,x, K α,,x are he aociaed procee, which belog, repecively, o H 2 ad H 2,ν. Noe ha Eα,,x,τ [ hτ,xτ α,,x ca be ake deermiiic modulo a chage of i value o a P -ull e. 3 For each iiial ime ad each iiial codiio x, we defie he value fucio a 2.6 u, x := up α A up τ E α,,x,τ [ hτ,xτ α,,x, which i a deermiiic fucio of ad x. For each α A, we iroduce he fucio u α defied a We hu ge u α, x := up τ E α,,x,τ [ hτ,xτ α,,x. 2.7 u, x = up u α, x. α A For each α, u α, x h, x, ad hece u, x h, x. Moreover, u α,x = u,x=gx. By [22, heorem 3.2, for each α, he value fucio u α i relaed o a refleced BSDE. More preciely, le Y α,,x,z α,,x,k α,,x S 2 H 2 H 2 ν, be he oluio of he refleced BSDE aociaed wih driver f α,,x := fα,,x α,,x,y,z,k, RCLL 3 Ideed, he oluio E.,τ α,,x [ hτ, Xτ α,,x = X. α,,x of he BSDE 2.5 i uique up o a P -ull e. Moreover, i value a ime i F -meaurable, ad F i equal o he σ-algebra geeraed by he P -ull e ha i, F 0. he ame propery hold for he oluio of he refleced BSDE 2.8. See alo he addiioal Remark 6 ad 8.

7 2096 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM obacle proce ξ α,,x filraio F,hai, := h, X α,,x, ermial codiio gx α,,x, ad wih 2.8 dyr α,,x = fα r,r,xr α,,x,yr α,,x,zr α,,x,kr α,,x dr + da α,,x Zr α,,x dwr E Kα,,x r, eñ dr, de, Y α,,x = gx α,,x ady α,,x ξ α,,x = h, X α,,x, 0 < a.., A α,,x i a RCLL odecreaig P -meaurable proce wih = 0 ad uch ha 0 Y α,,x ξ α,,x da α,,x,c = 0 a.. ad ΔA α,,x,d A α,,x = ΔA α,,x 1 {Y α,,x =ξα,,x } a.. Here A α,,x,c deoe he coiuou par of A ad A α,,x,d i dicoiuou par. I he paricular cae whe h,x gx, he obacle ξ α,,x aifie for all F - predicable oppig ime τ, ξ τ ξ τ a.., which implie he coiuiy of he proce A α,,x ee [22. I he followig, for each α A, Y α,,x will be alo deoed by Y α,,x, Noe ha i value a ime ca be ake a deermiiic modulo a chage of i value o a P -ull e. Uig [22, heorem 3.2, we ge ha for each α A, 2.9 u α, x =Y α,,x = Y α,,x, [gx α,,x. [gx α,,x. By uig hee equaliie, we ca reduce our mixed opimal oppig/corol problem 2.6 o a opimal corol problem for refleced BSDE. heorem 2 characerizaio of he value fucio. For each, x [0, R, he value fucio u, x of he mixed opimal oppig/ corol problem 2.6 aifie 2.10 u, x = up u α, x = up α A α A Y α,,x, [gx α,,x. hi key propery will be ued o olve our mixed problem. We poi ou ha i he claical cae of liear expecaio, hi approach allow u o provide aleraive proof of he DPP o hoe give i he previou lieraure. Remark 3. Some mixed opimal corol/oppig problem wih oliear expecaio have bee udied i [5, 22. I hee paper, he reward proce doe o deped o he corol, which yield he characerizaio of he value fucio a he oluio of a RBSDE. hi i o he cae here. 3. Weak DPP. I hi ecio, we prove a weak DPP for our mixed opimal corol/oppig problem 2.6. o hi purpoe, we fir provide ome pliig properie for he forward-backward yem We he how ome meaurabiliy properie of he fucio u α, x, defied by 2.9, wih repec o boh he ae variable x ad he corol α. Uig hee reul, we how he exiece of ε-opimal corol aifyig ome appropriae meaurabiliy properie. Moreover, we eablih a Faou lemma for RBSDE, where he limi ivolve boh ermial codiio ad ermial ime. Uig hee reul, we he prove a ub- rep., uper- opimaliy priciple of dyamic programmig, ivolvig he u..c. rep., l..c. evelope of he value fucio.

8 WEAK DPP WIH E f -EXPECAIONS Spliig properie. Le [0,. For each ω, le ω := ω r 0 r ad ω := ω r ω r. We hall ideify he pah ω wih ω, ω, which mea ha a pah ca be pli io wo par: he pah before ime ad he -ralaed pah afer ime. Le α be a give corol i A. We how below he followig: a ime, forfixed pa pah ω := ω, he proce α ω,. which oly deped o he fuure pah ω i a -admiible corol, ha i, α ω,. A ;furhermore,he crierium Y α,0,x ω,. from ime coicide wih he oluio of he refleced BSDE drive by W ad Ñ, corolled by α ω,. ad aociaed wih iiial ime ad iiial ae codiio X α,0,x ω. We iroduce he followig radom variable defied o Ω by S : ω ω ; : ω ω. Noe ha hey are idepede. For each ω Ω, we have ω = S ω+ ω1,, or equivalely ω r = ω r + ωr1, r, for all r [0,. For all pah ω, ω Ω, ω, ω deoe he pah uch ha he pa rajecory before i ha of ω, adhe-ralaed rajecory afer i ha of ω. hi ca alo be wrie a ω, ω := ω + ω 1,. Noe ha for each ω Ω, we have ω, ω = ω. Lemma 4. Le [0,. Le Z H 2. here exi a P -ull e N uch ha for each ω i he compleme N c of N, eig ω := ω = ω., he proce Z ω, deoed alo by Z ω,. defied by Z ω, :Ω [, R ;ω,r Z r ω, ω belog o H 2. Moreover, if Z A,heZ ω, A. hi propery alo hold for all iiial ime [0,. More preciely, le [,. Le Z H 2 rep., A. For a.e. ω Ω, he proce Z ω,. =Z r ω, r belog o H 2 rep., A. Proof. Claically, we have E[ Z2 r dr =E[E[ Z2 r dr F < +. Uig he idepedece of wih repec o F ad he meaurabiliy of S wih repec o F,wederiveha [ [ E Zr 2 dr F = E Z r S, 2 dr F = F S < +, P-a.., where F ω :=E[ Z r ω, 2 dr. Le u ow prove ha he proce Z ω, : ω,r Z r ω, ω i P - meaurable. here exi a proce idiiguihable of Z r, ill deoed by Z r, which i meaurable wih repec o he predicable σ-algebra aociaed wih he filraio geeraed by W ad N ee [13, IV, ecio 79. We ca hu uppoe i hi proof wihou lo of geeraliy ha F rep., F i he filraio geeraed by W ad N rep., W ad N, ad P rep., P i i aociaed predicable σ-algebra. Suppoe we have how ha he map ψ :Ω [, Ω [0,; ω,r ω, ω,rip, P-meaurable. Now, we have Z ω, ω,r=z ψω,r for each ω,r Ω [,. Sice Z i P-meaurable, by compoiio, we derive ha Z ω, ip -meaurable. I remai o how he P, P-meaurabiliy of ψ. Recall ha he σ-algebra P i geeraed by he e H v,, where v [0,[adH i of he form H =

9 2098 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM {B i A i, 1 i }, where A i BR 2 ad 1 < 2 < v. I i hu ufficie o how ha ψ 1 H v, P. Noe ha ψ 1 H v, = H v,, where H = {ω Ω, ω, ω H}. Ifhereexii uch ha i ad ω i A i,he H =. Oherwie, we have H = {ω i ω A i for all i uch ha i >}. Hece H Fv, which implie ha ψ 1 H v, P. he proof i hu complee. Le Z H 2. Le u give a iermediary ime [0, ad a fixed pa pah ω. Noe ha he Lebegue iegral u Z rdr ω,. i equal a.. o he iegral u Z r ω,.dr. We ow how ha he ochaic iegral u Z rdw r ω,. coicide wih he ochaic iegral of he proce Z ω,. wih repec o he ralaed Browia moio W,hai, u Z r ω,.dwr. Lemma 5 pliig properie for ochaic iegral. Le [0,. Le Z H 2 ad K H 2 ν.hereexiap-ull e N which deped o uch ha for each ω N c,ad ω := ω, we have Z r ω, r H 2 ad K r ω, r H 2,ν, ad u u 3.1 Z r dw r ω, = Z r ω, dwr, P-a.., u u K r eñdr, de ω, = K r ω,,eñ dr, de, P-a.. E Remark 6. I he lieraure, he -ralaed Browia moio i ofe defied by W v := W +v W = W+v,0 v. ForeachZ H2 ad for each u, we have u Z rdwr = u Z 0 +r dw r a.. he ue of W hu allow u o avoid a chage of ime. he ame remark hold for he Poio radom meaure. Noe ha equaliy 3.1 i equivale o u Z rdw r ω, ω = u Z r ω, dwr ω forp-almo every ω Ω. he ame remark hold for he ecod equaliy. Proof. We hall oly prove he fir equaliy wih he Browia moio. he ecod oe wih he Poio radom meaure ca be how by imilar argume. Le u fir how ha equaliy 3.1 hold for a imple proce. Le a< ad le H L 2 F a. For each ω ω, ω =S ω, ω Ω, we have u H1 a, dw r ω, ω =H ω, ω ωu ωa u u = H ω, 1 a, dwr ω. Le ow Z H 2.LeuhowhaZaifie equaliy 3.1. he idea i o approximae Z by a appropriae equece of imple procee Z N o ha he equece Z N coverge i H 2 o Z ad ha, for almo every pa pah ω, he equece Z ω, N coverge o Z ω, ih 2.Foreach N, defie 1 i Z r := i=1 i 1 E Z u du 1 i, i+1 By iequaliy A.2 i he appedix, we have u Z r ω2 dr u Z rω 2 dr, adfor each ω Ωad u, u Z r ω Z rω 2 dr 0. Sice u Z2 r dr L1 Ω, i follow, by he Lebegue heorem for he codiioal expecaio, ha [ u 3.2 E Zr Z r 2 dr F 0 r.

10 WEAK DPP WIH E f -EXPECAIONS 2099 excep o a P -ull e N. Sice S i F -meaurable ad i idepeda of F, here exi a P -ull e icluded i he previou oe, uch ha for each ω N c, eig ω = ω,wehave [ u [ u E Zr Z r 2 dr F ω =E Zr ω, Z r ω, 2 dr [ u u = E Zr ω, dwr Z r ω, dwr. he ecod equaliy follow by he claical iomery propery. Now, for each quare iegrable marigale M, M 2 M i a marigale. Hece, for each ω N c,where N i a P -ull e icluded i he previou oe, eig ω = ω,wehave 3.4 [ u E Zr Z r 2 dr F ω [ u u 2 = E Zr dw r Z r dw r F ω [ u u 2 = E Zr dw r ω, Z r dw r ω,. For each N,iceZ i a imple proce, i aifie equaliy 3.1 everywhere, ha i, u Z r dw r ω, = u Z r ω, dwr. By he covergece propery 3.2, equaliie 3.3 ad 3.4, ad he uiquee propery of he limi i L 2,wederiveha for each ω N c, eig ω = ω, equaliy 3.1 hold. he proof i hu complee. Uig he above lemma, we ow how ha for each, for almo every ω Ω, eig ω = ω, he proce Y α,,x ω, coicide wih he oluio of he refleced BSDE o Ω [,, aociaed wih driver f α ω,,,η ω wih obacle hr, X α ω,,,x α ω,,x ω r ad filraio F, ad drive by W ad Ñ. o implify oaio, will be replaced by i he followig. I paricular Y α,,x ω, will be imply deoed by Y α,,x ω,.. heorem 7 pliig properie for he forward-backward yem. Le [0,, α A,ad [,. here exi a P -ull e N which deped o ad uch ha for each ω N c, eig ω = ω, he followig properie hold: here exi a uique oluio Xr α ω,,,η ω r i S 2 of he followig SDE: 3.5 X α ω,,,η ω r = η ω+ + + r r r bxv α ω,,,η ω,α v ω, dv σx α ω,,,η ω v E,α v ω, dw v βx α ω,,,η ω v,α v ω,,eñ dv, de, where η ω :=X α ω,,x ω. We alo have Xr α,,x ω,. =Xr α ω,,,η ω, r P-a.. here exi a uique oluio Yr α ω,,,η ω,zr α ω,,,η ω,kr α ω,,,η ω i S 2 H 2 H 2,ν of he refleced BSDE o Ω [, drive by W ad Ñ r

11 2100 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM ad aociaed wih filraio F, driver f α ω,.,,η ω, ad obacle hr, Xr α ω,,,η ω. We have he followig: Yr α,,x ω,. =Yr α ω,.,,η ω, r, P-a.., ω,. =Z α ω,,,η ω Z α,,x r Y α,,x r ad K α,,x r ω,. = Kr α ω,,,η ω, r, dp dr-a.., ω,. =Y α ω,,,η ω = u α ω,, η ω, P-a.. Proof. Recall ha by Lemma 4, here exi a P -ull e N uch ha for each ω N c, he proce α ω, :=α r ω, r belog o A. Le u how he fir aerio. o implify he expoiio, we uppoe ha here i o Poio radom meaure. here exi a P -ull e, ill deoed by N, icluded i he above oe uch ha for each ω N c, eig ω = ω, Xr α,,x ω,. =η ω+ r r bxv α,,x ω,.,α v ω,.dv + σxv α,,x,α v dw v ω,., o [,, P -a.. Now, by he fir equaliy i Lemma 5, here exi a P -ull e N uch ha for each ω N c, eig ω = ω,wehave r σx α,,x v,α v dw v ω,. = r σx α ω,,,η ω v,α v ω, dw v, P-a.., which implie ha he proce Xr α,,x ω, r [, i a oluio of SDE 3.5, ad he, by uiquee of he oluio of hi SDE, we have Xr α,,x ω,. =Xr α ω,.,,η ω, r, P-a.. Le u how he ecod aerio. Fir, oe ha ice he filraio F i he compleed filraio of he aural filraio of W ad Ñ wih repec o he iiial σ-algebra BΩ, we have a marigale repreeaio heorem for F - marigale wih repec o W ad Ñ. Hece, here exi a uique oluio Yr α ω,,,η ω,zr α ω,,,η ω,kr α ω,,,η ω r i S 2 H 2 H 2,ν of he refleced BSDE o Ω [, drivebyw ad Ñ ad aociaed wih filraio F ad wih obacle hr, Xr α ω,,,η ω. Equaliie 3.6 he follow from imilar argume a above ogeher wih he uiquee of he oluio of a Lipchiz RBSDE. Equaliy 3.7 i obaied by akig r = i equaliy 3.6 ad by uig he defiiio of u α ω,.. Remark 8. Iheaboveproof,wehavereaedheP-ull e iue carefully. We re ha all he filraio are compleed wih repec o BΩ ad P. he uderlyig probabiliy pace i hu alway he compleio of he iiial probabiliy pace Ω, BΩ,P ha i, Ω, F,P. Noe ha he P -ull e remai alway he ame, which i paricularly impora for ochaic iegral ee Lemma 5 ad alo for BSDE becaue he oluio of a BSDE i uique up o a P -ull e. Moreover, i he proof of Lemma 5, he choice of he equece of ep fucio approximaig he proce Z i appropriae o hadle he iue of P -ull e. Noe ha heorem 7 applied o he impler cae whe α A eure ha he oluio of 2.8 wih replaced by coicide o [, Ω wih he oluio i S 2 H 2 H 2 ν of he refleced BSDE imilar o 2.8 bu drive by W ad Ñ iead of W ad Ñ, ad aociaed wih F.

12 WEAK DPP WIH E f -EXPECAIONS Meaurabiliy properie ad ε-opimal corol. We eed o how a meaurabiliy propery of he fucio u α, x wih repec o corol α ad iiial codiio x. o hi purpoe, we fir provide a prelimiary reul, which will allow u o hadle he olieariy of he expecaio. Propoiio 9. Le Ω, F,P be a probabiliy pace. For each q 0, wedeoe by L q he e L q Ω, F,P. Suppoe ha he Hilber pace L 2 equipped wih he uual calar produc i eparable. Le g : R R be a Borelia fucio uch ha gx C1 + x p for each real x wih p 0. hemapϕ g defied by ϕ g : L 2p L 2 L 2 ; ξ g ξ =gξ i he B L 2p L 2 /BL 2 -meaurable, where BL 2 i he Borelia σ-algebra o L 2, ad B L 2p L 2 i he σ-algebra iduced by BL 2 o L 2p L 2. he proof of hi propoiio i popoed uil he appedix. Uig hi reul, we ow how he followig meaurabiliy propery. heorem 10. Le [0,. he map α, x u α, x; A R R, i B A BR/BR-meaurable, where B A deoe he σ-algebra iduced by BH2 o A. Proof. Recall ha u α, x =Y α,,x, [gx α,,x i alo deoed by Y α,,x, [ h., X. α,,x. Le x 1,x 2 R, adα 1,α 2 A. By claical eimae o diffuio procee ad he aumpio made o he coefficie, we ge [ 3.8 E up X α1,,x 1 r X α2,,x 2 r 2 C α 1 α 2 2 H + 2 x1 x 2 2. r We iroduce he map Φ : A R S2 L2 S2 ;α, x, ζ,ξ Y α,,x, [η,ξ, where Y α,,x, [ζ,ξ deoe here he oluio a ime of he refleced BSDE aociaed wih driver f α,,x := fα r,r,xr α,,x,.1 r, obacle η <, ad ermial codiio ξ. By he eimae o RBSDE ee he appedix i [14, uig he Lipchiz propery of f w.r.. x, α ad eimae 3.8, for all x 1,x 2 R, α 1,α 2 A, η1,η2 S 2 ad ξ 1,ξ 2 L 2,wehave Y α1,,x 1, [η 1,ξ 1 Y α2,,x 2, [η 2,ξ 2 2 C α 1 α 2 2 H 2 + x1 x η 1 η 2 2 S 2 + ξ1 ξ 2 2 L 2. he map Φ i hu Lipchiz-coiuou wih repec o he orm. 2 H S L. 2 Recall ha by aumpio, h, x C1 + x p, ad ha h i Lipchiz coiuou wih repec o x uiformly i. Oe ca derive ha he map S 2p S 2 S2, η h., η i Lipchiz-coiuou for he orm. 2 S ad hu Borelia, S 2 2 beig equipped wih he Borelia σ-algebra BS 2 ad i ubpace S2p S 2 wih he σ-algebra iduced by BS 2. Moreover, by Lemma 25, he Hilber pace L 2 i eparable. We ca hu apply Propoiio 9 ad ge ha he map L 2p L 2 L 2, ξ gξ i Borelia. We hu derive ha he map α, x α, x, h., X. α,,x,gx α,,x defied o A R ad valued i A R S2 L2 i B A BR/B A BR BS2 BL2 - meaurable. By compoiio, i follow ha he map α, x Y α,,x, [h., X. α,,x, = u α, x i meaurable. gx α,,x

13 2102 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM For each, wih, we iroduce he e A of rericio o [, ofhe corol i A. hey ca alo be ideified o he corol α i A which are equal o 0o[,. Le η L 2 F. Sice η i F -meaurable, up o a P -ull e, i ca be wrie a a meaurable map, ill deoed by η, of he pa rajecory ω. See he argume ued i he proof of Lemma 25 for deail. For each ω Ω, by uig he defiiio of he fucio u, wehave 3.9 u, η ω = up u α, η ω. α A By heorem 10 ogeher wih a meaurable elecio heorem, we how he exiece of early opimal corol for 3.9 aifyig ome pecific meaurabiliy properie. heorem 11 exiece of ε-opimal corol. Le [0,, [, [, ad η L 2 F. Le ε>0. here exi αε A uch ha, for almo every ω Ω, α ε ω, i ε-opimal for Problem 3.9, i he ee ha u, η ω u αε ω,, η ω + ε. Proof. Wihou lo of geeraliy, we may aume ha = 0. We iroduce he pace Ω:={ω r 0 r ; ω Ω}, equipped wih i Borelia σ-algebra deoed by B Ω, ad he probabiliy meaure P, which correpod o he image of P by S :Ω Ω; ω ω r r. he Hilber pace H 2 of quare-iegrable predicable procee o Ω [,, equipped wih he orm H 2, i eparable ee Lemma 25. Moreover, A i a cloed ube of H2. Alo, he pace Ω of pah RCLL before i Polih for he Skorohod meric. Now, a ee above, ice η i F -meaurable, up o a P -ull e, we ca uppoe ha i i of he form η S,whereη i B Ω- meaurable. Moreover, by heorem 10, he map ω, α u α, η ω i B Ω BA -meaurable wih repec o x, α. We ca hu apply [6, Propoiio 7.50 o he problem 3.9. Hece, here exi a map α ε : Ω A ; ω α ε ω,, which i uiverally meaurable, ha i U Ω/BA -meaurable, ad uch ha u, η ω u αε ω,, η ω + ε ω Ω. Here, U Ω deoe he uiveral σ-algebra o Ω. Le u ow apply Lemma 26 o X = Ω, o E = H 2, ad o probabiliy Q = P. By defiiio of U Ω ee, e.g., [6, we have U Ω B Q Ω, where B Q Ω deoe he compleio of B Ω wih repec o Q. Hece, here exi a map ˆα ε : Ω A ; ω ˆα ε ω, whichi Borelia, ha i B Ω/BA -meaurable, ad uch ha ˆα ε ω, =α ε ω, for P -almo every ω Ω. Sice H 2 i a eparable Hilber pace, for each ω, wehaveˆα ε u ω, ω = i βi,ε ωe i uω dp ω du-a.., where β i,ε ω =< ˆα ε ω,,e i > H 2 ad {e i,i N} i a couable orhoormal bai of H 2.Noehaβ i,ε i Borelia, ha i, B Ω/BR-meaurable. Le ᾱ ε : Ω A ; ω ᾱε ω, = i βi,ε ωe i. I i Borelia, ha i, B Ω/BA - meaurable. We ow defie a proce α ε o [0, Ωbyα ε r ω := i βi,ε S ωe i ω. I remai o prove ha i i P-meaurable. Noe ha β i,ε S i F -meaurable by compoiio. Sice he proce e i u u i P -meaurable, he proce β i,ε S e i u

14 WEAK DPP WIH E f -EXPECAIONS 2103 i P-meaurable. Ideed, if we ake e i of he form e i u = H1 r, u wih r ad H a radom variable Fr -meaurable, he he radom variable β i,ε S H i F r - meaurable ad hece he proce β i,ε S H1 r, i P-meaurable. he proce α ε i hu P-meaurable. Noe alo ha α ε ω, ω = i βi,ε ωe i ω, ω. Now, we have e i ω, ω = e i ω becaue e i ω deped o ω oly hrough ω. Hece, α ε ω, ω = ᾱ ε ω, ω, which complee he proof A Faou lemma for refleced BSDE. We eablih a Faou lemma for refleced BSDE, where he limi ivolve boh ermial codiio ad ermial ime. hi reul will be ued o prove a uper- rep., ub- opimaliy priciple ivolvig he l..c. rep., u..c. evelope of he value fucio u ee heorem 17. We fir iroduce ome oaio. A fucio f i aid o be a Lipchiz driver if f : [0, Ω R 2 L 2 ν R ω,, y, z, k fω,, y, z, k i P BR 2 BL 2 ν -meaurable, uiformly Lipchiz wih repec o y, z, k, ad uch ha f., 0, 0, 0 H 2. A Lipchiz driver f i aid o aify Aumpio 12 if he followig hold. Aumpio 12. Aume ha dp d-a. for each y, z, k 1,k 2 R 2 L 2 ν 2, f, y, z, k 1 f, y, z, k 2 γ y,z,k1,k2,k 1 k 2 ν wih γ :[0, Ω R 2 L 2 ν 2 L 2 ν ;ω,, y, z, k 1,k 2 γ y,z,k1,k2 ω,., uppoed o be P BR 2 BL 2 ν 2 -meaurable, uiformly bouded i L 2 ν, ad aifyig dp ω d dνe-a.., for each y, z, k 1,k 2 R 2 L 2 ν 2, he iequaliy γ y,z,k1,k2 ω, e 1. hi aumpio eure he compario heorem for BSDE wih jump ee [21, heorem 4.2. Le η be a give RCLL obacle proce i S 2 ad le f be a give Lipchiz driver. I he followig, we will coider he cae whe he ermial ime i a oppig ime ad he ermial codiio i a radom variable ξ i L 2 F. I hi cae, he oluio, deoed Y., ξ,z., ξ,k., ξ, of he refleced BSDE aociaed wih ermial oppig ime, driverf,obacleη <, ad ermial codiio ξ i defied a he uique oluio i S 2 H 2 H 2 ν of he refleced BSDE wih ermial ime, driver f, y, z, k1 { }, ermial codiio ξ, ad obacle η 1 < + ξ1.noeha Y, ξ =ξ,z, ξ =0,k, ξ =0for. We fir prove a coiuiy propery for refleced BSDE where he limi ivolve boh ermial codiio ad ermial ime. Propoiio 13 a coiuiy propery for refleced BSDE. Le > 0. Le η be a RCLL proce i S 2. Le f be a give Lipchiz driver. Le N be a o icreaig equece of oppig ime i, covergig a.. o a ed o. Leξ N be a equece of radom variable uch ha E[up ξ 2 < +, ad for each, ξ i F -meaurable. Suppoe ha ξ coverge a.. o a F -meaurable radom variable ξ a ed o. Suppoe ha 3.10 η ξ a.. Le Y., ξ ; Y., ξ be he oluio of he refleced BSDE aociaed wih driver f, obacle η < rep.. η <, ermial ime rep.,, ad ermial codiio ξ rep., ξ. We have Y 0, ξ = lim Y 0, + ξ a.. Whe for each, = a.., he reul ill hold wihou aumpio 3.10.

15 2104 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM By imilar argume a i he Browia cae ee, e.g., [16, oe ca prove he followig eimae o refleced BSDE, which will be ued i he proof of he above propoiio. Lemma 14. Le ξ 1,ξ 2 L 2 F ad η 1, η2 S2. Le f 1,f 2 be Lipchiz driver wih Lipchiz coa C>0. Fori =1, 2, ley i,z i,k i,a i be he oluio of he refleced BSDE wih driver f i,ermialime, obacle η i,adermial codiio ξ i. For [0,, ley := Y 1 Y 2, η := η 1 η2, ξ := ξ1 ξ 2 ad f :=f 1, Y 2,Z 2,k 2 f, Y 2,Z 2,k. 2 he, we have [ 3.11 Y 2 S E[ξ K 2 +E f 2 d + φ 2 up η 0 < 0 L 2, where he coa K i uiveral, ha i, deped oly o he Lipchiz coa C ad, ad where he coa φ deped oly o C,, η i S 2, ξ i L 2, ad f i, 0, 0, 0 H 2, i =1, 2. Proof of Propoiio 13. Le N. We apply 3.11 wih f 1 = f1, f 2 = f1, ξ 1 = ξ, ξ 2 = ξ, η 1 = η 1 < + ξ 1 <,adη 2 = η 1 < + η 1 < + ξ1 <. We have Y 1 = Y., ξ a.. Moreover, ice by aumpio η ξ a.., we have Y 2 = Y., ξ a.. NoehaY 2,Z2,k2 =ξ,0, 0 a.. o { }. Wehu obai [ Y 0, ξ Y 0, ξ 2 K E[ξ ξ 2 +E f 2, ξ, 0, 0d φ up η η < L 2, where he coa K deped oly o he Lipchiz coa C of f ad he ermial ime, ad where he coa φ deped oly o C,, η S 2,up ξ L 2,ad f, 0, 0, 0 H 2. Sice he obacle η i righ-coiuou ad a.., we have lim + up η η L 2 =0. he righ member of 3.12 hu ed o 0 a ed o +. he reul follow. Remark 15. Compared wih he cae of orefleced BSDE ee [21, Propoiio A.6, here i a exra difficuly due o he preece of he obacle ad he variaio of he ermial ime. he addiioal aumpio 3.10 o he obacle i here required o obai he reul. Uig Propoiio 13, we derive a Faou lemma i he refleced cae, where he limi ivolve boh ermial codiio ad ermial ime. Propoiio 16 a Faou lemma for refleced BSDE. Le > 0. Le η be a RCLL proce i S 2. Le f be a Lipchiz driver aifyig Aumpio 12. Le N be a oicreaig equece of oppig ime i, covergig a.. o a ed o. Le ξ N be a equece of radom variable uch ha E[up ξ 2 < +, adforeach, ξ i F -meaurable. Le Y., ξ ; Y., lim if + ξ ad Y., lim up + ξ be he oluio of he refleced BSDE aociaed wih driver f, obacle η < rep., η <, ermial ime rep.,, ad ermial codiio ξ rep., lim if + ξ ad lim up + ξ.

16 3.13 Suppoe ha WEAK DPP WIH E f -EXPECAIONS 2105 lim if + ξ η he Y 0, lim if + ξ rep., lim if + Y 0, ξ rep., Y 0, lim up ξ η + lim up + a.. ξ lim up Y 0, ξ. + Whe for each, = a.., he reul ill hold wihou aumpio Proof. We pree oly he proof of he fir iequaliy, ice he ecod oe i obaied by imilar argume. For all, we have by he moooiciy of refleced BSDE wih repec o ermial codiio, Y 0, if p ξ p Y 0, ξ. We derive ha lim if Y 0, + ξ lim if Y 0, + if p ξp = Y 0, lim if + ξ, where he la equaliy follow from aumpio 3.13 ogeher wih Propoiio A weak DPP. We will ow provide a weak DPP ha i boh a weak ub- ad uper-opimaliy priciple of dyamic programmig, ivolvig, repecively, he u..c. evelope u ad he l..c. evelope u of he value fucio u, defied by u, x := lim up u,x ; u, x := lim if,x,x,x,x u,x, x [0, R. We ow defie he map ū ad ū for each, x [0, R by ū, x :=u, x1 < + gx1 = ; ū, x :=u, x1 < + gx1 =. Noe ha he fucio ū ad ū are Borelia. We have ū u ū ad ū,.= u,.=ū,. =g.. Noe ha ū rep., ū i o ecearily upper rep., lower emicoiuou o [0, R, ice he ermial reward g i oly Borelia. o prove he weak DPP, we will ue he pliig properie heorem 7, he exiece of ε-opimal corol heorem 11, ad he Faou lemma for RBSDE Propoiio 16. heorem 17 a weak DPP. he value fucio u aifie he followig weak ubopimaliy priciple of dyamic programmig: for each [0, ad for each oppig ime, we have 3.14 u, x up up E α,,x [, τ hτ,x α,,x τ 1 τ< +ū, X α,,x 1 τ. α A τ Moreover, he followig weak uperopimaliy priciple of dyamic programmig hold: for each [0, ad for each oppig ime, we have 3.15 u, x up α A up τ E α,,x, τ [ hτ,x α,,x τ 1 τ< +ū, X α,,x 1 τ. Remark 18. he proof give below alo how ha hi weak DPP ill hold wih replaced by α i iequaliie 3.14 ad 3.15, give a family of oppig ime idexed by corol { α,α A }.

17 2106 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM Noe ha o regulariy codiio i required o g o eure hi weak DPP, eve hi i o he cae i he lieraure eve for claical expecaio ee [8, [7, [4. Moreover, our DPP are roger ha hoe give i hee paper, where iequaliy 3.14 rep., 3.15 i eablihed wih u rep., u iead of ū rep., ū. Now, ū u ad ū u. Before givig he proof, we iroduce he followig oaio. For each ad each ξ i L 2 F, we deoe by Y α,,x., ξ,z α,,x., ξ,k α,,x., ξ he uique oluio i S 2 H 2 H 2 ν of he refleced BSDE wih driver f α,,x 1 { }, ermial ime,ermial codiio ξ, ad obacle hr, Xr α,,x 1 r< + ξ1 r. Proof. By eimae for refleced BSDE ee [14, Propoiio 5.1, he fucio u ha a mo polyomial growh a ifiiy. Hece, he radom variable ū, X α,,x ad ū, X α,,x are quare iegrable. Wihou lo of geeraliy, o implify oaio, we uppoe ha =0. We fir how he ecod aerio which i he mo difficul, or equivalely: [ 3.16 up Y α,0,x 0, ū, X α,0,x u0,x. α A Le. For each N, we defie 3.17 := 2 1 k=0 k 1 Ak + 1 =, where k := k+1 2 ad A k := { k 2 < k+1 2 }.Noeha ad. O { = } we have = for each. Wehugeū,X α,0,x =ū, X α,0,x for each o { = }. Moreover, o { <}, he lower emicoiuiy of ū o [0,[ R ogeher wih he righ coiuiy of he proce X α,0,x implie ha ū, X α,0,x lim if ū,x α,0,x + a.. Hece, by he compario heorem for refleced BSDE, we ge O { <}, wehave [ Y α,0,x 0, ū, X α,0,x Y α,0,x 0, [ lim if ū,x α,0,x + lim if ū,x α,0,x lim if h,x α,0,x = lim h,x α,0,x =h, X α,0,x a.. by he regulariy properie of h o [0,[ R. O { = }, = ad ū,x α,0,x =ū,x α,0,x =gx α,0,x = h,x α,0,x. Hece, we have lim if + ū,x α,0,x h, X α,0,x a.. Codiio 3.13 i hu aified wih ξ =ū,x α,0,x adξ = h, X α,0,x. We ca hu apply he Faou lemma for refleced BSDE Propoiio 16. We hu ge [ [ Y α,0,x 0, ū, X α,0,x Y α,0,x 0, lim if ū,x α,0,x [ lim if Y α,0,x 0, ū,x α,0,x..

18 WEAK DPP WIH E f -EXPECAIONS 2107 Le ε>0. Fix N. For each k<2 1, le A k be he e of he rericio o [ k,ofhecorolα i A. By heorem 11, here exi a P -ull e N which deped o ad ε uch ha for each k<2 1, here exi a ε-opimal corol corol α,ε,k i A 0 k = A k for he corol problem a ime k wih iiial codiio η = X α,0,x k ha i aifyig he iequaliy 3.19 u k,x α,0,x k k ω u α,ε,k k ω, k,x α,0,x k ω + ε for each ω N c. Uig he defiiio of he map u α,ε,k k ω, ogeher wih he pliig propery for refleced BSDE 3.7, we derive ha here exi a P -ull e N which coai he above oe uch ha for each ω N c ad for each k<2 1, we have u α,ε,k k ω, k,x α,0,x k k ω = Y α,ε,k k ω,, k,x α,0,x k, k k k ω = Y α,ε,k, k,x α,0,x k k, k ω. Here, Y α,ε,k, k,x α,0,x k., =Y f α,ε,k,k,x α,0,x k., [ hr, X α,ε,k, k,x α,0,x k r deoe he oluio of he refleced BSDE aociaed wih ermial ime,obacle hr, X α,ε,k, k,x α,0,x k r k r, ad driver f α,ε,k, k,x α,0,x k r, y, z, k :=fα,ε,k r,r,x α, k,x α,0,x k r,y,z,k. Se α,ε := k<2 1 α,ε,k 1 Ak + α 1 {= }. Sice for each k, A k F k,here exi a P -ull e N uch ha, o N c,foreachk<2 1, we have he followig equaliie: Y α,ε,k, k,x α,0,x k k, 1 Ak = Y f α k,,ε,k,k,x α,0,x k 1Ak = Y f α,ε,,x k, α,0,x 1 Ak [ h [ h r, X α,ε,k, k,x α,0,x k r 1 Ak r, X α,ε,,x α,0,x r 1 Ak, where, for a give driver f, Y f1a k deoe he oluio of he refleced BSDE aociaed wih f1 Ak.WehugeY α,ε,k, k,x α,0,x k k, 1 Ak = Y α,ε,,x α,0,x, 1 Ak o N c.uig iequaliie 3.19, we ge ū,x α,0,x = u k,x α,0,x k 1 Ak + gx α,,x 1 {= } 0 k<2 1 Y α,ε,,x α,0,x, + ε o N c. We e α,ε := α 1 < + α,ε 1. Noe ha α,ε A. Uig he compario heorem ogeher wih he eimae o refleced BSDE ee [14, we obai [ Y α,0,x 0, [ū,x α,0,x Y α,0,x 0, Y α,ε,,x α,0,x, + Kε = Y α,ε,0,x 0, + Kε, where he la equaliy follow from he flow propery. Sice Y α,ε,0,x 0, u0,x, uig 3.18, we ge Y α,0,x 0, [ū, X α,0,x Y α,0,x 0, [ū,x α,0,x u0,x+kε. akig he upremum o α Aad leig ε ed o 0, we obai iequaliy I remai o how he fir aerio. I i ufficie o how ha for each, [ 3.20 u0,x up Y α,0,x 0, ū, X α,0,x. α A

19 2108 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM Le. Le α A. A above, we approximae by he equece of oppig ime N defied above. Le N. By applyig he flow propery for refleced BSDE, we ge Y α,0,x 0, = Y α,0,x 0, [Y α,,x α,0,x,. By imilar argume a i he proof of he uperopimaliy priciple bu wihou uig he exiece of ε-opimal corol, we derive ha Y α,,x α,0,x, ū,x α,0,x a.. By he compario heorem for refleced BSDE, i follow ha [ Y α,0,x 0, = Y α,0,x 0, Y α,,x α,0,x, Y α,0,x 0, [ū,x α,0,x. Uig he Faou lemma for refleced BSDE Propoiio 16, we ge Y α,0,x 0, lim up Y α,0,x 0, [ū,x α,0,x Y α,0,x 0, [ lim up ū,x α,0,x. Uig he upper emicoiuiy propery of ū o [0,[ R ad ū,x=gx, we obai [ Y α,0,x 0, Y α,0,x 0, lim up ū,x α,0,x Y α,0,x 0, [ū, X α,0,x. Sice α Ai arbirary, we ge iequaliy 3.20, which complee he proof. 4. Noliear HJB variaioal iequaliie Some exeio of compario heorem for BSDE ad refleced BSDE. We provide wo reul which will be ued o prove ha he value fucio u, defied by 2.6, i a weak vicoiy oluio of ome oliear HJB variaioal iequaliie ee heorem 22. We fir how a ligh exeio of he compario heorem for BSDE give i [21, from which we derive a compario reul bewee a BSDE ad a refleced BSDE. Lemma 19. Le 0 [0, ad le 0. Le ξ 1 ad ξ 2 L 2 F. Le f 1 be a driver. Le f 2 be a Lipchiz driver wih Lipchiz coa C>0, aifyig Aumpio 12. For i =1, 2, lex i,πi,li be a oluio i S2 H 2 H 2 ν of he BSDE aociaed wih driver f i,ermialime, ad ermial codiio ξ i.suppoe ha f 1, X 1,π 1,l 1 f 2, X 1,π 1,l 1 0, d dp a.., ad ξ 1 ξ 2 + ε a.., where ε i a real coa. he, for each [ 0,, we have X 1 X 2 + εe C a.. Proof. From iequaliy 4.22 i he proof of he compario heorem i [21, we derive ha X 1 0 X 2 0 e C E [H 0, ε F 0 a.., where C i he Lipchiz coa of f 2, ad H 0, [0, i he oegaive marigale aifyig dh 0, = H [β 0, dw + E γuñd, du wih H 0, 0 =1,β beig a predicable proce bouded by C. he reul follow. Propoiio 20 a compario reul bewee a BSDE ad a refleced BSDE. Le 0 [0, ad le 0.Leξ 1 L 2 F ad le f 1 be a driver. Le X 1,π 1,l 1 be a oluio of he BSDE aociaed wih f 1,ermialime, ad ermial codiio ξ 1.Leξ 2 S2 ad le f 2 be a Lipchiz driver wih Lipchiz coa C>0which aifie Aumpio 12. Le Y 2 be he oluio of he refleced BSDE aociaed

20 WEAK DPP WIH E f -EXPECAIONS 2109 wih f 2,ermialime, ad obacle ξ 2. Suppoe ha 4.1 f 1, X 1,π 1,l 1 f 2, X 1,π 1,l 1, 0, d dp -a.. ad X 1 ξ 2 + ε, 0 a.. he, we have X 1 Y 2 + εe C, 0 a.. Proof. Le [ 0,. By he characerizaio of he oluio of he RBSDE a he value fucio of a opimal oppig problem ee heorem 3.2 i [21, Y 2 = e up τ [, E f 2,τ ξτ 2. By Lemma 19, for each τ [,, X 1 E f 2,τ ξτ 2 +e C ε. akig he upremum over τ [,, he reul follow Lik bewee he mixed corol problem ad HJB equaio. We iroduce he followig HJB variaioal iequaliy HJBVI: 4.2 miu, x h, x, if α A u, x Lα u, x f u,x=gx,x R, α,, x, u, x, σ u, x,b α u, x =0,, x [0, R, where L α := A α + K α, ad for φ C 2 R, A α φx := 1 2 σ2 x, α 2 φ x+bx, α φ 2 x adbα φx :=φx + βx, α, φx, K α φx := φ E φx + βx, α, e φx xβx, α, eνde. Defiiio 21. Afuciou i aid o be a vicoiy uboluio of 4.2 if i i u..c. o [0, R, ad if for ay poi 0,x 0 [0,[ R ad for ay φ C 1,2 [0, R uch ha φ 0,x 0 =u 0,x 0 ad φ u aai i miimum a 0,x 0, we have 4.3 miu 0,x 0 h 0,x 0, φ if α A f 0,x 0 L α φ 0,x 0 α, 0,x 0,u 0,x 0, σ φ 0,x 0,B α φ 0,x 0 0. I oher word, if u 0,x 0 >h 0,x 0,he if α A φ 0,x 0 L α φ 0,x 0 f α, 0,x 0,u 0,x 0, σ φ 0,x 0,B α φ 0,x 0 0. A fucio u i aid o be a vicoiy uperoluio of 4.2 if i i l..c. o [0, R, ad if for ay poi 0,x 0 [0,[ R ad ay φ C 1,2 [0, R uch

21 2110 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM ha φ 0,x 0 =u 0,x 0 ad φ u aai i maximum a 0,x 0, we have miu 0,x 0 h 0,x 0, if α A φ 0,x 0 L α φ 0,x 0 f α, 0,x 0,u 0,x 0, σ φ I oher word, we have boh u 0,x 0 h 0,x 0 ad 4.4 if α A φ 0,x 0 L α φ 0,x 0 f α, 0,x 0,u 0,x 0, σ φ 0,x 0,B α φ 0,x ,x 0,B α φ 0,x 0 0. Uig he weak DPP give i heorem 17 ad Propoiio 20, we ow prove ha he value fucio of our problem i a weak vicoiy oluio of he above HJBVI. heorem 22. he value fucio u, defied by 2.6, i a weak vicoiy oluio of he HJBVI 4.2, i he ee ha i u..c. evelope u i a vicoiy uboluio of 4.2 ad i l..c. evelope u i a vicoiy uperoluio of 4.2 wih ermial codiio u,x=gx. Proof. We fir prove ha u i a uboluio of 4.2. Le 0,x 0 [0,[ R ad φ C 1,2 [0, R be uch ha φ 0,x 0 =u 0,x 0 adφ, x u, x for all, x [0, R. Wihou lo of geeraliy, we ca uppoe ha he miimum of u φ aaied a 0,x 0 i ric. Suppoe for coradicio ha u 0,x 0 >h 0,x 0 ad ha if α A φ 0,x 0 L α φ 0,x 0 f α, 0,x 0,φ 0,x 0, σ φ 0,x 0,B α φ 0,x 0 > 0. By uiform coiuiy of K α φ ad B α φ :[0, R L 2 ν wih repec o α, weca uppoe ha here exi ɛ>0,η ɛ > 0 uch ha for all, x uch ha η ɛ < ad x x 0 η ɛ,wehaveφ, x h, x+ɛ ad 4.5 φ, x Lα φ, x f α,, x, φ, x, σ φ, x,b α φ, x ɛ α A. We deoe by B ηε 0,x 0 he ball of radiu η ε ad ceer 0,x 0. By defiiio of u, here exi a equece,x i B ηε 0,x 0, uch ha,x,u,x 0,x 0,u 0,x 0. Fix N. Leα be a arbirary corol of A ad X α,,x he aociaed ae proce. We defie he oppig ime α, a α, := 0 + η ɛ if{, X α,,x x 0 η ɛ }.

22 WEAK DPP WIH E f -EXPECAIONS 2111 Le ψ α, x := φ, x +Lα φ, x. Applyig Iô lemma o φ, X α,,x, we derive ha φ, X α,,x, σ φ, X α,,x,b α φ, X α,,x ; [, α, i he oluio of he BSDE aociaed wih he driver proce ψ α, X α,,x, ermial ime α,, ad ermial value φ α,,x α,,x. By 4.5 ad by defiiio α, of α,,wege 4.6 ψ α, X α,,x f α,,x α,,x,φ, X α,,x, σ φ, X α,,x,bφ, X α,,x + ɛ for each [, α,. hi iequaliy give a relaio bewee he driver ψ α, X α,,x adfα, of wo BSDE. Now, ice he miimum 0,x 0 i ric, here exi γ ɛ uch ha 4.7 u, x φ, x γ ɛ o [0, R \ B ηɛ 0,x 0. We have φ α,, X α,,x α, =φ, Xα,,x 1 < α, + φ α,,x α,,x α, 1 α,, a.. o implify oaio, e δ ε := miɛ, γ ɛ. Uig 4.7 ogeher wih he defiiio of α,,wege φ, X α,,x h, X α,,x +δ ε 1 < α, +u α,,x α,,x α, +δ ε1 = α,, α, a.. hi, ogeher wih iequaliy 4.6 o he driver ad he above compario heorem bewee a BSDE ad a refleced BSDE ee Propoiio 20 lead o φ,x Y α,,x, α, [h, X α,,x 1 < α, + u α,,x α,,x 1 α, = α+δ ε K, where K i a poiive coa which oly deped o ad he Lipchiz coa of f. Now, recall ha,x,u,x 0,x 0,u 0,x 0 ad φ i coiuou wih φ 0,x 0 = u 0,x 0. We ca hu aume ha i ufficiely large o ha φ,x u,x δ ε K/2. Hece, u,x Y α,,x, α, [h, Xα,,x 1 < α, + u α,,x α,,x 1 α, = α+δ εk/2. A hi iequaliy hold for all α A ad ice u ū, we ge a coradicio of he ubopimaliy priciple of DPP 3.14 ee alo Remark 18. We ow prove ha u i a vicoiy uperoluio of 4.2. Le 0,x 0 [0,[ R ad φ C 1,2 [0, R be uch ha φ 0,x 0 =u 0,x 0 adφ, x u, x for all, x [0, R. Wihou lo of geeraliy, we ca uppoe ha he maximum i ric i 0,x 0. Sice he oluio Y α,0,x0 ay above he obacle, for each α A,wehaveu 0,x 0 h 0,x 0. Our aim i o how ha iequaliy 4.4 hold.

23 2112 R. DUMIRESCU, M.-C. QUENEZ, AND A. SULEM Suppoe for coradicio ha hi iequaliy doe o hold. By coiuiy, we ca uppoe ha here exi α A, ɛ>0, ad η ɛ > 0uch ha for all, x wih η ɛ < ad x x 0 η ɛ,wehave 4.8 φ, x Lα φ, x f α,, x, φ, x, σ φ, x,b α φ, x ɛ. We deoe by B ηε 0,x 0 he ball of radiu η ε ad ceer 0,x 0. Le,x be a equece i B ηε 0,x 0 uch ha,x,u,x 0,x 0,u 0,x 0. We iroduce he ae proce X α,,x aociaed wih he above coa corol α ad defie he oppig ime a := 0 + η ɛ if{, X α,,x x 0 η ɛ }. By Iô formula, he proce φ, X α,,x, σ φ, Xα,,x,B α φ, X α,,x ; [, i he oluio of he BSDE aociaed wih ermial ime, ermial value φ,x α,,x, ad driver ψ α, X α,,x. he defiiio of he oppig ime ad iequaliy 4.8 lead o 4.9 ψ α, X α,,x f α,, X α,,x,φ, X α,,x, σ φ, X α,,x,b α φ, X α,,x for d dp -a.. Now, ice he maximum 0,x 0 i ric, here exi γ ɛ which deped o η ɛ uch ha u, x φ, x +γ ɛ o [0, R \ B ηɛ 0,x 0 which implie φ,x α,,x u,x α,,x γ ɛ. Hece, uig iequaliy 4.9 o he driver, ogeher wih he compario heorem for BSDE, we derive ha φ,x =E ψα, [φ,x α,,x E α,,x, Eα,,x, [u,x α,,x γ ɛk, [u,x α,,x γ ɛ where he ecod iequaliy follow from a exeio of he compario heorem Lemma 19. We ca aume ha i ufficie large o ha φ,x u,x δ ε K/2. We hu ge 4.10 u,x E α,,x, [u,x α,,x γ ɛk/2. Sice u aifie he uperopimaliy DPP heorem 17, we have u,x E α,,x, [ū,x α,,x. Sice ū u, hi iequaliy wih 4.10 lead o a coradicio. Remark 23. Whe g i oly Borelia, he weak oluio of he HJB equaio 4.2 i geerally o uique, eve i he deermiiic cae a reed i [2, [1, [3. Noe ha whe g i l..c., he value fucio u of our problem ca be how o be he miimal l..c. vicoiy uperoluio of he HJB equaio 4.2, wih ermial value greaer ha g by uig imilar argume a i he proof of [15, heorem 6.5.

24 WEAK DPP WIH E f -EXPECAIONS 2113 Noe alo ha he paper [2 ee alo [1 provide a characerizaio of he u..c. evelope u of he value fucio of he deermiiic corol problem u, x := up α A gx,x,α, which correpod o our problem wih σ = f = 0 ad o oppig ime corol. More preciely, whe g i u..c., he map u i characerized a he uique u..c. vicoiy oluio of he HJB equaio i.e., aifie u,x=gx ad he aalogou of 4.3 bu wih a equaliy. he proof i baed o PDE argume ad deermiiic corol heory. A iereig furher developme of our paper ad of [8 would be o udy aalogou properie i he ochaic cae. Appedix A. Appedix. We give here ome meaurabiliy reul which are ued i ecio 3.2. We ar by he proof of Propoiio 9. o hi purpoe, we fir provide he followig lemma. Lemma 24. Le Ω, F, P be a probabiliy pace. Suppoe ha he Hilber pace L 2 := L 2 Ω, F,P equipped wih he uual calar produc i eparable. Le F L 2. Coider a equece of fucio g N uch ha for each, g : R R i Borelia wih g x C1 + x p. Suppoe ha equece g N coverge poiwie. Le g be he limi, defied for each x R by gx := lim + g x. Suppoe alo ha for each N, hemapψ g,f deoed alo by ψ g defied by ψ g : L 2p L 2 R; ξ E[g ξ F i Borelia, L 2p L 2 beig equipped wih he σ-algebra iduced by BL 2. he, he map ψ g deoed alo by ψ g,f defied by A.1 ψ g : L 2p L 2 R; ξ E[gξ F i Borelia. Proof. By he Lebegue heorem, for each ξ L 2p L 2, we have ψ g ξ = E[gξ F = lim + E[g ξ F = lim + ψ g ξ. Sice he poiwie limi of a equece of R-valued meaurable map i meaurable, we derive ha he map ψ g i Borelia. Proof of Propoiio 9. Sice by aumpio he Hilber pace L 2 i eparable, here exi a couable orhoormal bai {e i,i N} of L 2. For each ξ L 2p L 2, we have ϕ g ξ =gξ = i ψg,i ξ e i i L 2,whereψ g,i ξ :=E[gξ e i foreachi N. Hece, i order o how he meaurabiliy of he map ψ, i i ufficie o how he meaurabiliy of he map ψ g,f, F L 2. For hi purpoe, we iroduce he e H of bouded Borelia fucio g : R R uch ha for each F L 2,hemapψ g,f i Borelia. Noe ha H i a vecor pace. Suppoe we have how ha for all real umber a, b wih a<b, 1 a,b[ H. he, by Lemma 24 ogeher wih a moooe cla heorem, we derive ha H i equal o he whole e of bouded Borelia fucio. Whe g i o bouded, he reul follow by approximaig g by a equece of bouded Borelia fucio, ad by uig Lemma 24. I remai o how ha for all a, b R wih a<b,wehave1 a,b[ H. Sice 1 a,b[ i l..c., i follow ha here exi a odecreaig equece g N of Lipchiz coiuou fucio akig heir value i [0, 1 uch ha for each x R, 1 a,b[ x := lim + g x. For each, iceg i Lipchiz coiuou, by uig he Cauchy Schwarz iequaliy, oe ca derive ha he map ψ g : L 2 R; ξ E[g ξ F i Lipchiz coiuou for he orm L 2, ad hece Borelia. he reul he follow from Lemma 24.