Reflected Solutions of BSDEs Driven by G-Brownian Motion

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1 Refleced Soluion of BSDE Driven by G-Brownian Moion Hanwu Li Shige Peng Abdoulaye Soumana Hima Sepember 1, 17 Abrac In hi paper, we udy he refleced oluion of one-dimenional backward ochaic differenial equaion driven by G-Brownian moion (RGBSDE for hor). he reflecion keep he oluion above a given ochaic proce. In order o derive he uniquene of refleced G-BSDE, we apply a maringale condiion inead of he Skorohod condiion. Similar o he claical cae, we prove he exience by approximaion via penalizaion. Key word: G-expecaion, G-BSDE, refleced G-BSDE. MSC-claificaion: 6H1, 6H3 1 Inroducion El Karoui, Kapoudjian, Pardoux, Peng and Quenez [5] udied he problem of BSDE wih reflecion, which mean ha he oluion o a BSDE i required o be above a cerain given coninuou boundary proce, called he obacle. For hi purpoe, an addiional coninuou increaing proce hould be added in he equaion. Furhermore, hi addiional proce hould be choen in a minimal way o ha he Skorohod condiion i aified. An imporan obervaion i ha he oluion i he value funcion of an opimal opping problem. Due o he imporance in BSDE heory and in applicaion, he refleced problem ha araced a grea deal of aenion ince Many cholar ried o relax he condiion on he generaor and he obacle proce. Hamadene [8] and Lepelier and Xu [17] gave a generalized Skorohod condiion and proved he exience and uniquene when he obacle proce i no longer coninuou. Cvianic and Karaza [3] and Hamadene and Lepelier [9] udied he cae of wo reflecing obacle. hey alo eablihed he connecion beween hi problem and Dynkin game. Maoui [] and Kobylanki, Lepelier, Quenez and orre [16] exended he reul o he cae where he generaor i no a Lipchiz funcion. We hould poin ou ha he claical BSDE can only provide probabiliic inerpreaion for quailinear parial differenial equaion (PDE for hor). Beide, hi BSDE canno be applied o price pah-dependen coningen claim in he uncerain volailiy model (UVM for hor). Moivaed by hee fac, Peng [3, 4] yemeically inroduced a ime-conien fully nonlinear expecaion heory. One of he mo imporan cae i he G-expecaion heory (ee [7] and he reference School of Mahemaic, Shandong Univeriy, lihanwu@mail.du.edu.cn. School of Mahemaic and Qilu Iniue of Finance, Shandong Univeriy, peng@du.edu.cn. Li and Peng reearch wa parially uppored by he ian Yuan Projecion of he Naional Naure Science Foundaion of China (No and No ) and by he 111 Projec (No. B13). Li and Peng acknowledge graefully financial uppor by he German Reearch Foundaion (DFG) via CRC 183. Iniu de recherche mahémaique de Renne, Univerié de Renne 1 and Déparemen de mahémaique, Univerié de Maradi, oumanahima@yahoo.fr. A. Soumana Hima i graeful for parial financial uppor from he Lebegue Cener of Mahemaic ( Inveiemen d aveni Program) under gran ANR-11-LABX--1. 1

2 herein). In hi framework, a new ype of Brownian moion and he correponding ochaic calculu of Iô ype were conruced. I ha been widely ued o udy he problem of model uncerainy, nonlinear ochaic dynamical yem and fully nonlinear PDE. he backward ochaic differenial equaion driven by G-Brownian moion (i.e., G-BSDE) can be wrien in he following way Y = ξ + f(, Y, Z )d + g(, Y, Z )d B Z db (K K ). he oluion of hi equaion coni of a riple of procee (Y, Z, K). he exience and uniquene of he oluion are proved in [1]. In [11] he comparion heorem, Feymann-Kac formula and ome relaed opic aociaed wih hi kind of G-BSDE were eablihed. In hi paper, we udy he cae where he oluion of a G-BSDE i required o ay above a given ochaic proce, called he lower obacle. An increaing proce hould be added in hi equaion o puh he oluion upward, o ha i may remain above he obacle. According o he claical cae udied by [5], we may expec ha he oluion of refleced G-BSDE i a quadruple {(Y, Z, K, A ), } aifying (1) Y = ξ + f(, Y, Z )d + () (Y, Z, K) S α G (, ) and Y S, ; g(, Y, Z )d B Z db (K K ) + A A ; (3) {A } i coninuou and increaing, A = and (Y S )da =. he horcoming i ha he oluion of he above problem i no unique. hu, o ge he u- niquene of he refleced G-BSDE, we hould reformulae hi problem a he following. A riple of procee (Y, Z, A) i called a oluion of refleced G-BSDE if he following properie hold: (a) (Y, Z, A) S α G (, ) and Y S ; (b) Y = ξ + f(, Y, Z )d + g(, Y, Z )d B Z db + (A A ); (c) { (Y S )da } [, ] i a non-increaing G-maringale. Here, we denoe by SG α(, ) he collecion of proce (Y, Z, A) uch ha Y Sα G (, ), Z Hα G (, ), A i a coninuou nondecreaing proce wih A = and A SG α (, ). Noe ha we ue a maringale condiion (c) inead of he Skorohod condiion. Under ome appropriae aumpion, we can prove ha he oluion of he above refleced G-BSDE i unique. In proving he exience of hi problem, we hould ue he approximaion mehod via penalizaion. hi i a conrucive mehod in he ene ha he oluion of he refleced G-BSDE i proved o be he limi of a equence of penalized G-BSDE. Differen from he claical cae, he dominaed convergence heorem doe no hold under G-framework. Beide, any bounded equence in M p G (, ) i no longer weakly compac. he main difficuly in carrying ou hi conrucion i o prove he convergence propery in ome appropriae ene. I i worh poining ou ha he main idea i o apply he uniformly coninuou propery of he elemen in S p G (, ). Acually, he above equaion hold P -a.. for every probabiliy meaure P belong o a nondominaed cla of muually ingular meaure. herefore, he G-expecaion heory hare many imilariie wih econd order BSDE (BSDE for hor) developed by Cheridio, Soner, ouzi and Vicoir [1]. Maoui, Poamai and Zhou [1] howed he exience and uniquene of econd order refleced BSDE whoe oluion i (Y, Z, K P ) P P κ H aifying Y = ξ + ˆF (Y, Z )d Z db + (K P K P ), P -a..,

3 wih P P H (+,P ) EP Y S, K P k P = e inf P [K P k P ], P -a..,, P P κ H, where (y P, z p, k P ) denoe he unique oluion o he andard RBSDE wih daa (ξ, ˆF, S) under P. he main conribuion of our paper i ha he riple (Y, Z, A) i univerally defined wihin he G- framework uch ha he procee have rong regulariy propery. Due o hi propery, he oluion i ime-conien and he proce A can be aggregaed ino a univeral proce. Similar wih [5], when he refleced G-BSDE i formulaed under a Markovian framework, he oluion of hi problem provide a probabiliic repreenaion for he oluion of an obacle problem for nonlinear parabolic PDE. here ha been remendou inere in developing he obacle problem for parial differenial equaion ince i ha wide applicaion o mahemaical finance (ee [6]) and mahemaical phyic (ee [9]). he mehod in hi paper i called he Feynman-Kac formula which give a link beween probabiliy heory and PDE uing he language of vicoiy oluion. he oher approach i relaed o he variaional inequaliie and in hi cae he oluion belong o a Sobolev pace, ee in paricular [15] and [7]. We hould poin ou ha boh of he oluion udied by hee wo mehod are in weak ene. he re of paper i organized a follow. In Secion, we preen ome noaion and reul a preliminarie for he laer proof. he problem i formulaed in deail in Secion 3 and we ae ome eimae of he oluion from which we derive ome inegrabiliy properie of he oluion. In Secion 4, we eablih he approximaion mehod via penalizaion. We ae ome convergence properie of he oluion o he penalized G-BSDE. Our main reul are howed and proved in Secion 5. Furhermore, we prove a comparion heorem imilar o ha in [11], pecifically for nonrefleced G- BSDE. In Secion 6, we give he relaion beween refleced G-BSDE and he correponding obacle problem for fully nonlinear parabolic PDE. Finally, we ue he reul of he previou ecion o udy he pricing problem for American coningen claim under model uncerainy in Secion 7. In Appendix we inroduce he opional opping heorem under G-framework uing for pricing for American coningen claim. Preliminarie We recall ome baic noion and reul of G-expecaion, which are needed in he equel. relevan deail can be found in [1], [11], [5], [6], [7]. More.1 G-expecaion Definiion.1 Le Ω be a given e and le H be a vecor laice of real valued funcion defined on Ω, namely c H for each conan c and X H if X H. H i conidered a he pace of random variable. A ublinear expecaion Ê on H i a funcional Ê : H R aifying he following properie: for all X, Y H, we have (i) Monooniciy: If X Y, hen Ê[X] Ê[Y ]; (ii) Conan preerving: Ê[c] = c; (iii) Sub-addiiviy: Ê[X + Y ] Ê[X] + Ê[Y ]; (iv) Poiive homogeneiy: Ê[λX] = λê[x] for each λ. he riple (Ω, H, Ê) i called a ublinear expecaion pace. X H i called a random variable in (Ω, H, Ê). We ofen call Y = (Y 1,..., Y d ), Y i H a d-dimenional random vecor in (Ω, H, Ê). 3

4 Le Ω = C ([, ]; R d ), he pace of R d -valued coninuou funcion on [, ] wih ω =, be endowed wih he upremum norm, and B = (B i ) d i=1 be he canonical proce. For each >, denoe L ip (Ω ) := {ϕ(b 1,..., B n ) : n 1, 1,..., n [, ], ϕ C Lip (R d n )}. Denoe by S d he collecion of all d d ymmeric marice. For each given monoonic and ublinear funcion G : S d R, we can conruc a G-expecaion pace (Ω, L ip (Ω ), Ê, Ê). he canonical proce B i he d-dimenional G-Brownian moion under hi pace. In hi paper, we uppoe ha G i non-degenerae, i.e., here exi ome σ > uch ha G(A) G(B) 1 σ r[a B] for any A B. Le B be he d-dimenional G-Brownian moion. For each fixed a R d, {B a } := { a, B } i a 1-dimenional G a -Brownian moion, where G a : R R aifie G a (p) = G(aa )p + + G( aa )p. Le π N = { N,, N N }, N = 1,,, be a equence of pariion of [, ] uch ha µ(πn ) = max{ N i+1 N i : i =,, N 1}, he quadraic variaion proce of Ba i defined by N 1 B a = lim (B a µ(π N ) N j+1 j= B a ). N j For a, ā R d, we can define he muual variaion proce of B a and Bā by B a, Bā := 1 4 [ Ba+ā B a ā ]. Denoe by L p G (Ω ) he compleion of L ip (Ω ) under he norm ξ L p := (Ê[ ξ p ]) 1/p for p 1. For G. herefore i can be all [, ], Ê[ ] i a coninuou mapping on L ip (Ω ) w.r.. he norm L p G exended coninuouly o he compleion L p G (Ω ). Deni e al. [4] proved he following repreenaion heorem of G-expecaion on L 1 G (Ω ). heorem. ([4, 1]) here exi a weakly compac e P M 1 (Ω ), he e of all probabiliy meaure on (Ω, B(Ω )), uch ha P i called a e ha repreen Ê. Ê[ξ] = up E P [ξ] for all ξ L 1 G(Ω ). P P Le P be a weakly compac e ha repreen Ê. For hi P, we define capaciy c(a) := up P (A), A B(Ω ). P P Definiion.3 A e A B(Ω ) i polar if c(a) =. A propery hold quai-urely (q..) if i hold ouide a polar e. In he following, we do no diinguih wo random variable X and Y if X = Y q... For ξ L ip (Ω ), le E(ξ) = Ê[up [, ] Ê[ξ]], where Ê i he G-expecaion. For convenience, we call E G-evaluaion. For p 1 and ξ L ip (Ω ), define ξ p,e = [E( ξ p )] 1/p and denoe by L p E (Ω ) he compleion of L ip (Ω ) under p,e. he following eimae beween he wo norm L p and G p,e will be frequenly ued in hi paper. heorem.4 ([3]) For any α 1 and δ >, L α+δ G (Ω ) L α E (Ω ). More preciely, for any 1 < γ < β := (α + δ)/α, γ, we have ξ α α,e γ { ξ α L α+δ G where C β/γ = i=1 i β/γ,γ = γ/(γ 1) /γ C β/γ ξ (α+δ)/γ }, ξ L L α+δ ip (Ω ). G 4

5 . G-Iô calculu Definiion.5 Le MG (, ) be he collecion of procee in he following form: for a given pariion {,, N } = π of [, ], η (ω) = N 1 j= ξ j (ω)1 [j, j+1)(), where ξ i L ip (Ω i ), i =, 1,,, N 1. For each p 1 and η MG (, ) le η H p := G {Ê[( η d) p/ ]} 1/p, η M p := (Ê[ G η p d]) 1/p and denoe by H p G (, ), M p G (, ) he compleion of MG (, ) under he norm H p, G M p repecively. G For wo procee η MG (, ) and ξ M G 1 (, ), he G-Iô inegral ( η db) i and ( ξ d B i, B j ) are well defined, ee Li-Peng [19] and Peng [7]. Moreover, by Propoiion.1 in [19] and he claical Burkholder-Davi-Gundy inequaliy, he following propery hold. Propoiion.6 If η HG α(, ) wih α 1 and p (, α], hen we can ge up u [, ] u η db p L 1 G (Ω ) and σ p c p Ê [( η d) p/ ] Ê[ up u [, ] u η db p ] σ p C p Ê [( η d) p/ ]. Le SG (, ) = {h(, B 1,..., B n ) : 1,..., n [, ], h C b,lip (R n+1 )}. For p 1 and η SG (, ), e η S p = {Ê[up G [, ] η p ]} 1/p. Denoe by S p G (, ) he compleion of S G (, ) under he norm S p. We have he following coninuiy propery for any Y Sp G G (, ) wih p > 1. Lemma.7 ([18]) For Y S p G (, ) wih p > 1, we have, by eing Y := Y for >, F (Y ) := lim up(ê[ up ε up [, ] [,+ε] Y Y p ]) 1 p =. We now inroduce ome baic reul of G-BSDE. Conider he following ype of G-BSDE (here we ue Einein convenion) where Y = ξ + f(, Y, Z )d + g ij (, Y, Z )d B i, B j Z db (K K ), (.1) aifying he following properie: f(, ω, y, z), g ij (, ω, y, z) : [, ] Ω R R d R, (H1 ) here exi ome β > 1 uch ha for any y, z, f(,, y, z), g ij (,, y, z) M β G (, ), (H) here exi ome L > uch ha f(, y, z) f(, y, z ) + d g ij (, y, z) g ij (, y, z ) L( y y + z z ). i,j=1 For impliciy, we denoe by S α G (, ) he collecion of proce (Y, Z, K) uch ha Y Sα G (, ), Z HG α(, ; Rd ), K i a decreaing G-maringale wih K = and K L α G (Ω ). 5

6 heorem.8 ([1]) Aume ha ξ L β G (Ω ) and f, g ij aify (H1 ) and (H) for ome β > 1. hen for any 1 < α < β, equaion (.1) ha a unique oluion (Y, Z, K) S α G (, ). We alo have he comparion heorem for G-BSDE. heorem.9 ([11]) Le (Y l, Z, l K) l, l = 1,, be he oluion of he following G-BSDE: Y l = ξ l + f l (, Y l, Z)d l + gij(, l Y l, Z)d B l i, B j + V l V l ZdB l (K l K), l where {V l } (H), ξ l L β G (Ω ) wih β > 1. If ξ 1 ξ, f 1 f, gij 1 g ij increaing proce, hen Y 1 Y. are RCLL procee uch ha Ê[up [, ] V l β ] <, f l, g l ij aify (H1 ) and 1, for i, j = 1,, d, V V i an 3 Refleced G-BSDE wih a lower obacle and ome a priori eimae For impliciy, we conider he G-expecaion pace (Ω, L 1 G (Ω ), Ê) wih Ω = C ([, ], R) and σ = Ê[B 1] Ê[ B 1] = σ. Our reul and mehod ill hold for he cae d > 1. We are given he following daa: he generaor f and g, he obacle proce {S } [, ] and he erminal value ξ, where f and g are map f(, ω, y, z), g(, ω, y, z) : [, ] Ω R R. We will make he following aumpion: here exi ome β > uch ha (H1) for any y, z, f(,, y, z), g(,, y, z) M β G (, ); (H) f(, ω, y, z) f(, ω, y, z ) + g(, ω, y, z) g(, ω, y, z ) L( y y + z z ) for ome L > ; (H3) ξ L β G (Ω ) and ξ S, q..; (H4) here exi a conan c uch ha {S } [, ] S β G (, ) and S c, for each [, ]; (H4 ) {S } [, ] ha he following form S = S + b()d + l()d B + σ()db, where {b()} [, ], {l()} [, ] belong o M β G (, ) and {σ()} [, ] belong o H β G (, ). Le u now inroduce our refleced G-BSDE wih a lower obacle. A riple of procee (Y, Z, A) i called a oluion of refleced G-BSDE wih a lower obacle if for ome 1 < α β he following properie hold: (a) (Y, Z, A) S α G (, ) and Y S, ; (b) Y = ξ + f(, Y, Z )d + g(, Y, Z )d B Z db + (A A ); (c) { (Y S )da } [, ] i a non-increaing G-maringale. 6

7 Here we denoe by SG α(, ) he collecion of proce (Y, Z, A) uch ha Y Sα G (, ), Z Hα G (, ; R), A i a coninuou nondecreaing proce wih A = and A SG α (, ). For impliciy, we mainly conider he cae wih g and l. Similar reul ill hold for he cae g, l. Now we give a priori eimae for he oluion of he refleced G-BSDE wih a lower obacle. In he following, C will alway deignae a conan, which may vary from line o line. Propoiion 3.1 Le f aifie (H1) and (H). Aume Y = ξ + f(, Y, Z )d Z db + (A A ), where Y SG α(, ), Z Hα G (, ; R), A i a coninuou nondecreaing proce wih A = and A SG α (, ) for ome α > 1. hen here exi a conan C := C(α,, L, σ) > uch ha Ê [( Z d) α ] C{ Ê [ up Y α ] + (Ê[ up [, ] Ê [ A A α ] C{Ê[ up Y α ] + Ê[( [, ] [, ] Y α ]) 1/ (Ê[( f(,, ) d) α ]) 1/ }, (3.1) f(,, ) d) α ]}. (3.) Proof. he proof i imilar o ha of Propoiion 3.5 in [1]. So we omi i. Propoiion 3. For i = 1,, le ξ i L β G (Ω ), f i aify (H1) and (H) for ome β >. Aume Y i = ξ i + f i (, Y, Z )d Z i db + (A i A i ), where Y i SG α(, ), Zi HG α(, ), Ai i a coninuou nondecreaing proce wih A i = and A i SG α(, ) for ome 1 < α < β. Se Ŷ = Y 1 Y, Ẑ = Z 1 Z, Â = A 1 A. hen here exi a conan C := C(α,, L, σ) uch ha Ê[( Ẑ d) α ] Cα {(Ê[ up Ŷ α ]) 1/ [, ] + (Ê[( [(Ê[ up i=1 [, ] Y i α ]) 1/ f i (,, ) d) α ]) 1/ ] + Ê[ up Ŷ α ]}. [, ] Proof. he proof i imilar o ha of Propoiion 3.8 in [1]. So we omi i. Propoiion 3.3 For i = 1,, le ξ i L β G (Ω ) wih ξ i S i, where S i = S i + b i ()d + [, ] σ i ()db. Here {b i ()} M β G (, ), {σi ()} H β G (, ) for ome β >. Le f i aify (H1) and (H). Aume ha (Y i, Z i, A i ) SG α (, ) for ome 1 < α < β are he oluion of he refleced G-BSDE correponding o ξ i, f i and S i. Se Ỹ = (Y 1 S 1 ) (Y S ). hen here exi a conan C := C(α,, L, σ) uch ha Y i α CÊ[ ξ i α + up S i α + λ i, α d], Ỹ α CÊ[ ξ α + ( ˆλ α + ˆρ α + Ŝ α )d], where ξ = (ξ 1 S 1 ) (ξ S ), ˆλ = f 1 (, Y, Z ) f (, Y, Z ), ˆρ = b 1 () b () + σ 1 () σ (), Ŝ = S 1 S and = f i (,, ) + b i () + σ i (). λ i, 7

8 Proof. We will only how he econd inequaliy, ince he fir one can be proved in a imilar way. For any ε >, e ˆf = f 1 (, Y 1, Z 1 ) f (, Y, Z 1 ), ˆf = f 1 (, Y 1, Z 1 ) f 1 (, Y, Z ), Â = A 1 A, Z = (Z 1 σ 1 ()) (Z σ ()), ε α = ε(1 α/) + and Ȳ = Ỹ + ε α. Applying Iô formula o Ȳ α e r, where r > will be deermined laer, we ge Ȳ α/ e r + re r Ȳ α/ d + = (ε α + ξ ) α/ e r + α(1 α ) + α er Ȳ α/ 1 ( Z ) d B αe r Ȳ α/ 1 Ỹ ( ˆf + b 1 () b ())d + (ε α + ξ ) α/ e r + + α(1 α ) where M = Indeed, noe ha Conequenly, hen we obain αer Ȳ α/ 1 hu we can conclude ha e r Ȳ α/ (Ỹ) ( Z ) d B αe r Ȳ α/ 1 Ỹ dâ αe r Ȳ α 1 { ˆf 1 + b 1 () b () + ˆλ }d e r Ȳ α/ 1 ( Z ) d B (M M ), αe r Ȳ α/ 1 Ỹ Z db (3.3) (Ỹ Z db (Ỹ) + da 1 (Ỹ) da ). We claim ha {M } i a G-maringale. Ê[ Ỹ = Y 1 S 1 + S Y Y 1 S 1. (Ỹ) + (Y 1 S 1 ) + = Y 1 S 1. (Ỹ) + da 1 (Ỹ) + da 1 ] Ê[ (Y 1 S 1 )da 1. (Y 1 S 1 )da 1 ] =. I follow ha he proce {K 1 } [, ] = { (Ỹ) + da 1 } [, ] i a non-increaing G-maringale. Se {K } [, ] = { (Ỹ) da } [, ]. Boh {K 1 } and {K } are non-increaing G-maringale, o i { αer Ȳ α/ 1 (dk 1 + dk )}, which yield ha {M } [, ] i a G-maringale. From he aumpion of f 1, we derive ha αe r Ȳ α 1 ˆf 1 + b 1 () b () d αe r Ȳ α 1 {L( Ỹ + Z ) + (L 1)( Ŝ + ˆρ )}d (αl + αl σ (α 1) ) + (L 1) By Young inequaliy, we have 3(α 1) e r Ȳ α/ α(α 1) d + 4 αe r Ȳ α 1 { Ŝ + ˆρ }d. αe r Ȳ α 1 { ˆλ + Ŝ + ˆρ }d e r Ȳ α/ 1 ( Z ) d B e r Ȳ α/ d + e r { ˆλ α + ˆρ α + Ŝ α }d. 8 (3.4) (3.5)

9 By (3.3)-(3.5) and eing r = 3(L 1)(α 1) + αl + αl σ (α 1) + 1, we ge Ȳ α/ e r + (M M ) C{(ε α + ξ ) α/ e r + e r ( ˆλ α + ˆρ α + Ŝ α )d}. aking condiional expecaion on boh ide and hen by leing ε, we have he proof i complee. Ỹ α CÊ[ ξ α + ( ˆλ α + ˆρ α + Ŝ α )d]. Propoiion 3.4 Le (ξ, f, S) aify (H1)-(H4). Aume ha (Y, Z, A) SG α (, ), for ome α < β, i a oluion of he refleced G-BSDE wih daa (ξ, f, S). hen here exi a conan C := C(α,, L, σ, c) > uch ha Y α CÊ[1 + ξ α + f(,, ) α d]. Proof. For any r >, e Ỹ = Y c. Applying Iô formula o Ỹ α/ e r, noing ha S c and A i a nondecreaing proce, we have Ỹ α/ e r + = ξ c α e r + re r Ỹ α/ d + α e r Ỹ α/ 1 Z d B αe r Ỹ α/ 1 (Y c)f(, Y, Z )d + α(1 α ) αe r Ỹ α/ 1 (Y c)z db + ξ c α e r + αe r Ỹ α/ 1 (Y c)da e r Ỹ α/ (Y c) Z B αe r Ỹ α 1 f(, Y, Z ) d + α(1 α ) e r Ỹ α/ 1 Z B (M M ), where M = αe r Ỹ α/ 1 (Y c)z db αe r Ỹ α/ 1 (Y S )da. By condiion (c), M i a G-maringale. From he aumpion of f and by he Young inequaliy, we ge αe r Ỹ α 1 f(, Y, Z ) d Seing r = α + αl + αl σ (α 1) αe r Ỹ α 1 [ f(, c, ) + L Ỹ + L Z ]d (αl + αl σ (α 1) ) + α(α 1) 4 e r Ỹ α/ d + (α 1) e r Ỹ α/ 1 Z B + and by he above analyi, we have Ỹ α/ e r + M M ξ c α e r + aking condiional expecaion on boh ide yield ha Y c α CÊ[ ξ c α + e r f(, c, ) α d. f(, c, ) α d]. e r Ỹ α/ d e r f(, c, ) α d. Noing ha for p 1, we have a + b p p 1 ( a p + b p ). hen he proof i complee. (3.6) 9

10 Propoiion 3.5 Le (ξ 1, f 1, S 1 ) and (ξ, f, S ) be wo e of daa, each one aifying all he aumpion (H1)-(H4). Le (Y i, Z i, A i ) SG α (, ) be a oluion of he refleced G-BSDE wih daa (ξ i, f i, S i ), i = 1, repecively wih α < β. Se Ŷ = Y 1 Y, Ŝ = S 1 S, ˆξ = ξ 1 ξ. hen here exi a conan C := C(α,, L, σ, c) > uch ha Ŷ α C{Ê[ ˆξ α + where ˆλ = f 1 (, Y, Z ) f (, Y, Z ) and Ψ, = i=1 ˆλ α d] + (Ê[ up Ŝ α ]) 1 α 1 α Ψ α, }, [, ] Ê [ up Ê [1 + ξ i α + [, ] f i (r,, ) α dr]]. Proof. Se Ẑ = Z 1 Z, ˆf = f 1 (, Y 1, Z 1 ) f (, Y, Z ) and ˆf 1 = f 1 (, Y 1, Z 1 ) f 1 (, Y For any r >, by applying Iô formula o Ȳ α/ Ȳ α/ e r + re r Ȳ α/ α d + = ˆξ α e r + α(1 α ) + αe r Ȳ α/ 1 Ŷ ˆf d + ˆξ α e r + α(1 α ) + e r = ( Ŷ ) α/ e r, we have er Ȳ α/ 1 (Ẑ) d B e r Ȳ α/ (Ŷ) (Ẑ) d B αe r Ȳ α/ 1 Ŷ dâ e r Ȳ α/ 1 (Ẑ) d B + αe r Ȳ α 1 { ˆf 1 + ˆλ }d (M M ), αe r Ȳ α/ 1 Ŷ Ẑ db αe r Ȳ α/ 1 Ŝ dâ, Z ). where M = αer Ȳ α/ 1 Ŷ Ẑ db αer Ȳ α/ 1 (Ŷ Ŝ) da αer Ȳ α/ 1 (Ŷ Ŝ) + da 1. By a imilar analyi a he proof of Propoiion 3.3, we conclude ha {M } [, ] i a G-maringale. By Young inequaliy and he aumpion of f 1, imilar wih inequaliie (3.4) and (3.5), we have Se r = α + αl + αe r Ȳ α 1 { ˆf 1 α(α 1) + ˆλ }d 4 e r Ỹ α/ 1 Z B + + (α 1 + αl + αl σ (α 1) ) e r ˆλ α d e r Ỹ α/ d. αl σ (α 1). aking condiional expecaion on boh ide of (3.7), we obain Ŷ α C{Ê[ ˆξ α + By applying Hölder inequaliy, we ge Ê [ ˆλ α d] + Ê[ Ȳ α/ 1 Ŝ d(a 1 + A )]}. Ȳ α/ 1 Ŝ d(a 1 + A )] Ê[ up Ȳ α/ 1 Ŝ ( A 1 A 1 + A A )] [, ] (Ê[ up Ŝ α ]) 1 α ( Ê [ up [, ] [, ] Ȳ α/ ]) α α From Propoiion 3.1 and Propoiion 3.4, we finally ge he deired reul. ( Ê [ A i A i α ]) 1 α. i=1 (3.7) 1

11 Remark 3.6 If we require ha he oluion of a refleced G-BSDE i a quadruple {(Y, Z, K, A ), } aifying condiion (1)-(3) in he inroducion, he oluion i no unique. We can ee hi fac from he following example. 1 Le f 1, ξ = and S. I i eay o check ha (,,, ) and (,, σ σ (σ 1 B ), σ σ ( σ B )) are oluion of refleced G-BSDE wih daa (, 1, ) aifying all he condiion (1)-(3). 4 Penalized mehod and convergence properie In order o derive he exience of he oluion o he refleced G-BSDE wih a lower obacle, we hall apply he approximaion mehod via penalizaion. In hi ecion, we fir ae ome convergence properie of oluion o he penalized G-BSDE, which will be needed in he equel. For f and ξ aify (H1)-(H3), {S } [, ] aifie (H4) or (H4 ), we now conider he following family of G-BSDE parameerized by n = 1,,, Y n = ξ + f(, Y n, Z n )d + n (Y n S ) d Z n db (K n K n ). (4.1) Now le L n = n (Y n S ) d, hen (L n ) [, ] i a nondecreaing proce. We can rewrie refleced G-BSDE (4.1) a Y n = ξ + f(, Y n, Z n )d Z n db (K n K n ) + (L n L n ). (4.) We now eablih a priori eimae on he equence (Y n, Z n, K n, L n ) which are uniform in n. Lemma 4.1 here exi a conan C independen of n, uch ha for 1 < α < β, Ê[ up Y n α ] C, Ê[ K n α ] C, Ê[ L n α ] C, Ê[( Z n d) α ] C. [, ] Proof. For impliciy, fir we conider he cae S. he proof of he oher cae will be given in he remark. r, ε >, e Ỹ = (Y n ) + ε α, where ε α = ε(1 α/) +. Noe ha for any a R, a a. By applying Iô formula o Ỹ α/ e r yield ha Ỹ α/ e r + re r Ỹ α/ α d + er Ỹ α/ 1 (Z n ) d B = ( ξ + ε α ) α e r + α(1 α ) e r Ỹ α/ (Y n ) (Z n ) d B + αe r Ỹ α/ 1 Y n dl n + αe r Ỹ α/ 1 Y n f(, Y n, Z n )d αe r Ỹ α/ 1 (Y n Z n db + Y n dk n ) ( ξ + ε α ) α e r + α(1 α ) e r Ỹ α/ (Y n ) (Z n ) d B + where M = αe r Ỹ α/ 1/ f(, Y n, Z n ) d (M M ), αe r Ỹ α/ 1 (Y n Z n db +(Y n ) + dk n ) i a G-maringale. Similar wih inequaliy (3.6), 11

12 we have αe r Ỹ α 1 f(, Y n, Z n ) d e r f(,, ) α α(α 1) d + 4 Se r = α + αl + αl σ (α 1). We derive ha + (α 1 + αl + αl σ (α 1) ) Ỹ α/ e r + M M ( ξ + ε α ) α e r + e r Ỹ α/ 1 (Z n ) d B e r Ỹ α/ d. e r f(,, ) α d. aking condiional expecaion on boh ide and hen by leing ε, we obain Y n α CÊ[ ξ α + f(,, ) α d]. By heorem.4, for 1 < α < β, here exi a conan C independen of n uch ha Ê[up [, ] Y n α ] C. By Propoiion 3.1, we have Ê[( Z n d) α ] Cα {Ê[ up Y n α ] + (Ê[ up Y n α ]) 1 ( Ê[( [, ] [, ] Ê[ L n K n α ] C α {Ê[ up Y n α ] + Ê[( f(,, ) d) α ]}, [, ] f(,, ) d) α ]) 1 }, where he conan C α depend on α,, σ and L. hu we conclude ha here exi a conan C independen of n, uch ha for 1 < α < β, Since L n and Kn Ê[( Z n d) α ] C, Ê[ L n K n α ] C. are nonnegaive, i follow ha Ê[ K n α ] C, Ê[ L n α ] = n α Ê[( (Y n ) ) α ] C. Remark 4. If he obacle proce {S } [, ] aifie (H4), e Ỹ n ha = Y n c. I i imple o check Ỹ n = ξ c + f(, Ỹ n + c, Z n )d + n(ỹ n (S c)) d Z n db (K n K n ). By a imilar analyi a he proof of Lemma 4.1, we derive ha Ỹ n α CÊ[ ξ c α + f(, c, ) α d]. If S aifie (H4 ), for impliciy we uppoe ha l. Le Ỹ n we can rewrie (4.1) a he following: Ỹ n = ξ S + [f(, Ỹ n + S, Z n + σ()) + b()]d + n 1 = Y n S and Z n = Z n σ(), (Ỹ n ) d Z n db (K n K n ).

13 Uing he ame mehod, we ge Ỹ n α CÊ[ ξ S α + f(, S, σ()) + b() α d]. hu we conclude ha in he above wo cae, for 1 < α < β, here exi a conan C independen of n uch ha Ê[up [, ] Y n α ] C. By Propoiion 3.1, we have Ê[ K n α ] C, Ê[ L n α ] = n α Ê[( (Y n S ) d) α ] C, and Ê[( Z n d) α ] C. Lemma 4.1 implie ha (Y n S) in MG 1 (, ). he following lemma which correpond o Lemma 6.1 in [5] how ha hi convergence hold in SG α (, ), for 1 < α < β. I i of vial imporance o prove he convergence propery for (Y n ). Lemma 4.3 For ome 1 < α < β, we have lim Ê[ up (Y n S ) α ] =. n [, ] Proof. We now conider he following G-BSDE parameerized by n = 1,,, y n = ξ + f(, Y n, Z n )d + n(s y n )d z n db (k n k n ). By applying G-Iô formula o e n y n, we can ge y n = e n Ê [e n ξ + ne n S d + e n f(, Y n, Z n )d]. By he comparion heorem.9, we have for all n 1, Y n Y 1 and Y n S y n S = Ê[ S n + e n( ) f(, Y n, Z n )d], where S n = e n( ) (ξ S ) + ne n( ) (S S )d. I follow ha (Y n S ) (y n S ) Ê[ S n + e n( ) f(, Y n, Z n )d ]. Applying Hölder inequaliy yield ha e n( ) f(, Y n, Z n )d 1 ( f (, Y n, Z n )d) 1/ n By Lemma 4.1, for 1 < α < β, we have C ( up Y n n + [, ] f (,, ) + Z n d) 1/. Ê[ up [, ] e n( ) f(, Y n, Z n )d α ], a n. (4.3) 13

14 For ε >, i i raighforward o how ha S n = e n( ) (ξ S ) + ne n( ) (S S )d + +ε e n( ) ξ S + e nε up [+ε, ] For > δ >, from he above inequaliy we obain up S n e nδ up ξ S + e nε up [, δ] [, δ] +ε S S + up S S. [,+ε] up [, δ] [+ε, ] e nδ ( up S + ξ ) + e nε up S + up [, ] [, ] I i eay o check ha for each fixed ε, δ >, ne n( ) (S S )d S S + up up [, ] [,+ε] up [, δ] [,+ε] S S. S S Ê[ up S n β ] C{(e nβε nβδ + e )Ê[ up S β + ξ β ] + Ê[ up [, δ] [, ] CÊ[ up up [, ] [,+ε] S S β ], a n. up [, ] [,+ε] S S β ]} (4.4) For 1 < α < β and < δ <, we have Ê[ up (Y n S ) α ] Ê[ up [, ] Ê[ up {Ê[ S n + C{Ê[ [, δ] up [, δ] Ê [ up u [, δ] [, δ] (Y n S ) α ] + Ê[ up (Y n S ) α ] [ δ, ] e n( ) f(, Y n, Z n )d ]} α ] + Ê[ up [ δ, ] (Y 1 S ) α ] S u n α ]] + Ê[ up Ê [ up e n( ) f(, Y n, Z n )d α ]]} [, δ] u [, ] u + Ê[ up (Y 1 S ) α ] =: I + Ê[ up (Y 1 S ) α ]. [ δ, ] [ δ, ] (4.5) By Lemma.7, noing ha Y 1 S S α G (, ) and (Y 1 S ) =, we obain lim Ê[ up (Y 1 S ) α ] =. δ [ δ, ] By heorem.4 and combining (4.3), (4.4), we derive ha I C{Ê[ up up [, ] [,+ε] S S β ] + (Ê[ up up [, ] [,+ε] S S β ]) α/β }, a n. Now fir le n and hen le ε, δ in (4.5). By Lemma.7 again, he above analyi prove ha for 1 < α < β, lim Ê[ up (Y n S ) α ] =. n [, ] Now we how he convergence propery of equence (Y n ) n=1. Lemma 4.4 For ome β > α, we have lim Ê[ up Y n Y m α ] =. n,m [, ] 14

15 Proof. Wihou lo of generaliy, we may aume S in (4.1). For any r >, e Ȳ = Y n Y m, ˆf = f(, Y n, Z n ) f(, Y m, Z m ). By applying Iô formula o Ȳ α/ e r, we ge Ȳ α/ e r + re r Ȳ α/ α d + er Ȳ α/ 1 (Ẑ) d B where M = we have = α(1 α ) e r Ȳ α/ (Ŷ) (Ẑ) d B + + αe r Ȳ α/ 1 Ŷ ˆf d αe r Ȳ α/ 1 α(1 α ) e r Ȳ α/ (Ŷ) (Ẑ) d B + αer Ȳ α/ 1 αe r Ȳ α/ 1 Y n dl m αe r Ȳ α/ 1 Ŷ dˆl (ŶẐdB + Ŷd ˆK ) αe r Ȳ α 1 ˆf d αe r Ȳ α/ 1 Y m dl n (M M ), (ŶẐdB + (Ŷ) + dk m + (Ŷ) dk n ) i a G-maringale. Similar wih (3.4), αe r Ȳ α 1 ˆf d (αl + αl σ (α 1) ) Le r = 1 + αl + αl σ (α 1). By he above analyi, we have Ȳ α/ e r + (M M ) e r Ȳ α/ α(α 1) d + e r Ȳ α/ 1 (Ẑ) d B. 4 αe r Ȳ α/ 1 Y n dl m αe r Ȳ α/ 1 Y m dl n. hen aking condiional expecaion on boh ide of he above inequaliy, we conclude ha Oberve ha Ȳ α/ e r Ê[ αe r Ȳ α/ 1 Y n dl m αe r Ȳ α/ 1 Y m dl n ]. (4.6) Ê [ CÊ[ αe r Ȳ α/ 1 Y m dl n ] αe r Ê [ n (Y n ) α 1 (Y m ) d] + CÊ[ From (4.6) and aking expecaion on boh ide, we deduce ha Ȳ α/ 1 n(y n ) (Y m ) d] n (Y m ) α 1 (Y n ) d]. Ê[ up Y n [, ] Y m α ] CÊ[ up + Ê[ [, ] {Ê[ (n + m) (Y n ) α 1 (Y m ) d] (n + m) (Y m ) α 1 (Y n ) d]}]. (4.7) For α < β, here exi α, p, q, r, p, q > 1, uch ha 1 p + 1 q + 1 r = 1, 1 p + 1 q = 1, (α )α p < β, α q < β, α r < β, (α 1)α p < β and α q < β. Applying Lemma 4.1, Lemma 4.3 and he Hölder 15

16 inequaliy, here exi a conan C independen of m, n uch ha and Ê[( n (Y n ) α 1 (Y m ) d) α ] Ê[ up { (Y n ) (α )α (Y m ) α }( [, ] n(y n ) d) α ] (Ê[ up (Y n ) (α )α p ]) 1 p ( Ê[ up (Y m ) α q ]) 1 q ( Ê[( [, ] [, ] C(Ê[ up (Y m ) α q ]) 1 q, [, ] Ê[( m (Y n ) α 1 (Y m ) d) α ] Ê[ up (Y n ) (α 1)α ( [, ] m(y m ) d) α ] (Ê[ up (Y n ) (α 1)α p ]) 1 p (Ê[( m(y m ) d) α q ]) 1 q [, ] C(Ê[ up (Y n ) (α 1)α p ]) 1 p. [, ] hen by heorem.4 and Lemma 4.3, inequaliie (6.1)-(6.3) yield ha lim Ê[ up Y n Y m α ] =. n,m [, ] n(y n ) d) α r ]) 1 r (4.8) (4.9) 5 Exience and uniquene of refleced G-BSDE wih a lower obacle heorem 5.1 Suppoe ha ξ, f aify (H1)-(H3) and S aifie (H4) or (H4 ). hen he refleced G-BSDE wih daa (ξ, f, S) ha a unique oluion (Y, Z, A). Moreover, for any α < β we have Y SG α(, ), Z Hα G (, ) and A Sα G (, ). Proof. he uniquene of he oluion i a direc conequence of he a priori eimae in Propoiion 3., Propoiion 3.3 and Propoiion 3.5. o prove he exience, i uffice o prove he S cae. Recalling penalized G-BSDE (4.1), e Ŷ = Y n Y m, Ẑ = Z n Z m, ˆK = K n K m, ˆL = L n L m and ˆf = f(, Y n, Z n ) f(, Y m, Z m ). By Lemma 4.4, here exi Y SG α(, ) aifying lim n Ê[up [,] Y Y n α ] =. Applying Iô formula o Ŷ, we ge = Ŷ + L Ŷ ˆf d Ẑ d B Ŷd ˆK + [ Ŷ + Ŷ Ẑ ]d ŶdˆL Ŷd ˆK + 16 ŶẐdB ŶdˆL ŶẐdB.

17 Noe ha for each ε >, Chooing ε < σ, we have L Ŷ Ẑ d L /ε Ŷ d + ε Ẑ d. Ẑ d C( Ŷ d Ŷ d ˆK + Ŷ dˆl Ŷ Ẑ db ) C( up Ŷ + [, ] up [, ] Ŷ ( K n + K m + L n + L m ) Ŷ Ẑ db ). (5.1) By Propoiion.6, for any ε >, we obain Ê[( Ŷ Ẑ db ) α ] C Ê[( Ŷ d) α 4 ] C(Ê[ up Ŷ α ]) 1/ (Ê[( [, ] C Ê[ up Ŷ α ] + Cε Ê[( 4ε [, ] Ẑ d) α ]) 1/ Ẑ d) α ]. Applying Lemma 4.1 and he Hölder inequaliy, chooing ε mall enough, i follow from (5.1) ha I i raighforward o how ha Ê[( Z n Z m d) α ] C{ Ê[ up Ŷ α ] + (Ê[ up Ŷ α ]) 1/ }. [, ] [, ] lim Ê[( Z n Z m d) α ] =. n,m hen here exi a proce {Z } H α G (, ) uch ha Ê[( Z Z n d) α/ ] a n. Se A n = L n K n, i i eay o check ha (A n ) [, ] i a nondecreaing proce and A n A m = Ŷ Ŷ ˆf d + Ẑ db. By applying Propoiion.6 and he aumpion of f, i follow ha Ê[ up A n A m α ] CÊ[ up Ŷ α + ( ˆf d) α + up [, ] [, ] [, ] C{Ê[ up Ŷ α ] + Ê[( Ẑ d) α/ ]}. [, ] hen here exi a nondecreaing proce (A ) [, ] aifying ha lim Ê[ up A A n α ] =. n [, ] Ẑ db α ] 17

18 In he following i remain o prove ha Y, [, ] and { Y da } [, ] i a nonincreaing G-maringale. For he fir aemen, i can be deduced eaily from Lemma 4.3. Se K n := Y dk n. Since Y, and K n i a decreaing G-maringale, hen K n i a decreaing G-maringale. Noe ha where Ỹ m up [, ] Y da K n up [, ] + { up { [, ] Y da + (Y n Y )dk n + Y da n + Ỹ m d(a n A ) + Y n n(y n (Y n Y )da n ) d } (Y Ỹ m )d(a n A ) } + up Y Y n [ A n + K n ] + up (Y n ) L n [, ] [, ] =I + II + III + IV, = m 1 i= Y mi i [ m i,m i+1 ) () and m i = i m, i =, 1,, m. By imple analyi, we have m 1 Ê[I] Ê[ up Y ( A n A m m + An i+1 i+1 m i [, ] Ê[II] Ê[III] Ê[IV ] i= A m i )] m 1 (Ê[ up Y ]) 1/ {(Ê[ An A m m ]) 1/ + (Ê[ An i+1 i+1 m i [, ] i= (Ê[ up Y Ỹ m ]) 1/ {(Ê[ An ]) 1/ + (Ê[ A ]) 1/ }, [, ] (Ê[ up Y Y n ]) 1/ {(Ê[ An ]) 1/ + (Ê[ Kn ]) 1/ }, [, ] (Ê[ up (Y n ) ]) 1/ (Ê[ Ln ]) 1/. [, ] A m i ]) 1/ }, hen for each fixed m, fir le n end o infiniy, we conclude ha lim Ê[ up n [, ] Y da K n ] C(Ê[ up Y Ỹ m ]) 1/. [, ] By Lemma 3. in [1], ending m end o infiniy, we ge lim n Ê[up [, ] Y da K n ] =. I follow ha { Y da } i a non-increaing G-maringale. Furhermore, we have he following reul. heorem 5. Suppoe ha ξ, f and g aify (H1)-(H3), S aifie (H4) or (H4 ). hen he refleced G-BSDE wih daa (ξ, f, g, S) ha a unique oluion (Y, Z, A). Moreover, for any α < β we have Y SG α(, ), Z Hα G (, ) and A Sα G (, ). Proof. he proof i imilar o ha of heorem 5.1. We nex prove a comparion heorem, imilar o ha of [11] for non-refleced G-BSDE. he proof i baed on he approximaion mehod via penalizaion. 18

19 heorem 5.3 Le (ξ 1, f 1, g 1, S 1 ) and (ξ, f, g, S ) be wo e of daa. Suppoe S i aify (H4) or (H4 ), ξ i, f i and g i aify (H1)-(H3) for i = 1,. We furher aume in addiion he following: (i) ξ 1 ξ, q..; (ii) f 1 (, y, z) f (, y, z), g 1 (, y, z) g (, y, z), (y, z) R ; (iii) S 1 S,, q... Le (Y i, Z i, A i ) be a oluion of he refleced G-BSDE wih daa (ξ i, f i, g i, S i ), i = 1, repecively. hen Y 1 Y, q.. Proof. We fir conider he following G-BSDE parameerized by n = 1,,, y n = ξ 1 + f 1 (, y n, z n )d + g 1 (, y n, z n )d B + n(y n S 1 ) d z n db (K n K n ). By a imilar analyi a he proof of heorem 5.1, i follow ha lim n Ê[up [, ] Y 1 y n α ] =, where α < β. Noing ha (Y, Z, A ) i he oluion of he refleced G-BSDE wih daa (ξ, f, g, S ) and Y S,, we have Y = ξ + f (, Y, Z )d+ g (, Y, Z )d B + n(y S ) d Z db +(A A ). Applying heorem.9 yield Y y n, for all n N. Leing n, we conclude ha Y Y 1. Remark 5.4 Acually, A can be repreened a he um of wo nondecreaing procee A 1 and A uch ha and for any, Ê [ (Y S )da =, (S r Y r )da 1 r] = (S r Y r )da 1 r. (5.) Indeed, e A 1 = I {Y >S }da, A = I {Y =S }da. I i eay o check ha A = A 1 + A and A aifie he Skorohod condiion. We now how ha A 1 aifie (5.). Se K := (S Y )da. By heorem 5.1, K i a decreaing G-maringale and K L p G (Ω ) for ome 1 < p < β, [, ]. Chooe a equence of bounded, nonnegaive and Lipchiz coninuou funcion ϕ n (x) uch ha ϕ(x) I {x>}. Se K := ϕ n (Y S )d K = (S Y )ϕ n (Y S )da. Applying Lemma 3.4 in [1], We obain ha K i a decreaing G-maringale. Furhermore, we have ϕ n (Y S )d K (S Y )da 1 L 1 G (Ω ), 19

20 where L 1 G (Ω ) i defined in he Appendix. By he exended condiional G-expecaion defined in [13], we derive ha Ê [ (S r Y r )da 1 r] = lim Ê [ n = lim = n (S r Y r )ϕ n (Y r S r )da r ] (S r Y r )ϕ n (Y r S r )da r (S r Y r )da 1 r. 6 Relaion beween refleced G-BSDE and obacle problem for nonlinear parabolic PDE In hi ecion, we will give a probabiliic repreenaion for oluion of ome obacle problem for fully nonlinear parabolic PDE uing he refleced G-BSDE we have menioned in he above ecion. For hi purpoe, we need o pu he refleced G-BSDE in a Markovian framework. For each (, x) [, ] R d, le {X,x, } be he unique R d -valued oluion of he SDE driven by G-Brownian moion (here we ue Einein convenion): X,x = x + b(r, X,x r )dr + l ij (r, X,x r )d B i, B j r + We aume ha he daa (ξ, f, g, S) of he refleced G-BSDE ake he following form: ξ = φ(x,x ), f(, y, z) = f(, X,x, y, z), S = h(, X,x ), g ij (, y, z) = g ij (, X,x, y, z), σ i (r, X,x r )db i r. (6.1) where b : [, ] R d R d, l ij : [, ] R d R d, σ i : [, ] R d R d, φ : R d R, f, g ij : [, ] R d R R d R and h : [, ] R d R are deerminiic funcion and aify he following condiion: (A1) l ij = l ji and g ij = g ji for 1 i, j d; (A) b, l ij, σ i, f, g ij are coninuou in ; (A3) here exi a poiive ineger m and a conan L uch ha b(, x) b(, x ) + d l ij (, x) l ij (, x ) + i,j=1 d σ i (, x) σ i (, x ) L x x, φ(x) φ(x ) L(1 + x m + x m ) x x, d f(, x, y, z) f(, x, y, z ) + g ij (, x, y, z) g ij (, x, y, z ) i.j=1 i=1 L[(1 + x m + x m ) x x + y y + z z ]. (A4) h i uniformly coninuou w.r. (, x) and bounded from above, h(, x) Φ(x) for any x R d ; (A4 ) h belong o he pace C 1, Lip ([, ] Rd ) and h(, x) Φ(x) for any x R d, where C 1, Lip ([, ] R d ) i he pace of all funcion of cla C 1, ([, ] R d ) whoe parial derivaive of order le han or equal o and ielf are Lipchiz coninuou funcion wih repec o x.

21 We have he following eimae of G-SDE, which come from Chaper V of Peng [7]. Propoiion 6.1 ([7]) Le ξ, ξ L p G (Ω ; R d ) and p. hen we have, for each δ [, ], Ê [ up [,+δ] Ê [ up X,ξ [,+δ] where he conan C depend on L, G, p, d and. X,ξ p ] C ξ ξ p, Ê [ X,ξ +δ p ] C(1 + ξ p ), X,ξ ξ p ] C(1 + ξ p )δ p/, Proof. For he reader convenience, we give a brief proof here. I i eay o check ha {X,ξ } [, ], } [, ] M p G (, ; Rd ). By Propoiion.6, we have {X,ξ Ê [ up [,+δ] X,ξ CÊ[ ξ ξ p + C{ ξ ξ p + Ê[ C{ ξ ξ p + X,ξ p ] +δ +δ +δ X,ξ X,ξ Ê [ up Xr,ξ r [,] X,ξ p d + up [,+δ] +δ X,ξ p d] + Ê[( Xr,ξ p ]d}. (σ(r, Xr,ξ ) σ(r, Xr,ξ ))db r p ] X,ξ X,ξ d) p/ ]} By he Gronwall inequaliy, we ge he fir inequaliy. he oher can be proved imilarly. I follow from he previou reul ha for each (, x) [, ] R d, here exi a unique riple, Z,x, A,x ) [, ], which olve he following refleced G-BSDE: (Y,x (i) Y,x (ii) Y,x = φ(x,x ) + db r + A,x Z,x r f(r, X,x r h(, X,x ), ; (iii) {A,x non-increaing G-maringale., Y,x, Z,x )dr + r A,x, ; } i nondecreaing and coninuou, and { r g ij(r, X,x r (Y,x r, Y,x, Z,x )d B i, B j r r r h(r, X,x r ))da,x, } i a r We now conider he following obacle problem for a parabolic PDE. { min( u(, x) F (D xu, D x u, u, x, ), u(, x) h(, x)) =, (, x) (, ) R d, u(, x) = φ(x), x R d, (6.) where F (D xu, D x u, u, x, ) =G(H(D xu, D x u, u, x, )) + b(, x), D x u + f(, x, u, σ 1 (, x), D x u,, σ d (, x), D x u ), H(D xu, D x u, u, x, ) = D xuσ i (, x), σ j (, x) + D x u, l ij (, x) + g ij (, x, u, σ 1 (, x), D x u,, σ d (, x), D x u ). We need o conider oluion of he above PDE in he vicoiy ene. he be candidae o define he noion of vicoiy oluion i by uing he language of ub- and uper-je; (ee []). 1

22 Definiion 6. Le u C((, ) R d ) and (, x) (, ) R d. We denoe by P,+ u(, x) [he parabolic uperje of u a (, x)] he e of riple (p, q, X) R R d S d which are uch ha u(, y) u(, x) + p( ) + q, y x + 1 X(y x), y x + o( + y x ). Similarly, we define P, u(, x) [he parabolic ubje of u a (, x)] by P, u(, x) := P,+ ( u)(, x). hen we can give he definiion of he vicoiy oluion of he obacle problem (6.). Definiion 6.3 I can be aid ha u C([, ] R d ) i a vicoiy uboluion of (6.) if u(, x) φ(x), x R d, and a any poin (, x) (, ) R d, for any (p, q, X) P,+ u(, x), min(u(, x) h(, x), p F (X, q, u(, x), x, )). I can be aid ha u C([, ] R d ) i a vicoiy uperoluion of (6.) if u(, x) φ(x), x R d, and a any poin (, x) (, ) R d, for any (p, q, X) P, u(, x), min(u(, x) h(, x), p F (X, q, u(, x), x, )). u C([, ] R d ) i aid o be a vicoiy oluion of (6.) if i i boh a vicoiy uboluion and uperoluion. We now define u(, x) := Y,x, (, x) [, ] R d. (6.3) I i imporan o noe ha u(, x) i a deerminiic funcion. We claim ha u i a coninuou funcion. For impliciy, we only conider he cae g = in he nex hree lemma. he reul ill hold for he oher cae. Lemma 6.4 Le aumpion (A1)-(A3) and (A4 ) hold. For each [, ], x 1, x R d, we have u(, x 1 ) u(, x ) C(1 + x 1 m + x m ) x 1 x. Proof. From Propoiion 3.3, ince u(, x) i a deerminiic funcion, we have u(, x 1 ) u(, x ) C{Ê[ (φ(x,x1 + + f(, X,x1 + h(, X,x1 ) h(, X,x1, Y,x1, Z,x1 )) (φ(x,x ) h(, X,x )) ) f(, X,x, Y,x1, Z,x1 ) d b 1 () b () + l 1 ij() l ij() + σ 1 i () σ i () ) h(, X,x ) d] + h(, x 1 ) h(, x ) }, (6.4) where for k = 1,, b k () = h(, X,x k l k ij() = D x h(, X,x k ) + b(, X,x k ), D x h(, X,x k ), ), l ij (, X,x k σi k () = σ i (, X,x k ), D x h(, X,x k ). ) + 1 D xh(, X,x k )σ i (, X,x k ), σ j (, X,x k ),

23 Se ˆX = X,x1 X,x. By he aumpion (A3), (A4 ) and Propoiion 6.1, we have u(, x 1 ) u(, x ) C{Ê[(1 + he proof i complee. + k=1 Ê[(1 + X,x k m ) ˆX ] + k=1 Ê[(1 + k=1 X,x k m ) ˆX ]d X,x k ) ˆX ]d + Ê[ ˆX ]d + x 1 x } C(1 + x 1 m 4 m 4 + x ){(Ê[ up ˆX 4 ]) 1/ + x 1 x } [, ] C(1 + x 1 m 4 + x m 4 ) x 1 x. Lemma 6.5 Le aumpion (A1)-(A4) hold. For each [, ], x, x R d, we have u(, x 1 ) u(, x ) C{(1 + x 1 m + x m ) x 1 x + (1 + x 1 m+1 + x m+1 ) x 1 x }. Proof. From Propoiion 3.5 and Propoiion 6.1, by a imilar analyi wih he above lemma, we ge he deired reul. he following lemma ae ha u(, x) i coninuou wih repec o. Lemma 6.6 he funcion u(, x) i coninuou in. Proof. We only need o prove he cae where (A1)-(A3) and (A4 ) hold. he cae ha (A1)-(A4) hold can be proved in a imilar way. We define X,x := x, Y,x := Y,x, Z,x := and A,x := for. hen we define he obacle { S u,x h(, x) + u = b(, X,x )d + u l ij (, X,x )d B i, B j + u σ i (, X,x )db, i u (, ]; h(, x), u [, ], where b(, X,x lij (, X,x σ i (, X,x I i eay o check ha (Y,x ), f,x, S,x ) where (φ(x,x ) = h(, X,x ) + b(, X,x ), D x h(, X,x ), ) = D x h(, X,x ), l ij (, X,x ) + 1 D xh(, X,x )σ i (, X,x ), σ j (, X,x ), ) = σ i (, X,x ), D x h(, X,x )., Z,x, A,x ) [, ] i he oluion o he refleced G-BSDE wih daa f,x (, y, z) := I [, ] ()f(, X,x, y, z). Fix x R d, for 1, by Propoiion 3.3, we have where u( 1, x) u(, x) = Y 1,x Y,x C{Ê[ (φ(x1,x ) h(, X 1,x )) (φ(x,x ) h(, X,x )) + h( 1, x) h(, x) + ˆλ 1, () + ˆρ 1, () + h(, X 1,x ) h(, X,x ) d]}, ˆλ 1, () = I [1, ]()f(, X 1,x, Y,x, Z,x ) I [, ]()f(, X,x, Y,x, Z,x ), 3

24 and Se ˆX x = X 1,x ˆρ 1, () = I [1, ]() b(, X 1,x ) I [, ]() b(, X,x ) + I [1, ]() l ij (, X 1,x ) I [, ]() l ij (, X,x ) + I [1, ]() σ i (, X 1,x ) I [, ]() σ i (, X,x ). X,x. By Hölder inequaliy and aumpion (A3), (A4 ), we deduce ha u( 1, x) u(, x) C{Ê[(1 + X1,x m + X,x m ) ˆX x ] + h( 1, x) h(, x) + Ê[(1 + X 1,x m + X,x m ) ˆX x ]d Noe ha ˆX x,x 1,x = X + 1 Ê[1 + X 1,x (m+) 6 + Y,x ]d}. X,x, for [, ]. Applying Propoiion 6.1, i follow ha u( 1, x) u(, x) C{(1 + x (m+1) 3 ) h(, x) h( 1, x) }. he proof i complee. We will ue he approximaion of he refleced G-BSDE by penalizaion. For each (, x) [, ] R d, n N, le {(Y n,,x, Z n,,x, K n,,x ), } denoe he oluion of he G-BSDE: Y n,,x We define =φ(x,x ) + + n (Y n,,x r f(r, X,x r, Yr n,,x, Zr n,,x )dr + h(r, Xr,x )) dr g ij (r, X,x r u n (, x) := Y n,,x,, x R d., Y n,,x, Z n,,x )d B i, B j r r Zr n,,x db r (K n,,x K n,,x ),. By heorem 4.5 in [11], u n i he vicoiy oluion of he parabolic PDE { u n (, x) F n (Dxu n (, x), D x u n (, x), u n (, x), x, ) =, (, x) [, ] R d u n (, x) = φ(x), x R d, r (6.5) where F n (D xu, D x u, u, x, ) = F (D xu, D x u, u, x, ) + n(u h(, x)). heorem 6.7 he funcion u defined by (6.3) i he unique vicoiy oluion of he obacle problem (6.). Proof. From he reul of he previou ecion, for each (, x) [, ] R d, we obain u n (, x) u(, x), a n. By Propoiion 4., heorem 4.5 in [11] and Lemma 6.4-Lemma 6.6, u n and u are coninuou. hen by applying Dini heorem, he equence u n uniformly converge o u on compac e. We fir how ha u i a uboluion of (6.). For each fixed (, x) (, ) R d, le (p, q, X) P,+ u(, x). Wihou lo of generaliy, we may aume ha u(, x) > h(, x). By Lemma 6.1 in [], here exi equence n j, ( j, x j ) (, x), (p j, q j, X j ) P,+ u nj ( j, x j ), 4

25 uch ha (p j, q j, X j ) (p, q, X). Since u n i he vicoiy oluion o equaion (6.5), i follow ha for any j, p j F nj (X j, q j, u nj ( j, x j ), x j, j ). Noe ha u n i uniformly convergence on compac e, by he aumpion ha u(, x) > h(, x), we derive ha for j large enough u nj ( j, x j ) > h( j, x j ); herefore, ending j goe o infiniy in he above inequaliy yield: p F (X, q, u(, x), x, ). hen we conclude ha u i a uboluion of (6.). I remain o prove ha u i a uperoluion of (6.). For each fixed (, x) (, ) R d, and (p, q, X) P, u(, x). Noing ha {Y,x } [, ] i he oluion of refleced G-BSDE wih daa (ξ, f, g, S), where S = h(, X,x ), we have u(, x) = Y,x Applying Lemma 6.1 in [] again, here exi equence h(, x). n j, ( j, x j ) (, x), (p j, q j, X j ) P, u nj ( j, x j ), uch ha (p j, q j, X j ) (p, q, X). Since u n i he vicoiy oluion o equaion (6.5), we derive ha for any j, p j F nj (X j, q j, u nj ( j, x j ), x j, j ). herefore Sending j in he above inequaliy, we have p j F (X j, q j, u nj ( j, x j ), x j, j ). p F (X, q, u(, x), x, ), which implie ha u i a uperoluion of (6.). hu u i a vicoiy oluion of (6.). Analyi imilar o he proof of heorem 8.6 in [5] how ha here exi a mo one oluion of he obacle problem (6.) in he cla of coninuou funcion which grow a mo polynomially a infiniy. he proof i complee. 7 American opion under volailiy uncerainy Now le u conider he financial marke wih volailiy uncerainy. he marke model M i inroduced in [31] coniing of wo ae whoe dynamic are given by dγ = rγ d, γ = 1, ds = rs d + S db, S = x >, where r i a conan inere rae. he ae γ = (γ ) repreen a rikle bond. he ock price i decribed by a geomeric G-Brownian moion. Since he deviaion of he proce B from i mean i unknown, hi model how he ambiguiy under volailiy uncerainy. Definiion 7.1 ([31]) A cumulaive conumpion proce C = (C ) i a nonnegaive F -adaped proce wih value in L 1 G (Ω ), and wih nondecreaing, RCLL pah on (, ], and C =, C <, q... A porfolio proce π = (π ) i an F -adaped real valued proce wih value in L 1 G (Ω ). 5

26 Remark 7. Le (Ω, F, P ) be he claical probabiliy pace, where Ω = C ([, ], R) and F = B(Ω ). Furhermore, here exi a proce W = {W } which i a Brownian moion wih repec o P. hen he filraion in he above definiion i given by F = σ{w } N, where N denoe he collecion of all P -null ube. Definiion 7.3 ([31]) For a given iniial capial y, a porfolio proce π, and a cumulaive conumpion proce C, conider he wealh equaion dx = X (1 π ) dγ γ wih iniial wealh X = y. Or equivalenly γ 1 X = y + X π ds S = rx d + π X db dc γ 1 u dc u + dc γ 1 u X u π u db u,. If hi equaion ha a unique oluion X = (X ) := X y,π,c, i i called he wealh proce correponding o he riple (y, π, C). Definiion 7.4 ([31]) A porfolio/conumpion proce pair (π, C) i called admiible for an iniial capial y R if (i) he pair obey he condiion of Definiion 7.1 and 7.3; (ii) (π X y,π,c ) MG (, ); (iii) he oluion X y,π,c aifie X y,π,c L,, q.. where L i a nonnegaive random variable in L G (Ω ). We hen wrie (π, C) A(y). We denoe by, he e of all opping ime aking value in [, ], for any. hen he American coningen claim may be defined by he following: Definiion 7.5 ([14]) An American coningen claim i a financial inrumen coniing of (i) an expiraion dae (, ]; (ii) he elecion of an exercie ime τ, ; (iii) a payoff H τ a he exercie ime. We hould require ha he payoff proce {H } [, ] aify (H4) or (H4 ) in Secion 3. Since he financial marke under volailiy uncerainy i incomplee, i i naural o conider he uperhedging price for he American coningen claim. Definiion 7.6 Given an American coningen claim (, H) we define he uperhedging cla U := {y (π, C) A(y) : for any opping ime τ, X y,π,c τ he uperhedging price i defined a h up := inf{u y U}. H τ, q..}. 6

27 heorem 7.7 Given he financial marke M and an American coningen claim (, H), we have h up = Y, where Y = (Y ) i he oluion o he refleced G-BSDE wih parameer (H, f, H ) wih f(y) = ry. Proof. Le y U. By he definiion of U, here exi a pair (π, C) A(y) uch ha for any opping ime τ, Xτ y,π,c H τ. Applying Lemma 3.4, Lemma 4. and Lemma 4.3 in [19], we derive ha for any η MG (, ) τ Ê[ η db ] =. hen we obain y = Ê[y + τ Ê[y + τ = Ê[γ 1 τ γu 1 Xτ y,π,c γu 1 Xu y,π,c π u db u ] Xu y,π,c π u db u ] Ê[γ 1 τ H τ ]. τ γ 1 u dc u ] I follow ha h up up τ, Ê[γτ 1 H τ ]. Now we urn o prove he invere inequaliy. Conider he following refleced G-BSDE { Y = H ry d Z db + (A A ), Y H. By heorem 5.1, here exi a unique oluion (Y, Z, A) o he above equaion. Le C = A, π = Z Y. hen H τ Y τ = Xτ Y,π,C, which implie Y U. I follow ha h up Y. Applying Iô formula o Ỹ = γ 1 Y, we conclude ha Ỹ i a oluion o he refleced G-BSDE wih daa (γ 1 H,, γ 1 H). By he following propoiion, we finally ge he deired reul. Propoiion 7.8 Le (Y, Z, A) be a oluion of he refleced G-BSDE wih daa (ξ, f, S). hen we have Y = up Ê[ τ, τ Proof. Le τ,. Noe he fac ha hen we have f(, Y, Z )d + S τ I {τ< } + ξi {τ= } ]. τ Ê[ Z db ] =. τ Y = Ê[ f(, Y, Z )d + Y τ + A τ ] τ Ê[ f(, Y, Z )d + S τ I {τ< } + ξi {τ= } ]. We are now in a poiion o how he invere inequaliy. By he definiion of he oluion of he refleced G-BSDE, we may define K := (Y S )da, 7

28 hen K i a non-increaing G-maringale. Le D n = inf{ : Y S < 1 n }. By example A.4, D n i a -opping ime for n 1. I i eay o check ha D n D, where D = inf{ : Y S = }. Noing ha A i nondecreaing, by heorem A.5, i follow ha D n = Ê[K D n] = Ê[ (Y S )da ] 1 nê[ A Dn], which yield Ê[ A D n] =. By he coninuiy propery of A, we have Ê[ A D] =. hen i i eay o check ha D Y = Ê[ f(, Y, Z )d + S D I {D< } + ξi {D= } ]. Hence, he reul follow. A In hi ecion, we mainly inroduce he exended condiional G-expecaion and opional opping heorem under G-framework. More deail can be found in [13]. Le (Ω, L ip (Ω), Ê[ ]) be he G-expecaion pace and P be a weakly compac e ha repreen Ê. We e L (Ω) := {X : Ω [, ] and X i B(Ω)-meaurable}, L(Ω) := {X L (Ω) : E P [X] exi for each P P}. We exend G-expecaion Ê o L(Ω) and ill denoe i by Ê. For each X L(Ω), we define hen we give ome noaion Ê[X] = up E P [X]. P P L p (Ω) := {X L (Ω) : Ê[ X p ] < } for p 1, L 1 G (Ω) := {X L 1 (Ω) : X n L 1 G(Ω) uch ha X n X, q..}, L 1 G (Ω) := {X L1 (Ω) : X n L 1 G (Ω) uch ha X n X, q..}, L 1 G (Ω) := {X L1 (Ω) : X n L 1 G (Ω) uch ha Ê[ X n X ]}. Se Ω = {ω : ω Ω} for >. Similarly, we can define L (Ω ), L(Ω ), L p (Ω ), L 1 G (Ω ), L 1 G (Ω ) and L 1 G (Ω ) repecively. hen we can exend he condiional G-expecaion o pace L 1 G (Ω). Propoiion A.1 ([13]) For each X L 1 G (Ω), we have, for each P P, Ê [X] = e up P E Q [X F ], Q P(,P ) P -a.., where P(, P ) = {Q P : E Q [X] = E P [X], X L ip (Ω )}. 8

29 We now give he definiion of opping ime under G-expecaion framework. Definiion A. A random ime τ : Ω [, ) i called a -opping ime if I {τ } L 1 G (Ω ) for each. Definiion A.3 For a given -opping ime τ and ξ L 1 G (Ω), we define Êτ [ξ] := M τ, where M = Ê [ξ] for. We hen give an example of -opping ime. Example A.4 Le (X ) [, ] be an d-dimenional righ coninuou proce uch ha X L 1 G (Ω ) for [, ]. For each fixed cloed e F R d, we define hen τ i a -opping ime. τ = inf{ : X / F }. Now we inroduce he following opional opping heorem under G-framework. heorem A.5 ([13]) Suppoe ha G i non-degenerae. Le M = Ê[ξ] for, ξ L p G (Ω ) wih p > 1 and le σ, τ be wo -opping ime wih σ τ. hen M τ, M σ L 1 G (Ω) and M σ = Êσ[M τ ], q... Acknowledgmen he auhor are graeful o Yongheng Song and Falei Wang for heir help and many ueful dicuion. Reference [1] P. Cheridio, H. M. Soner, N. ouzi, and N. Vicoir, Second order backward ochaic differenial equaion and fully nonlinear parabolic PDE. Comm. Pure Appl. Mah., 6 (7), [] M. Crandall, H. Ihii and P. L. Lion, Uer guide o he vicoiy oluion of econd order parial differenial equaion. Bull. Amer. Mah. Soc., 7 (199), [3] J. Cvianic and I. Karaza, Backward ochaic differenial equaion wih reflecion and Dynkin game. Ann. Probab., 4(4) (1996), [4] L. Deni, M. Hu and S. Peng, Funcion pace and capaciy relaed o a ublinear expecaion: applicaion o G-Brownian moion pahe. Poenial Anal., 34 (11), [5] N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng and M.C. Quenez, Refleced oluion of backward SDE, and relaed obacle problem for PDE. he Annal of Probabiliy, 3() (1997), [6] N. El Karoui, E. Pardoux and M.C. Quenez, Refleced backward SDE and American opion. Numerical Mehod in Finance (Cambridge Univ. Pre), (1997), [7] D. Friedman, Variaional Principle and Free-boundary Problem. Second ediion. Rober E.Krieger Publihing Co., Inc., Malabar, FL, [8] S. Hamadene, Refleced BSDE wih diconinuou barrier and applicaion. Sochaic Sochaic Rep. 74(3-4) (),