Research Article A Two-Mode Mean-Field Optimal Switching Problem for the Full Balance Sheet

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1 Hndaw Publhng Corporaon Inernaonal Journal of Sochac Analy Volume 14 Arcle ID page hp://dx.do.org/1.1155/14/ Reearch Arcle A wo-mode Mean-Feld Opmal Swchng Problem for he Full Balance Shee Boualem Djehche and Al Hamd Deparmen of Mahemac KH-Royal Inue of echnology 1 44 Sockholm Sweden Correpondence hould be addreed o Boualem Djehche; boualem@mah.kh.e Receved February 14; Acceped 4 May 14; Publhed 5 May 14 Academc Edor: Qng Zhang Copyrgh 14 B. Djehche and A. Hamd. h an open acce arcle drbued under he Creave Common Arbuon Lcene whch perm unrerced ue drbuon and reproducon n any medum provded he orgnal work properly ced. We conder he problem of wchng a large number of producon lne beween wo mode hgh producon and low producon. he wchng baed on he opmal expeced prof and co yeld of he repecve producon lne and conder boh de of he balance hee. Furhermore he producon lne are all aumed o be nerconneced hrough a couplng erm whch he average of all opmal expeced yeld. Inuvely h mean ha each ndvdual producon lne compared o he average of all peer whch ac a a benchmark. Due o he complexy of he problem we conder he aggregaed opmal expeced yeld where he couplng erm approxmaed wh he mean of he opmal expeced yeld. h urn he problem no a womode opmal wchng problem of mean-feld ype whch can be decrbed by a yem of Snell envelope where he obacle are nerconneced and nonlnear. he man reul of he paper a proof of a connuou mnmal oluon o he yem of Snell envelope a well a he full characerzaon of he opmal wchng raegy. 1. Inroducon Conder a company wh N dfferen producon lne whch all have wo mode of producon hgh mode and low mode where each mode of producon ha own balance hee of expeced prof and co. For each producon lne j n mode ley +j denoe he opmal expeced prof yeld a me and le he correpondng opmal expeced co yeld be denoed by Y j. Aume ha we wan o wch beween he wo mode of producon eher f he curren mode unprofable or f we can expec beer prof n he oher mode. Aume furher ha he wchng baed on boh de of he balance hee o ha we for example wch f we can expec lower co n he oher mode. hen h problem can be modeled a a wo-mode opmal wchng problem for each producon lne j whchcanbedecrbedbyhe followng yem of Snell envelope: Y +j = e up E [ ψ +j () d + S +j 1 {<} +ξ + 1 {=} F ] Y j = e nf E [ ψ j () d + S j 1 {<} +ξ 1 {=} F ] for each producon lne j = 1...Nwhereψ ±j () are he prof and co rae per un me (he generaor) and where S +1j S +j S 1j S j =(Y +j =(Y +1j =(Y j =(Y 1j l 1 ()) (Y 1j l ()) (Y j +l 1 ()) (Y +1j +l ()) (Y +j a 1 ()) a ()) +b 1 ()) +b ()) are he obacle of he wchng problem. Here ξ ± are he fnal prof and co of each mode a ome fxed me and he funcon l a andb repreen wchng co. (1) ()

2 Inernaonal Journal of Sochac Analy Now aume ha he producon lne all have he ame generaor bu hey are nerconneced hrough a couplng erm.ifhecouplngermheaverageofheprofandco yeld of all he projec N Y ±j j=1 1 (3) N hen h nuvely mean ha each producon lne compared o a benchmark conued of he average of peer. In h cae he correpondng generaor become ψ +j () =ψ + ( Y +j ψ j () =ψ ( Y j N Y +k k=1 1 ) N N Y k k=1 1 ). N Wh h aumpon olvng he yem (1) becomea hghly complex ak for large N ncehesnellenvelope are all nerconneced hrough he couplng erm (3). Bu nead of olvng (1) we can conder he expeced prof andcoyeldonanaggregaedlevelwhereweuehe mean-feld approxmaon EY ± for he couplng erm (3). he correpondng yem of Snell envelope become Y + Y = e up = e nf for =1where E [ E [ ψ + ( Y + EY + )d +S + 1 {<} +ξ + 1 {=} F ] ψ + ( Y EY )d +S 1 {<} +ξ 1 {=} F ] S +1 =(Y + l 1 ()) (Y 1 a 1 ()) S + =(Y +1 l ()) (Y a ()) S 1 =(Y +l 1 ()) (Y +1 +b 1 ()) S =(Y 1 +l ()) (Y + +b ()). In h paper we wll how he exence of a connuou mnmal oluon of h yem. In a forhcomng paper we wll how convergence of he yem (1) o our yem of Snell envelop of mean-feld ype. he e of counerexample derved n [1] can be ued o argue ha unquene may no hold n general. (4) (5) (6) In erm of BSDE he yem equvalen o he followng yem of mean-feld refleced BSDE (MF-RBSDE): =ξ + Y + ψ + ( Y + Z + db =ξ Y ψ ( Y Z db Y + (Y + (S EY + )d+(k + EY )d (K S + Y S S + )dk + = Y )dk =. K+ ) K ) For deal ee for example []. In h paper we conder he followng lghly more general yem of MF-RBSDE: =ξ + Y + ψ + ( Y + Z + db =ξ Y ψ ( Y Z db Y + (Y + (S EY + EY Z + Z S + Y S S + )dk + = Y )dk =. )d+(k + )d (K K+ ) K ) We follow a procedure mlar o he one ued n [1]; ha we ue an ncreang equence of approxmang meanfeld refleced ochac dfferenal equaon (MF-RBSDE) o how he exence of a connuou mnmal oluon of he yem. However due o he added generaly n he problem we have o prove a comparon reul and an upper bound for h ype of MF-RBSDE gven he n-daa. Mean-feld relaed problem have been uded no only n he eng of backward ochac dfferenal equaon bu n many oher feld a well. Example of area where mean-feldapproxmaonhavebeenuccefulncludeacal mechanc quanum mechanc quanum chemry economc fnance and game heory. Recen work nclude for example [3] where he auhor conder he problem (7) (8)

3 Inernaonal Journal of Sochac Analy 3 of ecor-we allocaon n a porfolo conng of a very large number of ock. Anoher paper on mean-feld approxmaon he emnal work by Lary and Lon [4] whch concern applcaon of mean-feld approxmaon o problem n economc and fnance. For an accoun on recen work relaed o our paper we refer o[5] and he reference heren. Backward ochac dfferenal equaon of he meanfeld ype have been uded by everal auhor ncludng [5 8]. o he be of our knowledge he work of Buckdahn e al. [6] he fr paper o ackle h cla of problem. hey udy an equaon of he form Y =ξ+ E f(ω ωy (ω )Z (ω )Y (ω) Z (ω))d Z dw where he mean-feld neracon lnear n he generaor obaned a a mean-feld lm of BSDE equaon drven by SDE of mean-feld ype. h ype of mean-feld backward ochac dfferenal equaon (MF-BSDE) furher uded n [6] where he auhor oban exence and unquene for a general drver under Lpchz condon. Exenon of h work o refleced BSDE nclude [7 8]. he equaon hey udy are of he form Y =ξ+ E f(ω ωy (ω ) +K K Z dw Y S Z (ω )Y (ω) Z (ω))d (Y S )dk =. (9) (1) he auhor prove exence and unquene of he MF- BSDEawellaacomparonheoremunderomeaddonal condon for he generaor. hee reul ealy exend o our cae where he mean-feld neracon of he MF- RBSDE nonlnear n he generaor. For compleene we dplay n he Appendce an adapaon of he proof of [7 8] o our eng. he oulne of h paper a follow. Secon ae he neceary noaon and prelmnare. In Secon 3 we ae andprovehemanreulofhpaper.. Noaon and Prelmnare For he re of he paper we fx a probably pace denoed by (Ω F P) on whch defned a a andard d-dmenonal Brownan moon B = (B ) whoe naural flraon (F := σ{b }).LeF = (F ) be he flraon (F ) compleed wh he P-null e of F. h mple ha F afe he uual condon; ha rgh connuou and complee. For fuure reference we nroduce he followng pace: () P he σ-algebra on [ ] Ω of F-progrevely meaurable procee () M d he e of P-meaurable and R d -valued procee w=(w ) uch ha w M d 1/ := E[ w d] < (11) () S (rep. S c )heeofp-meaurable and càdlàg (rep. connuou) R-valued procee w=(w ) uch ha w S 1/ := E[ up w ] < (1) (v) K (rep. K c ) a ube of S (rep. S c ) on nondecreang càdlàg (rep. connuou) procee (K ) uch ha K =. Le ξ be an F -meaurable L -random varable le f be an R-valued funcon and le S be an F-adaped proce. o reamlne he preenaon of he reul we nroduce he followng noaon. () If here ex a par of procee (Y Z) uch ha Y S (rep. S c ) Z Md Y =ξ+ f(ωy EY Z )d Z db hen we ay ha (13) (Y Z) =F(ξ f) (rep. (Y Z) =F c (ξ f)). (14) () If here ex a rple of procee (YZK)uch ha Y S (rep. S c ) Z Md K K (rep. K c ) Y =ξ+ f(ωy EY Z )d+k K Z db Y S (Y S )dk = hen we ay ha (rep. (Y S )dk =) (15) (Y Z K) =F + (ξ f S) (rep. (Y Z K) =F c + (ξ f S)). (16)

4 4 Inernaonal Journal of Sochac Analy () If here ex a rple of procee (Y Z K) uch ha Y S (rep. S c ) Z Md K K (rep. K c ) Y =ξ+ f(ωy EY Z )d (K K ) Z db Y S (S Y )dk = hen we ay ha (rep. (Y S )dk =) (17) (Y Z K) =F (ξ f S) (rep. (Y Z K) =F c (ξ f S)). (18) hee mean-feld backward ochac dfferenal equaon (MF-BSDE) and mean-feld refleced backward ochac dfferenal equaon (MF-RBSDE) are ad o be andard f he followng condon hold. (H1) he generaor f Lpchz wh repec o (y 1 y z) unformly n ( ω). (H) he proce (f( ω )) F-progrevely meaurable and d dp-quare negrable. (H3) he random varable ξ n L (Ω F P). (H4) he barrer S càdlàg F-adaped and afe and S ξ P-a.. E [ up S+ ]< (19) MoreonMF-BSDEcanbefoundn[5 6]. For furher reference on MF-RBSDE ee [7 8]. Fnally a key ool ued n h paper he noon of he Snell envelope. Le θ denoe he cla of F-oppng me uch ha θfor ome F-oppng me θ. Propoon 1. Le U=(U ) be an F-adaped R-valued càdlàg proce uch ha he e of random varable {U } unformly negrable. hen here ex an F-adaped R- valued càdlàg proce Z:=(Z ) uch ha Z he malle upermarngale whch domnae U.he proce Z called he Snell envelope of U and ha he followng propere. () For any F-oppng me θ hold ha Z θ = e up E [U F θ ] (and hence Z =U ). θ () () he Doob-Meyer decompoon of Z mple he exence of a connuou marngale (M ) and wo nondecreang predcable procee (A ) and (B ) whch are repecvely connuou and purely dconnuou uch ha for all one ha Z =M A B (wh A =B =). (1) () For any {ΔB > } {ΔU <} {Z =U }. () Hence f U only ha pove jump hen Z a connuou proce. (v) If θ an F-oppng me hen θ := nf { θ : Z =U } (3) opmal afer θ;ha Z θ = e up θ E [U F θ ] = E [U θ F θ]=e [Z θ F θ]. (4) For furher reference on he Snell envelope we refer o [9 1]or[11]. We fnally collec reul regardng exence unquene boundandcomparonformf-rbsde.heeareadapaon of reul n [7 8] o our cae. Proof are deferred o he Appendce. Propoon. Le (ξ f S) be ome n-daa whch afe (H1) (H4). hen here ex a unque rple (YZK)whch olve (Y Z K) =F + (ξ f S). (5) Amlarreulholdfor(Y Z K) = F (ξ f S). Proof. See Appendx A. Propoon 3. Le (ξ f S) be a e of daa afyng aumpon (H1) (H4) andle(y Z K) = F ± (ξ f S). hen here ex a conan C uch ha E [ up Y Z d + K ] (6) CE [ξ f ( ) d + up (S + )]. Proof. See Appendx B. Nex we dplay a comparon reul for oluon of (Y Z K) = F + (ξ f S). Amlarreulholdfor(Y Z K) = F (ξ f S). Propoon 4. Le (ξ f S) and (ξ f S ) be wo e of daa each one afyng aumpon (H1) (H4)and le (Y Z K) =F + (ξ f S) (Y Z K )=F + (ξ f S ). If he followng condon hold: (7) () ξ ξ a.. () f( y 1 y z) f ( y 1 y z) dp d-a.e. and (y 1 y z) R R R d

5 Inernaonal Journal of Sochac Analy 5 () S S a.. (v) S and S are connuou (v) aleaoneofhewogeneraorf and f nondecreang n y hen P-a.. Proof. See Appendx C. Y Y (8) Propoon 5. Propoon 4 hold rue even when S=S a.. and S only afe (H4); ha need no be connuou. Proof. See Appendx D. 3. he Syem of MF-RBSDE Conder he followng yem of equaon: for =1where (Y + Z + K + )=F + (ξ + ψ+ S+ ) (Y Z K )=F (ξ ψ S ) S +1 =(Y + l 1 ()) (Y 1 a 1 ()) S + =(Y +1 l ()) (Y a ()) S 1 =(Y +l 1 ()) (Y +1 +b 1 ()) (9) (3) S =(Y 1 +l ()) (Y + +b ()). Furher aume he followng. (A1) ψ ± are Lpchz n (y 1 y z)unformly n ( ω);ha here ex a C > uch ha for any ( ω) [ ] Ω ψ± ( ω y 1 y z) ψ ± (ωy 1 y z) C( y y + z z ). (31) In addon he procee ψ ± () := ψ ± (ω) are F-progrevely meaurable and d dp-quare negrable. (A) he procee (a ( ω)) (b ( ω)) and (l ( ω)) belong o S c. In addon l () > P- a.. (A3) he random varable ξ ± are F -meaurable and quare negrable. Furhermore P-a.. hold ha ξ + 1 (ξ+ l 1 ()) (ξ 1 a 1 ()) ξ + (ξ+ 1 l ()) (ξ a ()) ξ 1 (ξ +l 1 ()) (ξ + 1 +b 1 ()) ξ (ξ 1 +l ()) (ξ + +b ()). (3) (A4) heprocee(b ()) and (l ()) are of Iôype; ha b () =b () l () =l () U () d V () db U d V db (33) where (U U) and (V V) are ome F-progrevely meaurable procee whch are d P-quare negrable. I worh nong a few hng here. Fr he obacle procee a need no be of Iô-ype. Second he e of oluon o he yem (3) nonempy. An example of oluon o e ξ + =ξ =1andle l () =e 4 a () =b () = (34) for =1.LalywhleY + a upermarngale Y a ubmarngale. he aumpon (A4) needed o prove he connuy of he ncreang proce K. h n urn ued o prove connuy of Y whch fnally ued o derve connuy of Y +. heorem 6. Le he generaor n he yem (9) be nondecreangnhehrdargumen;ha y ψ ± ( y 1 y z) = 1 (35) are nondecreang funcon. hen under he aumpon (A1) (A4) he yem (9) adm a mnmal oluon uch ha Y ± = 1 connuou. he oluon mnmal n he ene ha f ( Y ± Z ± K ± ) =1 anoher oluon hen Y ± Y ± a..heoluonoheyem(3) no unque n general. Proof. he heorem proved ung an approxmang cheme. Le and denoe hen (Y + Z + )=F(ξ + ψ+ ) (36) L L =ξ+ ψ + ( Y + EY + := Y+ +b (). (37) Z + )d +b () U () d V () db =L ψ ( L EL Z )d Z db Z + db (38)

6 6 Inernaonal Journal of Sochac Analy where Z := Z+ +V () ψ ( L EL Z ) where := ψ + ( L b () EL Eb () Z V ()) U (). (39) Now le ( Y Z) be he unque oluon o he BSDE Y = Y α(ωy 1 y z) α( Y EY Z )d := (ψ 1 ψ ψ 1 ψ )(ωy 1y z) Z db S (4) Y := (ξ + 1 +b 1 ()) (ξ + +b ()) ξ 1 ξ. By Propoon 4 hold ha Hence (41) Y L =Y+ +b (). (4) Y (Y + +b ()) ( Y +l ()) (43) nce l () > a.. Conder now he procee where (Y 1 Z 1 K 1 )=F ( Y ψ S 1 ) (Y +1 Z +1 K +1 )=F + (ξ + ψ+ S+1 ) S 1 and for n 1 where =(Y + +b ()) ( Y +l ()) S +11 =(Y + l 1 ()) (Y 11 a 1 ()) S +1 =(Y +1 l ()) (Y 1 a ()) (Y n+1 Z n+1 K n+1 )=F (ξ ψ S n+1 ) (Y +n+1 Z +n+1 K +n+1 )=F + (ξ + ψ+ S+n+1 ) S 1n+1 =(Y n +l 1 ()) (Y +1n +b 1 ()) (44) (45) (46) Snce (Y + Z + ) he oluon of a andard MF-BSDE he exence and unquene have been eablhed n [6] and he exence and unquene of he procee (Y 1 Z 1 K 1 ) were eablhed n Propoon. Wh h n mnd ealy hown by he ue of nducon ha for any n 1he rple (Y +n Z +n K +n ) ex are unque and belong o he approprae pace. follow ha Y + Y +1.MoreoverbyPropoon 4 agan follow ha Y Y 1.Hence From Propoon 4 and he fac ha K +1 ( Y +l 1 ()) (Y +1 +b 1 ()) K +1 (Y 1 +l 1 ()) (Y +11 +b 1 ()) ( Y +l ()) (Y + +b ()) Propoon 4 hen yeld Y 1 Now aume ha hen S +1n+1 S +n+1 =(Y +n (Y +n+1 =(Y +1n (Y +1n+1 (Y 11 +l ()) (Y +1 +b ()). Y +n Y n+1 Y. Y +n+1 Y n+. l 1 ()) (Y 1n+1 l 1 ()) (Y 1n+ l ()) (Y n+1 l ()) (Y n+ a 1 ()) (48) (49) a 1 ()) =S +1n+ a ()) a ()) =S +n+ (5) from whch follow by Propoon 4 agan ha Y +n+1.hencealoholdha Y +n+ S 1n+ S n+ =(Y n+1 (Y n+ =(Y 1n+1 (Y 1n+ +l 1 ()) (Y +1n+1 +l 1 ()) (Y +1n+ +l ()) (Y +n+1 +l ()) (Y +n+ Y n+3.bynduc- hu Propoon 4 yeld ha Y n+ on +b 1 ()) +b 1 ()) =S 1n+3 +b ()) +b ()) =S n+3. (51) S n+1 S +1n+1 =(Y 1n =(Y +n +l ()) (Y +n l 1 ()) (Y 1n+1 +b ()) a 1 ()) (47) Y +n Y n+1 Y +n+1 Y n+ (5) S +n+1 =(Y +1n l ()) (Y n+1 a ()). for n 1.

7 Inernaonal Journal of Sochac Analy 7 By Propoon 3 hee equence are bounded and nce hey are alo ncreang he lm ex. Denoe hee lm by Y + Y := lm n Y+n := lm n Y n. (53) In wha follow we wll prove ha hee lm are n fac connuouandolveheyem(9). o do h we wll ue he followng clam: here ex a pove conan C uch ha for all n 1 E [ ( dk n ) d] C. (54) d he fr ep oward provng h clam o prove he abolue connuy of dk 1 wh repec o d.nongha S 1 =L ( Y +l ()) =L (L Y l ()) + (55) hen n vew of (A4) and he Iô-anaka formula we ge S 1 =S 1 f () d g () db 1 L (56) where L he local me a zero of he connuou emmarngale L Y l and where f () := ψ ( L EL Z )+1 {L > Y +l ()} (ψ ( L EL Z ) α( Y EY Z )+U ()) g () := Z 1 {L > Y +l ()} (Z Z V ()). I follow ha (57) where Λ he local me a for S 1 Y 1.Snce(S 1 Y 1 ) + S 1 he dfferenal mu concde whch yeld ha 1 dλ = 1 {Y 1 Hence dk 1 = 1 {Y 1 Y 1 =S 1 =S 1 } ((f () +ψ ( Y 1 1 dl dk 1 ) } ((f () +ψ ( Y 1 1 dl ) dk (dλ + 1 {Y 1 =S 1 } dl ) = 1 {Y 1 =S 1 from whch follow ha dk 1 } (f () +ψ ( Y 1 1 {Y 1 =S 1 } (f () +ψ ( Y 1 ( f () + ψ ( Y 1 EY 1 EY 1 Z 1 EY 1 EY 1 EY 1 Z 1 ) )d. Z 1 )) d Z 1 )) d Z 1 )) d (6) )) d (61) (6) Now n vew of (A1) (A4) andpropoon 3 here ex a conan C uch ha E [ ( f () + ψ ( Y 1 EY 1 + g () )d] C Z 1 ) (63) d(s 1 Y 1 ) =(f () +ψ ( Y 1 EY 1 Z 1 )) d +(g () Z 1 )db 1 dl dk 1. From Iô-anaka agan we ge ha d(s 1 Y 1 ) + (58) whch ogeher wh (6)prove(54)forn=1. For he nducon ep we only conder he cae =1 nce he oher cae follow n a mlar fahon. Aume ha dk n S 1n+1 /d afe (54) andconder heobacle proce =(Y n =Y +1n +l 1 ()) (Y +1n +b 1 () (Y +1n +b 1 ()) +b 1 () Y n l 1 ()) +. (64) = 1 {S {S 1 1 {S 1 >Y 1 } (f () +ψ ( Y 1 EY 1 >Y 1 } (g () Z 1 )db Z 1 )) d >Y 1 } (dk dl )+1 dλ (59) By (A4) and he Iô-anaka formula we ge S 1n+1 =S 1n+1 1 L1n f n+1 () d g n+1 () db 1 {Y +1n +b 1 () Y n +l 1 ()} dk+1n (65)

8 8 Inernaonal Journal of Sochac Analy where L 1n he local me a for he connuou emmarngale Y +1n +b 1 () Y n l 1 () and where f n+1 () := ψ {Y +1n ( Y+1n EY +1n +b 1 ()>Y n +l 1 ()} Z +1n )+U 1 () akng he weak lm n each de of h equaon along he ubequence menoned earler yeld Y =Y φ d k d Z db P-a.. (71) ( ψ + 1 +ψ ( Y+1n ( Y n EY +1n EY n U 1 () dk n ) d f n+1 () := Z +1n +V 1 () Z +1n Z n ) )+U 1 () (66) he procee on each de of he equaly beng oponal we can ue he oponal econ heorem (ee e.g. [1heorem 86 page 138]) o conclude ha Y =Y φ d k d Z db P-a.. (7) 1 {Y +1n (Z +1n +b 1 ()>Y n +l 1 ()} +V 1 () Z n V 1 ()). In vew of (A1) (A4) andpropoon 3 ogeher wh he nducon aumpon here ex a conan C > ndependen of nuchha E [ ( f n+1 () + ψ 1 ( Y 1n+1 + g n+1 () d)] C. EY 1n+1 Z 1n+1 ) (67) Followng he ame ep a we dd earler for dk 1 can be hown ha dk 1n+1 ( f n+1 () + ψ 1 ( Y 1n+1 EY 1n+1 Z 1n+1 ) )d (68) whch yeld ha here ex a conan C>ndependen of n uch ha E [ ( dk 1n+1 ) d] C. (69) d Hence clam (54)rueforalln 1. Propoon 3 and emae (54) ell u ha here a ubequence along whch he equence of procee /d) (ψ ( Y n EY n Z n )) and (Z n ) convergeweaklynherrepecvepacem 1 M 1 andm d o he procee (k ) (φ )and (Z ). Now for any n and any F-oppng me wehave (dk n Y n =Y n +K n ψ ( Y n Z n db. EY n Z n )d (7) herefore he proce Y connuou. Relyng on boh Dn heorem and Lebegue domnaed convergence heorem we fnd ha lm E [ up n Y n Y ]=. (73) In wha follow we wll characerze he lm procee Y + of he equence {Y +n } n a Snell envelope of he procee n he ene ha P-a.. and for each Y + = e up E [ ψ + 1 ( Y+ EY + Z + )d +ξ + 1 {=} +S + 1 {<} F ] (74) andhenwedervehermeconnuy. In vew of Propoon 3 and applyng Peng monoone lm heorem (ee [13]) o he equence (Y +n Z +n K +n ) we oban ha Y + càdlàg. Moreover here ex a càdlàg nondecreang proce K + K and a proce Z + M d uch ha Y + Y + =ξ ψ + ( Y + S +. Z + db EY + Z + )d+k + K+ (75) Hence n vew of Propoon 1 we arrve a (74)oncewehow ha (Y + S+ )dk+ =. (76)

9 Inernaonal Journal of Sochac Analy 9 Conder he malle ψ + -upermarngale Y + lower obacle S + whcholve Y + =ξ + + K + ( Y + ψ + ( Y + K + Y + EY + Z + Z + db S + S+ + )d K =. )d wh (77) By defnon we have ha Y + Y +.UngPropoon 5 we ee ha Y + Y +n n 1.Pangohelmwegeha Y + = Y + =1.Hence(76) afed. I reman o how ha Y + are connuou. Nong ha > and ha Y + afe (74) we fnd ha Y + Y + or equvalenly ha ΔY +.huuffcenoproveha he e {ΔY + <} empy. By he Doob-Meyer decompoon of he Snell envelope (ee Propoon 1) here ex for each a connuou marngale (M ) a connuou nondecreang proce (C ) and a purely dconnuou proce (D ) wh C =D =uchha Y + ψ + ( Y + EY + Z + )d=m C D (78) whch mean ha ΔY + = ΔD. Now n vew of Propoon 1 we have he followng propere of he jump of D.Ifhereajumpame n D 1 henhmeanha he proce (Y + l ) (Y 1 a 1 ) alo jump. Snce l 1 and Y 1 a 1 are connuou h can only mean ha here a jump n Y +. h n urn mean ha here a jump n D. Andconverelybyheameypeofreaonngwecandeduce ha f here a jump n D herealoajumpnd 1.Hence D 1 and D alwayjumpaheameme. Wh h n mnd denoe X:=(Y + l 1 ) and F:=Y 1 a 1.Wehave {Δ (X F) <} ={X F <X F } = {{X F <X F } {X <F <X }} {{X F <X F } {X <X <F }} {{X F <X F } {F <X <X }} = {{F <X } {X <F <X }} {{F <F } {X <X <F }} {{X <X } {F <X <X }} = {{F <X } {X <F } {F <X }} {{X <X } {F <X } {X <X }} = {{F <X } {X <F }} {{X <X } {F <X }} = {{F <X } {X <F } {X <X }} {{X <X } {F <X } {F <X }} = {{X <X } {F <X }} {{X <F } {F <X }} ={X <X } {F <X } ={ΔX <} {F <X } ={ΔY + <(Y 1 a 1 ()) <(Y + l 1 ())}. (79) WhahelluhafS +1 jump a me henweknow wo hng. One ha Y + alo ha a jump a me and he oher ha herefore nce ΔY + oban ha {ΔD 1 >} {ΔD >} {Y+1 ={ΔD >} {Y+1 ={ΔD >} {Y+1 Smlarly we have {ΔD >} {ΔD 1 >} {Y+ ={ΔD 1 >} {Y+ ={ΔD 1 >} {Y+ Y 1 a 1 () <Y + l 1 (). (8) = ΔD nvewofpropoon 1 we =S+1 } =(Y+ l 1 ()) (Y 1 a 1 ())} =Y+ =S+ } l 1 ()}. (81) =(Y+1 l ()) (Y a ())} =Y+1 l ()}. (8) Fnally nce we know ha D 1 and D alway jump a he ame me hold ha {ΔD 1 >} {ΔD1 >} ={ΔD 1 >} {ΔD >} ={Y +1 {Y + =Y+ l 1 ()} =Y+1 l ()} = (83)

10 1 Inernaonal Journal of Sochac Analy nce for any l 1 () + l () >. I follow ha he procee D 1 and D are conan and dencally equal o nce D 1 =D =.HenceheproceeY+1 and Y + are connuou. h n urn gven (75) yeld he connuy of he ncreang procee K +. herefore (Y + Z + K + ) a oluon o he fr par of he yem (9). By Dn heorem and Lebegue domnaed convergence heorem agan we alo conclude ha he convergence of (Y +1n ) n 1 o Y +1 hold n S c ;ha lm E [ up n Y+n Y + ]=. (84) Furhermore nce Y connuou we can rely on andard argumen n parcular by applyng Iô formula o (Y +n Y +m ) (m n )oclamha(z +n ) a Cauchy equence and herefore converge o Z + n M d ;ha lm E [ n Z+n Z + d] =. (85) Combnng h wh (84) and he defnon of he procee Y +n and Y + yeld ha lm E [ up n K+n In oal we have hown ha lm E [ up n Y+n Z+n Y + Z + d + up K + ]=. (86) K+n K + ]=. (87) Ung (84)andapplyngIô formula o (Y n Y m ) (m n )followha(z n ) n 1 a Cauchy equence and herefore converge o Z n M d ;ha lm E [ n Z n Z d] =. (88) Ung h and akng no accoun he decompoon obaned n (7) we arrve a Y =ξ ψ ( Y EY Z )d k d Z db Y 1 (Y +l 1 ()) (Y +1 +b 1 ()) Y (Y 1 +l ()) (Y + +b ()). Ahownabovealoholdha lm E [ up n Y n + up K n Y Z n k d ]=. Z d (89) (9) Moreover due o he weak convergence of (dk n /d n 1)o he proce k and he rong convergence of (Y +n ) and (Y n ) n S followha = (S n Y n ) dk n (S a n whch mple ha f we defne K Y ) k d (91) := k d (9) hen (Y Z K ) a oluon o he econd par of (9). Iremanohowhaheobanedoluonmnmal. o do h we compare Y + wh he malle ψ + - upermarngale wh lower obacle S + denoed by Y + and we compare Y + wh he malle ψ -upermarngale wh upper obacle S denoed by Y ; ha le (Y ± Z ± K ± ) be he malle oluon o he yem (Y + Z + K + )=F + (ξ + ψ+ S+ ) = 1 (Y Z K )=F (ξ ψ S ) = 1. (93) Ung Propoon 4 and he mnmaly of he oluon we can conclude ha Y ± Y ±. On he oher hand Propoon 4 aloelluhaforalln we have Y ±n Y ±.NownceY ±n are ncreang n n we can ake he lm n and oban ha Y ± Y ±.huy ± =Y ±. Fnally o eablh ha he oluon no unque n general he counerexample found n [1] vald here a well. Appendce A. Proof of Propoon he proof ue echnque found n [ 7 8]. Le S be he pace of progrevely meaurable {(Y Z ) : } akng value n R R d uch ha Z M d and Y S for. ake any (U V) S and defne f() := f( U EU V ). hen by Propoon 5.1 n [] here a unque rple (Y Z K) uch ha Y S Z M d K K K L (Ω F P) Y =ξ+ f () d + K K Z db K = Y S (Y S )dk =. (A.1) We defne a mappng Φ from S ono elf by ayng ha (Y Z) = Φ(U V) he unque elemen of S uch ha f we defne K =Y Y f () d Z db (A.)

11 Inernaonal Journal of Sochac Analy 11 hen (YZK) he unque rple whch olve he yem above. Now le (U V ) be anoher elemen of S and le (Y Z )= Φ(U V ). In addon le U=U U V=V V Y=Y Y Z=Z Z f= f f where f () := f( U EU V ). For any β> applyng Iô formula o e β Y akng expecaon we fnd ha E [e β Y βe β Y d e β Z d] Moreover nce =E [ e β Y f () d e β Y dk ]. E [ e β Y dk ] hold ha =E [ e β (Y Y )d(k K )] (A.3) and hen =E [ e β [(Y S ) (Y S )] (dk dk )] = E [ e β (Y S )dk ] E [ e β (Y S )dk ] E [ βe β Y d e β Z d] E [ e β Y f () d]. (A.4) (A.5) (A.6) Ung he Lpchz propery of f and hen Young nequaly yeld ha E [ βe β Y d e β Z d] E [ e β Y f () d] =E [ e β Y (f ( U EU V ) f ( U EU V )) d] KE [ e β Y ( U U + EU EU + V V )d] KE [ e β Y ( U U + E U U 1K E [ e β Y d] + V V )d] E [ e β ( U U + E U U 1K E [ e β Y d] E [ = 1K E [ e β Y d] + V V ) d] e β ( U U + E U U + V V )d] e β (EU + E V )d 1K E [ e β Y d] + 1 E [ e β (U + V )d]. Seng β = 1K +1we fnd (A.7) E [ e β (Y + Z )d] (A.8) 1 E [ e β (U + V )d]. hu he mappng Φ a rc conracon on S wh he norm 1/ (Y Z) β =(E e β (Y + Z )d). (A.9) he Banach fxed pon heorem hen yeld ha Φ ha a unque fxed pon whch he ough oluon.

12 1 Inernaonal Journal of Sochac Analy B. Proof of Propoon 3 Applyng Iô formula o Y yeld ha Y Z d =ξ + Y f(y EY Z )d + Y dk Y Z db =ξ + Y f(y EY Z )d + S dk Y Z db and o by akng expecaon we ge E [Y Z d] (B.1) = E [ξ + S dk ]+E[ Y f(y EY Z )d]. (B.) Ung he Lpchz propery of f yeld ha E [ Y f(y EY Z )d] Hence E [ Y f ( ) +K Y +K Y E [ Y ]+K Y Z d] E [ Y ]+E[ f ( ) ]+KE[ Y ] +KE[ Y ] +K E [ Y ]+ 1 E [ Z ]d (1 + 4K + K ) E [ Y ]+E[ f( ) ]d + 1 E [ Z ]d. E [ Y ]+ 1 E [ Z d] E [ξ + S dk f ( ) d] +(1+4K+K ) E [ Y ]d. (B.3) (B.4) herefore E [ Y ] E[ξ + S dk f ( ) d] +(1+4K+K ) E [ Y ]d (B.5) 1 E [ Z d] E [ξ + S dk f ( ) d] +(1+4K+K ) E [ Y ]d (B.6) E [ξ + S + dk f ( ) d] +(1+4K+K ) E [ Y ]d. Ung Gronwall nequaly on (B.5) we oban ha E [ Y ] CE[ξ + S dk f ( ) d] CE [ξ + S + dk f ( ) d]. (B.7) Inerng no (B.6) yeld ha 1 E [ Z d] E [ξ + S + dk f ( ) d] +(1+4K+K ) CE [ξ + S + u dk u f (u ) du] d E [ξ + S + dk f ( ) d] +(1+4K+K ) CE [ (ξ + S + dk f ( ) d)]. (B.8) Hence E [ Z d] C E [ξ + S + dk f ( ) d]. (B.9)

13 Inernaonal Journal of Sochac Analy 13 Conder now he equaly =4K E [ f ( ) d] K =Y ξ f(y EY Z )d+ Z db. (B.1) +8K E [ Y d] + 4K E [ Z d] (B.1) We have E [K ]=E[(Y ξ f(y EY Z )d Z db ) ] 4E [Y +ξ +( f(y EY Z )d) +( Z db ) ] ={Iô ruleonhequaredochacnegral} =4E [Y +ξ +( f(y EY Z )d) Z d] 4E [Y +ξ f( Y EY Z ) d Z d] CE [ξ + S + dk f ( ) d] +4E [ξ f( Y EY Z ) d] (B.11) where he la nequaly follow from ung (B.7) and(b.9). Now from he Lpchz propery of f we ge E [ f( Y EY Z ) d] K E [( f ( ) + Y + EY + Z ) ]d 4K E [ f ( ) + Y + EY + Z ]d 4K E [ f ( ) + Y + E [ Y ]+ Z ]d whch by nerng no he above and hen ung (B.7) and (B.9) agan yeld ha E [K ] CE [ξ + S + dk f ( ) d] +4E [ξ ]+8K E [ f ( ) d] CE [ξ + S + dk f ( ) d] +8(1+K ) E [ξ f ( ) d + S + dk ] =CE [ξ f ( ) d + S + dk ] CE [ξ f ( ) d] + E [C ( up S + )K ] CE [ξ f ( ) d] +C E [ up (S + ) ]+ 1 E [K ]. (B.13) hu E [K ] C[ξ f ( ) d + up (S + ) ]. (B.14) Ung he Burkholder-Dav-Gundy nequaly we fnd ha E [ up Y ] CE[ Z d]. (B.15) Hence agan n vew of (B.9) we ge E [ up Y Z d + K ] E [(1+C) Z d + K ] CE [ξ f ( ) d + up (S + ) ]. (B.16)

14 14 Inernaonal Journal of Sochac Analy C. Proof of Propoon 4 Iô formula appled o (Y Y )+ yeld ha (Y Y )+ = (Y Y )+ (f ( Y EY Z ) f ( Y EY Z )) d + (Y Y )+ (dk dk ) (Y Y )+ (Z Z )db (Y Y )+ dl 1 {Y >Y } Z Z d (Y Y )+ (f ( Y EY Z ) f ( Y EY Z )) d + (Y Y )+ (dk dk ) (Y Y )+ (Z Z )db 1 {Y >Y } Z Z d. akng expecaon we fnd ha E (Y Y )+ + E 1 {Y >Y } Z Z d E (Y Y )+ (f ( Y EY Z ) f ( Y EY Z )) d +E (Y Y )+ (dk dk ). (C.1) (C.) Hence E (Y Y )+ + E 1 {Y >Y } Z Z d E (Y Y )+ (f(y EY Z ) f ( Y EY Z )) d =E (Y Y )+ E (Y Y )+ (f(y EY Z ) f(y EY Z ) +f(y EY Z ) f ( Y EY Z )) d (f(y EY Z ) f(y EY Z )) d (C.4) by aumpon. Whou lo of generaly we may aume ha f he generaor whch nondecreang n he hrd argumen o ha by he Lpchz propery of f we ge E (Y Y )+ + E 1 {Y >Y } Z Z d KE (Y Y )+ ( Y Y +(EY EY )+ + Z Z )d KE (Y Y )+ ( Y Y + Z Z )d +KE (Y Y )+ E [(Y Y )+ ]d (C.5) where he la nequaly follow from Jenen nequaly. For he fr negral we have Bu on {Y >Y } hold ha Y >Y S S fromwhch follow ha (Y Y )+ (dk dk )= (Y Y )+ dk. (C.3) KE (Y Y )+ ( Y Y + Z Z )d =KE (Y Y )+ d + E K(Y Y )+ 1 {Y >Y } Z Z d

15 Inernaonal Journal of Sochac Analy 15 KE (Y Y )+ d +K E (Y Y )+ d + 1 E 1 {Y >Y } Z Z d = (K + K ) E (Y Y )+ d + 1 E 1 {Y >Y } Z Z d. (C.6) Inerng no he above and cancelng ou erm yeld ha E (Y Y )+ D. Proof of Propoon 5 Conder he penalzed equaon of he MF-RBSDE for n 1: Leng Y n Y n =ξ+ f(y n EYn Zn )d +n (Y n S ) d Z n db =ξ f ( Y n +n (Y n EY n Z n )d S ) d Z n db. f n ( y 1 y z):=f(y 1 y z)+n(y z) f n ( y 1 y z):=f ( y 1 y z)+n(y z) (D.1) (D.) + 1 E 1 {Y >Y } Z Z d (K + K ) E (Y Y )+ d +KE (Y Y )+ E [(Y Y )+ ]d = (K + K ) E (Y Y )+ d (C.7) we have wo equence of unrefleced connuou MF-BSDE wh generaor f n and f n where f n ( y 1 y z) f n ( y 1 y z). (D.3) Followng he ep n he proof of Propoon 4 we ee ha a.. akng he lm n weoban he reul. Y n Y n Conflc of Inere he auhor declare ha here no conflc of nere regardng he publcaon of h paper. +K E[(Y Y )+ ] d (K + K ) E (Y Y )+ d +K E [ (Y Y )+ ]d = (4K + K ) E (Y Y )+ d. Hence hold ha E (Y Y )+ (4K+K ) E (Y Y )+ d. (C.8) Ung Gronwall nequaly yeld ha and hence Y Y P-a.. E (Y Y )+ = (C.9) Acknowledgmen he fnancal uppor from he Swedh Expor Cred Corporaon (SEK) graefully acknowledged. Reference [1] B. Djehche and A. A. Hamd Full balance hee wo-mode opmal wchng problem Preprn 11. [] N. El Karou C. Kapoudjan E. Pardoux S. Peng and M. C. Quenez Refleced oluon of backward SDE and relaed obacle problem for PDE he Annal of Probablyvol.5 no. pp [3] V. S. Borkar and K. S. Kumar McKean-Vlaov lm n porfolo opmzaon Sochac Analy and Applcaon vol.8no. 5pp [4] J.M.LaryandP.L.Lon Meanfeldgame Japanee Journal of Mahemacvol.no.1pp [5] R. Buckdahn B. Djehche J. L and S. Peng Mean-feld backward ochac dfferenal equaon: a lm approach he Annal of Probablyvol.37no.4pp [6] R.BuckdahnJ.LandS.Peng Mean-feldbackwardochac dfferenal equaon and relaed paral dfferenal equaon Sochac Procee and her Applcaon vol.119no. 1 pp

16 16 Inernaonal Journal of Sochac Analy [7] J. L Refleced mean-feld backward ochac dfferenal equaon. Approxmaon and aocaed nonlnear PDE Journal of Mahemacal Analy and Applcaonvol.413no. 1pp [8] Z. L and J. Luo Mean-feld refleced backward ochac dfferenal equaon Sac & Probably Leer vol.8 no.11pp [9] J. Cvanć and I. Karaza Backward ochac dfferenal equaon wh reflecon and Dynkn game he Annal of Probablyvol.4no.4pp [1] S. Hamadène Refleced BSDE wh dconnuou barrer and applcaon Sochac and Sochac Reporvol.74no.3-4 pp [11] I. Karaza and S. E. Shreve Mehod of Mahemacal Fnance vol. 39 Sprnger New York NY USA [1] C. Dellachere and P. Meyer Probable and Poenal chaper 1 4 Hermann Par France [13] S. Peng Monoonc lm heorem of BSDE and nonlnear decompoon heorem of Doob-Meyer ype Probably heory and Relaed Feldvol.113no.4pp

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