Dual Representation of Minimal Supersolutions of Convex BSDEs

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1 Dual Repreenaion of Minimal Superoluion of Convex BSDE Samuel Drapeau a,1,, Michael Kupper b,2,, Emanuela Roazza Gianin c,3, Ludovic Tangpi a,4, December 15, 214 ABSTRACT We give a dual repreenaion of minimal uperoluion of BSDE wih non-bounded, bu inegrable erminal condiion and under weak requiremen on he generaor which i allowed o depend on he value proce of he equaion. Converely, we how ha any dynamic rik meaure aifying uch a dual repreenaion em from a BSDE. We alo give a condiion under which a uperoluion of a BSDE i even a oluion. KEYWORDS: Convex Dualiy; Superoluion of BSDE; Cah-Subaddiive Rik Meaure. AUTHORS INFO a Humbold-Univeriä Berlin, Uner den Linden 6, 199 Berlin, Germany b Univeriä Konanz, Univeriäraße 1, Konanz, Germany c Univeriy of Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 2126 Milan, Ialy 1 drapeau@mah.hu-berlin.de 2 kupper@uni-konanz.de 3 emanuela.roazza1@unimib.i 4 angpi@mah.hu-berlin.de MATHEON projec E.11 Funding: Berlin Mahemaical School, phae II cholarhip PAPER INFO AMS CLASSIFICATION: 6H2; 6H3 1. Inroducion Since heir inroducion by Pardoux and Peng [15], nonlinear Backward Sochaic Differenial Equaion (BSDE) have found numerou applicaion in mahemaical finance. For inance, hey are ued o conrucively decribe he opimal oluion of ome uiliy maximizaion problem, ee Hu e al. [11]. Through he g-expecaion of Peng [16], BSDE offer a framework o udy nonlinear expecaion and ime conien dynamic rik meaure a decribed by Roazza Gianin [19] and Delbaen e al. [4]. Mainly driven by i financial applicaion, he udy of BSDE ha been exended in variou way beyond he queion of exience and uniquene of oluion. Many auhor have been inereed in queion uch a numerical approximaion, rucural and pah properie of BSDE oluion, ee for inance he urvey of El Karoui e al. [9] for an overview. The ubjec of hi paper i o udy BSDE by convex dualiy heory. Deviaing from he uual quadraic growh or Lipchiz aumpion on he generaor of he BSDE, Drapeau e al. [6] how exience of he minimal uperoluion of a BSDE. They udy he properie of minimal uperoluion and give he link o cah-ubaddiive rik meaure of El Karoui and Ravanelli [8]. Our main objecive are, on he one hand, o derive a dual repreenaion of minimal uperoluion of BSDE, and, on he oher hand, o udy condiion under which an operaor aifying uch a repreenaion i he minimal uperoluion or a oluion of a BSDE. Dual repreenaion of oluion of BSDE wih quadraic growh in he conrol variable, linear growh in he value proce and bounded erminal condiion are by now well underood, ee for inance Barrieu and El Karoui [1] and El Karoui and Ravanelli [8]. In hi work we give he dual repreenaion of he minimal uperoluion funcional of a BSDE in he framework of Drapeau e al. [6]. The H 1 -L dualiy urn ou o be he righ candidae o coniue he We hank he anonymou referee for valuable commen and uggeion. 1

2 bai of our repreenaion. A a aring poin, we conider he e of eenially bounded erminal condiion. In hi cae, we obain a dual repreenaion of he minimal uperoluion a ime and a poinwie robu repreenaion in he dynamic cae. We how ha when he generaor of he equaion i decreaing in he value proce, he minimal uperoluion define a ime conien cah-ubaddiive rik meaure. I allow for a dual repreenaion on he pace of eenially bounded random variable, which agree wih he repreenaion of El Karoui and Ravanelli [8] obained for BSDE oluion. Our dual repreenaion i obained by howing ha he repreenaion of El Karoui and Ravanelli [8] can be rericed on a maller e. Then we can ue runcaion and approximaion argumen o obain he repreenaion in he general cae, due o monoone abiliy of minimal uperoluion. A direc conequence of our repreenaion i he idenificaion of BSDE oluion and minimal uperoluion in he cae of linear growh generaor. Noe ha our runcaion echnique appear already in he work of Delbaen e al. [4] where i i ued o conruc a equence of µ-dominaed rik meaure. Furhermore, prior o u Barrieu and El Karoui [1] and Bion-Nadal [2] already ued he BM O-maringale heory in he udy of financial rik meaure, bu in differen eing from our. Uing andard convex dualiy argumen uch a he Fenchel-Moreau heorem and he properie of he Fenchel-Legendre ranform of a convex funcional, we exend our dual repreenaion o he e of random variable ha can be idenified o H 1 -maringale. Noice ha hi repreenaion i obained in he aic cae. Our repreenaion reul can be een a exenion of he dual repreenaion of he minimal uperreplicaing co of El Karoui and Quenez [7] o he cae where we allow for a nonlinear co funcion in he dynamic of he wealh proce. The econd heme of hi work i o give condiion baed on convex dualiy under which a dynamic cah-ubaddiive rik meaure wih a given repreenaion can be een a he oluion, or he minimal uperoluion of a BSDE. The cah-addiive cae ha been udied by Delbaen e al. [5]. Their reul are baed on m-abiliy of he dual pace, ome upermaringale propery and Dood-Meyer decompoiion of he rik meaure. We hall how ha in he cah-ubaddiive cae, dicouning he rik meaure yield imilar reul, hence howing an equivalen relaionhip beween exience of he minimal uperoluion and he dual repreenaion. The re of he paper i rucured a follow: The nex ecion i dedicaed o he eing of he probabiliic framework of our udy. We alo inroduce he noaion and gaher ome reul on minimal uperoluion of BSDE. Our repreenaion reul are aed and proved in Secion 3. The queion of deriving a BSDE from he repreenaion i deal wih in he la ecion. 2. Minimal Superoluion of Convex BSDE Given a fixed ime horizon T >, le (Ω, F, (F ) [,T ], P ) be a filraed probabiliy pace. We aume ha he filraion (F ) i generaed by a d-dimenional Brownian moion W and i aifie he uual condiion. We furher aume ha F T = F. The e of F meaurable random variable i denoed by L where random variable are idenified in he P -almo ure ene. For 1 p <, we denoe by L p he e of random variable in L which are p-inegrable and e L p = L p T, and L i he e of eenially bounded random variable in L T. Saemen concerning random variable or procee like inequaliie and equaliie are o be underood in he P -almo ure or P d-almo ure ene, repecively. The e of opping ime wih value in [, T ] i denoed by T. We conider he e of procee S := {Y : Ω [, T ] R; Y i adaped and càdlàg} ; L := Z : Ω [, T ] Rd ; Z i predicable, and Z 2 d < + ; 2

3 H p := { X S : X i a coninuou maringale wih BMO := { M : M H 1 uch ha M BMO < }, } up X L p ; [,T ] where M BMO := up τ T E [ M T M τ F τ ] 1 2. The e H+ 1 denoe he e of non-negaive maringale in H 1. Furher, le L + and L ++ be he e of non-negaive and ricly poiive random variable in L, repecively. Noice ha X = E[X T F ] for all T and every X H 1. Therefore, H 1 will be idenified wih he e of random variable X L 1, aifying up [,T ] E[X F ] L 1. The dual of he Banach pace H 1 can be idenified wih BMO, ee Kazamaki [13, Theorem 2.6]. We furher conider he e Q := q L : exp q u dw u 1 q u 2 du L 2, D := β : Ω [, T ] R; β predicable, βu du L and β u + du <. In our eing, he dual variable will appear o be cloely relaed o he e D and Q. The idea of defining he e Q wih ochaic exponenial in L i moivaed by he fac ha he repreenaion will rely on he H 1 -L dualiy. For q Q, we denoe by Q q he probabiliy meaure whoe deniy proce i given by he ochaic exponenial M q := exp( q u dw u 1 2 q 2 u du) and for β D we denoe by D β, := exp( β udu), T he dicouning facor wih repec o β. In he cae where β D + := {β D : β }, he meaure wih deniy M q D β, wa referred o by El Karoui and Ravanelli [8] a ubprobabiliy meaure. A generaor i a joinly meaurable funcion g : Ω [, T ] R R d (, + ] where Ω [, T ] i endowed wih he predicable σ-field, and uch ha (y, z) g (ω, y, z) i P d-almo urely lower emiconinuou. We denoe by g he poinwie Fenchel-Legendre ranform of g, ha i g (ω, β, q) = up { yβ + qz g (ω, y, z)}, (β, q) R R d, (y,z) R R d where he calar produc beween wo vecor q, z R d i denoed by qz := q z. For any (β, q) R R d, he proce g (β, q) i predicable, ee Rockafellar and We [18, Propoiion 14.4]. Following Drapeau e al. [6], a uperoluion of he BSDE wih erminal condiion X L and driver g i defined a a couple (Y, Z) S L uch ha Y g u (Y u, Z u )du + Z u dw u Y, for every T (2.1) Y T X. The following equivalen formulaion of (2.1) will omeime be ueful: a pair (Y, Z) i a uperoluion if and only if here exi a càdlàg, increaing and adaped proce K wih K = uch ha Y = X + g u (Y u, Z u )du + (K T K ) Z u dw u, for every T. (2.2) The conrol proce Z of a uperoluion (Y, Z) i aid o be admiible if he coninuou local maringale Z dw i a upermaringale. Given a driver g we define A (X) := {(Y, Z) S L : (Y, Z) fulfill (2.1) and Z i admiible}, X L. 3

4 A uperoluion (Ȳ, Z) A(X) i aid o be minimal if Ȳ Y for every (Y, Z) A(X). A generaor g i aid o be (POS) poiive, if g ; (DEC) decreaing, if g(y, z) g(y, z) whenever y y ; (CONV) convex, if (y, z) g(y, z) i convex; (LSC) lower emiconinuou, if (y, z) g(y, z) i lower emiconinuou. Theorem 2.1. Le g be a driver aifying (CONV), (LSC) and (POS). For any X X := {X L : X L 1 } uch ha A(X), here exi a unique minimal uperoluion (Ȳ, Z) A(X) which aifie Ȳ = e inf {Y : (Y, Z) A(X)} for all [, T ]. Proof. See Appendix A. For a generaor g which aifie (CONV), (LSC) and (POS) we define he operaor E : X S { } a {Ȳ if A(X) E : X + ele, where Ȳ i defined in Theorem 2.1 and depend on X. We conclude hi ecion by he following rucural properie and abiliy reul for E. Propoiion 2.2. Le g aifying (CONV), (LSC) and (POS), le X, X L and m R. I hold (i) Monooniciy: if X X hen E(X ) E(X); (ii) Convexiy: E (λx + (1 λ)x ) λe (X) + (1 λ)e (X ), for all λ (, 1); (iii) Cah-ubaddiiviy: if g i (DEC) and m, hen E (X + m) E (X) + m; (iv) Cah-addiiviy: if g : (y, z) g(z), hen: E (X + m) = E (X) + m; (v) Normalizaion: for every y R uch ha g(y, ) = i hold E (y) = y. Furhermore, for any equence of random variable (X n ) L uch ha inf n X n L 1, i hold (vi) Monoone convergence: lim E (X n ) = E (X) whenever (X n ) i increaing and converge P -a.. o X L ; (vii) Faou: E (lim inf X n ) lim inf E (X n ). A a rericion on L 1 he operaor E i L 1 -lower emiconinuou. Proof. See Appendix A. 4

5 3. Dual Repreenaion 3.1. The Bounded Cae The following propoiion provide he dual repreenaion of g-expecaion, ee alo [9, Propoiion 3.3]. Noe ha uch a repreenaion wa already obained in [8] in he more general quadraic cae, where he value funcion of he BSDE wa wrien a a upremum over a e of meaure wih uniformly inegrable deniie. Here, we how ha under he linear growh aumpion he repreenaion can be rericed o a e of meaure wih deniie in L. Propoiion 3.1. Le X L and f be a driver aifying (CONV), (LSC) and (POS), a well a he linear growh condiion f(y, z) a + b y + c z, a, b, c >. Then he oluion (Y, Z) of he BSDE admi he dual repreenaion Y = Y = X + e up E Q q (β,q) D Q f u (Y u, Z u ) du Z u dw u, [, T ], (3.1) D β,t X D,uf β u(β u, q u ) du F, [, T ]. (3.2) Before going hrough he proof, le u provide he following well known lemma, ee [8]. Lemma 3.2. Le f : R R d (, ] be a funcion aifying (LSC), (CONV) a well a f(y, z) a + b y + c z, (y, z) R R d for ome poiive conan a, b and c. Then, f admi for all (y, z) R R d he dual repreenaion f(y, z) = for ome β b and q c. max { βy + qz f (β, q)} = βy + qz f ( β, q) (3.3) β R,q R d Proof. We horly preen he argumen. Fir, he dual repreenaion of f i a conequence of he Fenchel-Moreau heorem, ince he growh condiion implie ha f i proper. Second, he growh condiion on f implie f (β, q) βy + qz f(y, z) a βy + qz b y c z for all (y, z) R R d. In paricular f (β, q) a + m β ( β b) + n q ( q c) for every n, m N, howing ha f (β, q) = for all b < β or c < q. Hence, he upremum in (3.3) can be rericed o β b and q c. Finally, f being lower emiconinuou and having a domain conained in a compac e, he upremum i herefore a maximum. Proof (Proof of Propoiion 3.1). Fir noice ha by Lemma 3.2, f i globally Lipchiz due o he boundedne of q and β. Thi enure exience and uniquene of a rong oluion for he BSDE wih bounded erminal condiion, ee [15]. Le (β, q) D Q. Wih he ame argumen a in [8, 9], uing Iô formula applied o D β,uy u beween and T where (Y, Z) i he oluion of he Lipchiz BSDE wih 5

6 bounded erminal condiion (3.1), i hold Y = D β,t X D β,u ( β u Y u + q u Z u f u (Y u, Z u )) du D β,uz u dw Qq u = E Q q D β,t X D β,u ( β u Y u + q u Z u f u (Y u, Z u )) du F for all (β, q) D Q. ince β u Y u + q u Z u f u (Y u, Z u ) f u(β u, q u ), i follow Y e up E Q q (β,q) D Q D β,t X D,uf β u(β u, q u ) du F. (3.4) For he oher inequaliy, ince f aifie he condiion of Lemma 3.2, for all (ω, ) Ω [, T ] he ubgradien f(ω,, Y, Z ) wih repec o (Y, Z ) are non-empy for all (ω, ) Ω [, T ]. Therefore, by mean of [18, Theorem 14.56], we can apply a meaurable elecion heorem, ee for inance [17, Corollary 1C], o aer he exience of a predicable R R d -valued proce ( β, q) uch ha and β b and q c. Hence, f(y, Z) = βy + qz f ( β, q), P d-a.., (3.5) Y = E Q q D β,t X D β,uf u( β u, q u ) du F. (3.6) Bu even hough q i bounded i i no guaraneed ha he deniy of Q q belong o L. Thu, we inroduce he following localizaion by defining σ n := inf > : q u dw u n T, n N, and pu q n := q1 [,σ n ] Q and β n := β1 [,σ n ] D. Then, ince q u c, he deniy proce of Q qn i bounded and he equence of poiive random variable (D β n o D β,t dq q /dp. Furhermore, for any p > 1 i hold E [ dq qn dp p] = E exp p q n udw u 1 2 ( ) p(p 1) exp c 2 T. 2 p q n u 2 du +,T dq qn /dp ) converge P -almo urely p(p 1) 2 q u n 2 du Hence (D β n,t dq qn /dp ) i uniformly inegrable. Therefore, ince X i bounded i hold [ lim E D β n n Q qn,t X ] F = E Q q [ D β,t X F ]. (3.7) 6

7 Le u how ha lim n E Q qn D,uf β n u( β u, n q u) n du F = E Q q D,uf β u( β u, q u ) du F. (3.8) For almo all ω Ω and u T, by definiion of β n and q n, i hold ( β u(ω), n q u(ω)) n = ( β u (ω), q u (ω)) for n large enough. Hence, he equence (D,uf β n u( β u, n q u)) n converge P d-almo urely o D,uf β u( β u, q u ). Since he procee β and q are bounded, by Equaion (3.5) and he linear growh aumpion on f, we can find wo poiive number C 1 and C 2 uch ha f u( β u, q u ) du C 1 Y u du + C 2 Z u du. (3.9) I i known ha if X i bounded and f i Lipchiz, hen he oluion (Y, Z) of he BSDE i uch ha Y i bounded and Z dw i in BMO, ee for inance [14] and [1, Propoiion 7.3] 1. Equaion (3.9) and BMO H p for all 1 p <, ee [13], ogeher wih Hölder inequaliy imply E dq qn dp 2 f u ( β u, n q u) n du [ (dq q ) n 2 ] [ (dq q ) n 4 ] 1/2 C 1 E + dp C 2 E E dp Z u 2 du 2 1/2 where C i a poiive real number independen of n. Recalling ha D βn i bounded, we ge he required uniform inegrabiliy o derive (3.8). Now, from Equaion (3.6), we obain Y = E Q q = lim n E Q qn D β,t X e up E Q q (β,q) D Q D β n,t X D β,uf u( β u, q u ) du F D β n,uf u( β n u, q n u) du F D β,t X D,uf β u(β u, q u ) du F. C, Togeher wih Equaion (3.4), hi conclude he proof. Remark 3.3. Equaion (3.6) enable u already o obain he repreenaion of he g-expecaion wih repec o meaure wih quare inegrable deniie. Thi i a well-known reul. The role of he ubequen localizaion procedure i o prove ha he repreenaion can, in fac, be wrien wih repec o meaure wih bounded deniie. Thi urn ou o be imporan for he repreenaion in he non-bounded cae, ince we work on he H 1 -L dualiy. 1 Noice ha in [1], he generaor doe no depend on y, bu he ame proof carrie over o he general cae a menioned in [8]. 7

8 Conidering a more general driver, we can build on he reul above o repreen he minimal uperoluion funcional defined on he e of eenially bounded random variable. Theorem 3.4. Le g be a driver aifying (CONV), (LSC) and (POS). Then, he operaor E : L R { } admi he dual repreenaion { ] } E (X) = E Q q [D β,t X α,t (β, q), (3.1) where he penaly funcion α i given by α, (β, q) := E Q q up (β,q) D Q for every T. In addiion, for any [, T ], and X L uch ha E (X) <, { E (X) = E Q q e up (β,q) D Q D,ug β u(β u, q u ) du F, (β, q) D Q (3.11) [D β,t X F ] α,t (β, q) }. (3.12) Proof. Fir inequaliy: Le X be a bounded erminal condiion. If A(X), hen we fix a uperoluion (Y, Z) A(X). Le [, T ] and (β, q) D Q. Le u define he localizing equence of opping ime (τ n ) by τ n := inf > : Z u dw u > n T, n N. We apply Iô formula o Ȳu = D β,uy u for u. Since (Y, Z) aifie he equivalen formulaion (2.2), here exi a nondecreaing proce K uch ha dȳu = β u D β,uy u du + D β,u (Z u dw u g u (Y u, Z u ) du dk u ). Hence, K being nondecreaing, i follow Ȳ τn Ȳ τ n D β,u ( β u Y u + q u Z u g(y u, Z u )) du + τ n D β,uz u dw Qq u. Applying Giranov heorem, i follow ha τ n D β,uz u dwu Qq i a Q q -maringale beween and T. Taking condiional expecaion on boh ide, uing he definiion of g, he fac ha Y τn E [X F τn ] and g, we are led o τ n Y E Q q D β,τ n E [X F τn ] D,ug β u(β u, q u ) du F. Since X i bounded, aking he limi on he righ hand ide we obain by dominaed convergence Y E Q q D β,t X D,ug β u(β u, q u ) du F, 8

9 o ha aking he upremum wih repec o β and q and by he fac ha Y wa choen arbirary, we have E (X) e up E Q q D β,t X D,ug β u(β u, q u ) du F. (3.13) (β,q) D Q If A(X) =, hen Equaion (3.13) i obviou. Second inequaliy: Le n N, and define g n (y, z) := up { βy + qz g (β, q)}. { β n; q n} For every n N, he funcion g n aifie he aumpion of Propoiion 3.1. Namely, g n i proper, ha linear growh in y and z and aifie (CONV), (LSC) and (POS). Moreover, he equence (g n ) i nondereaing and by he Fenchel-Moreau heorem, i converge poinwie o g. By Propoiion 3.1, he oluion (Y n, Z n ) of he BSDE wih generaor g n and erminal condiion X ha he dual repreenaion Y n = e up E Q q (β,q) D Q D β,t X D,ug β u n, (β u, q u ) du F Le u denoe by (Ȳ n, Z n ) he minimal uperoluion 2 of he BSDE wih driver g n and erminal condiion X. Since for every n N we have g n g, i hold g n, g, and, by minimaliy of Ȳ n we have Ȳ n Y n. Thu, for all n N Ȳ n e up E Q q D β,t X D,ug β u(β u, q u ) du F. (3.14) (β,q) D Q If =, aking he limi a n goe o infiniy and uing he monoone abiliy of minimal uperoluion of BSDE, ee Theorem A.1, we obain E (X) up E Q q D β,t X D β,u g u(β u, q u ) du. (β,q) D Q Therefore Equaion (3.1) hold rue. If [, T ] and E (X) <, hen i hold, by monooniciy, lim n Ȳ n <. Hence, aking he limi in Equaion (3.14), by Theorem A.1 we have E (X) e up E Q q D β,t X D,ug β u(β u, q u ) du F, (β,q) D Q which end he proof. In he nex corollary, we exend he reul of Theorem 3.4 by giving condiion under which he repreenaion i valid on he whole pace L even in he dynamic cae. Corollary 3.5. Le g be a driver aifying (CONV), (DEC), (LSC) and (POS). Then eiher E (X) + for all X L, [, T ], or E : L S admi he dual repreenaion { E (X) = E Q q [D β,t X ] } F α,t (β, q), X L, [, T ], (3.15) up (β,q) D + Q where he penaly funcion α i defined in Theorem A explained in Remark 3.6 we canno enure a hi poin ha Y n = Ȳ n.. 9

10 Proof. If for every X L he e A(X) i empy, hen he domain of E i empy. On he oher hand, if here exi ξ L uch ha A(ξ), hen A(X) for all X L. In fac, uing ξ ξ we have A( ξ ) and by (DEC), ee he argumen of he proof of Propoiion 2.2, we have A( ξ + c) for all c. Hence A(X) for all X L, ince X X and A( X ) for all X L. The re of he proof i imilar o ha of Theorem 3.4. Becaue g aifie (DEC), he domain of g i concenraed on R + R d, o ha he repreenaion can be rericed o D + Q. Remark 3.6. For a given BSDE, i i no a priori clear ha he minimal uperoluion oluion and he oluion agree, ince he meaure induced by he proce K appearing in he definiion of he minimal uperoluion can be ingular o he Lebegue meaure. Propoiion 3.1 and Corollary 3.5 how ha if he erminal condiion i bounded and he generaor i of linear growh boh in y and z, hen he minimal uperoluion of a BSDE coincide wih i oluion. In paricular, E(X) i a coninuou proce, compare [6, Propoiion 4.4] The Exenion o H 1 The goal of hi ecion i o exend he dual repreenaion of E o he pace H 1. We define SQ = { M L ++ : E[M] 1 }. We denoe by E he convex conjugae of E, defined a E (M) := up X H 1 {E [MX] E (X)}, M L. The following lemma i a conequence of he Fenchel-Moreau heorem and he rucural properie of E. Lemma 3.7. Le g be a driver aifying (CONV), (DEC), (LSC), (POS) and uch ha E i proper. 3 Then, he operaor E : H 1 ], ] i σ(h 1, L )-lower emiconinuou, and admi he dual repreenaion E (X) = up {E[MX] E (M)}, M SQ X H 1. (3.16) Proof. E i proper, convex ince g fulfill (CONV), and σ(l 1, L )-lower emiconinuou by [6, Theorem 4.9] and herefore, ince H 1 L 1, i i σ(h 1, L )-lower emiconinuou. By he Fenchel-Moreau heorem, i follow E (X) = up {E [MX] E (M)}, X H 1. (3.17) M L A andard argumen how ha we can reric he previou upremum from L o SQ. On he one hand, le M L wih E[M] > 1 and ξ H 1 uch ha E (ξ ) <. By cah-ubaddiiviy, ee Propoiion 2.2, i hold E (M) up {E[M(ξ + n)] E (ξ + n)} n N up {n (E[M] 1)} + E[Mξ ] E (ξ ) = +. n N 3 E i proper for inance if here exi y R wih g(y, ) =. In fac, in ha cae, he pair (y, ) i in A(y ) and herefore E (y ) y <. And by (POS), E (X) E[X] > for all X H 1. 1

11 On he oher hand, le M L \ L + and ξ H 1 uch ha E (ξ ) <. There i X H+ 1 uch ha E [ M X ] < ince L + i he polar cone of H+. 1 By monooniciy of E, we have E ( n X + ξ ) E (ξ ). Hence, E { [ (M) up ne M X] + E[Mξ ] E ( n X + ξ ) } n N { [ up ne M X]} + E[Mξ ] E (ξ ) = +. n N Therefore, we have E (X) = up {E [MX] E M L + : E[M] 1 (M)}. Now, le M L + uch ha E[M] 1, and for all λ (, 1), we pu M λ = (1 λ)m + λ. Then, M λ SQ. Since for any X H 1 we have E (X) E[X], i follow from he definiion of E ha E (1) o ha by convexiy, i hold lim up λ E (M λ ) E (M). Le X H 1, applying dominaed convergence heorem o (M λ X) implie E[MX] E { [ (M) lim inf E M λ X ] E (M λ ) }. λ Hence, E (X) up M SQ {E[MX] E (M)}. The oher inequaliy follow by e incluion. Thu, Equaion (3.16) hold rue. We oberve ha here i a relaionhip beween he e SQ and D + Q, and he dual repreenaion of E. Remark 3.8. Any elemen of SQ may be paramerized by elemen of D + Q and vice vera. Indeed, for every M SQ, ince M/E[M] i a ricly poiive random variable wih expecaion 1, here exi a unique proce q Q uch ha M q T = M/E[M], wih M q = exp( q u dw u 1 2 q 2 u du) and aking β D + uch ha exp( β d) = E[M] (, 1], we have M = exp( β d)m q T. Converely, given (β, q) D + Q, i hold exp( β d) exp( q u dw u 1 T 2 q 2 u du) SQ. Thi underline he imporance of working wih probabiliy meaure wih bounded deniie in he previou ecion. Remark 3.9. To every M SQ correpond a unique q Q. Hence, for all X L, Corollary 3.5 yield [ ] dq q E (X) = up (β,q) D + Q E dp Dβ,T X E Q q D β,u g u(β u, q u ) du = up up E [MX] E Q q D β,u g u(β u, q u ) du for he penaly funcion defined on SQ. M SQ {β D +: D β,t =E[M]} = up {E [MX] α min (M)}, M SQ α min (M) := inf {β D +: D β,t =E[M]} E Q q D β,u g u(β u, q u ) du (3.18) 11

12 We may now preen he main reul of hi ecion, he exenion o H 1 of he dual repreenaion Theorem 3.5. Theorem 3.1. Le g be a driver aifying (CONV), (DEC), (LSC) and (POS) and uch ha E i proper. Then he operaor E : H 1 ], + ] admi he dual repreenaion { ] } E (X) = E Q q [D β,t X α (β, q), X H 1, (3.19) up (β,q) D + Q where α (β, q) := E Q q D β,u g u(β u, q u ) du, (β, q) D + Q. (3.2) Proof. Due o Lemma 3.7 and Remark 3.9, i uffice o how ha E = α min on SQ, where α min i he penaly funcion defined by Equaion (3.18). Fir inequaliy. For all X H 1, i hold E (X) up E Q q (β,q) D + Q D β,t X D β,u g u(β u, q u ) du. (3.21) In fac, le X H 1. If A(X) =, hen he reul i rivial. Suppoe ha A(X), and ake (Y, Z) A(X). Le (β, q) D + Q, arguing exacly like in he fir par of he proof of Theorem 3.4 we obain a localizing equence of opping ime (τ n ) uch ha Y E Q q D β,τ n E[X F τn ] τ n D β,u g u(β u, q u ) du for all n N. (3.22) Since X H 1, he equence of maringale (N n ) given by N n := E[E[X F τn ] F ] = E[X F τn ] i in H 1, and i uch ha ( up [,T ] N n ) i uniformly inegrable. Therefore, by [3, Theorem 4.9], n ee alo [13, Lemma 2.5], (N n ) admi a ubequence again denoed by (N n ) which converge weakly in H 1. Thu, he equence of produc (D β,τ nnn T ) converge weakly in H1 o D β,t X, ince (Dβ,τ n) i bounded by 1. Now, a a conequence of he boundedne of he maringale M q = E[dQ q /dp F ], he funcion X E[M q T X] from H1 o R i linear and coninuou, and herefore σ(h 1, BMO)-coninuou. Hence, aking he limi on boh ide of Equaion (3.22) lead o Y E Q q D β,t X D β,u g u(β u, q u ) du. Thi implie, by mean of Remark 3.9, ha E (X) up {E [MX] α min (M)}, M SQ ha i, for every M SQ we have α min (M) E [MX] E (X) o ha aking he upremum wih repec o X H 1, we obain by definiion of E α min (M) E (M). 12

13 Second inequaliy. The main argumen for he econd inequaliy i o how ha he penaly funcion α min defined by Equaion (3.18) i minimal, ha i, EL (M) := up {E [MX] E (X)} = α min (M), X L M SQ. In fac, ha would imply E (M) E L (M) = α min(m), where he fir inequaliy i obained by e incluion. To ha end, i uffice o how ha for every c he e {M SQ : α min (M) c} i convex and cloed in L 1, ince by convexiy, i would hen be σ(l 1, L )-cloed and herefore σ(l, L )-cloed. Convexiy: Le λ [, 1], M 1, M 2 SQ and q i Q uch ha M qi T = M i /E[M i ], i = 1, 2. Pu M λ = λm 1 + (1 λ)m 2. For a given ε >, here exi β i D + uch ha D βi,t = E[M i ] and Applying Iô formula o log ( λm q1 have λm q1 D β1, q2 + (1 λ)m D β2 and D βλ,t = E[M λ ], wih ε + α min (M i ) E Q q i D β1, = exp + (1 λ)m q2,u g (βu, i qu) i du. D βi q λ u dw u 1 2 ) D β2 uch a in he proof of [5, Lemma 2.1] we qu λ 2 du βu λ du = M qλ D βλ, q λ = λm q 1 D β1, q1 + (1 λ)m q2 λm q1 D β1, D β2, q2 q2 + (1 λ)m D β2,, β λ = λm q 1 D β1, β1 + (1 λ)m q2 λm q1 D β1, D β2, β2 q2 + (1 λ)m D β2,. Thi follow from he fac ha M qλ T E[M λ ] = M λ = M qλ T Dβλ,T and M qλ >. Therefore, join convexiy of g and he definiion of (β λ, q λ ) lead u o 2ε + λα min (M 1 ) + (1 λ)α min (M 2 ) E = E Q q λ ) (λmu q1 D β1 q2,u + (1 λ)mu D β2,u gu(β u, λ qu) λ du,u g u(βu, λ qu) λ du. Therefore, aking fir he infimum for β D + uch ha D β,t = E[M λ ] on he righ hand ide, and hen he limi on he lef hand ide a ε goe o we have D βλ λα min (M 1 ) + (1 λ)α min (M 2 ) α min (M λ ). Cloedne: Le c and (M n ) be a equence in SQ converging o M SQ in L 1 and uch ha α min (M n ) c for every n N. Le u how ha α min (M) c. For all n N le q n be uch ha 13

14 M qn T = M n /E[M n ] and q be uch ha M q T = M/E[M]. Le ε > be fixed. For every n N, here exi β n D + uch ha D βn,t = E[M n ] and ε + α min (M n ) E Q q n,u g u(βu, n qu) n du. Since (M n ) converge o M in L 1, he equence (E[M n ]) converge o E[M], wih E[M n ] > and E[M] >. Therefore, (M qn ) converge o M q in L 1 for all [, T ]. We alo inroduce he maringale M n := E [M n F ] and M := E [M F ], [, T ]. We chooe a fa ubequence (M n,m ) uch ha P ( M n,m T M T 1) < 2 n /m and for all m N, define he opping ime τ m := inf { [, T ] : M n,m M m for ome n}. D βn Then, (τ m ) i a localizing equence of opping ime ince P (τ m = T ) 1 P ( M n,m T M T 1 for ome n) P ( M T m 1) 1 1 m E[ M T ] 1. m 1 For every m, he equence (M n,m τ M n,m m τ m) i bounded, herefore (Mτ ) converge o (M m τ m) in L2. I follow by Burkholder-Davi-Gundy and Doob inequaliie ha here exi a poiive conan C uch ha τ m E M n,m u Mu qn,m M q u Mu q 2 du = E [ M n,m M 2 ] τ m q n,m ( CE up M n,m M [,τ m ] ) 2 n. Thu, up o a ubequence, (q n,m M qn,m 1 [,τ m ]) converge P d-a.. o qm q 1 [,τ m ]. Bu ince he equence of ricly poiive maringale ( M qn,m ) converge P d-a.. o M q >, i follow ha lim n qn,m 1 [,τ m ] = q1 [,τ m ] P d-a.. Since (τ m ) converge P -a.. o T we obain, by a diagonalizaion argumen, anoher ubequence again denoed (q n ) which converge P d-a.. o q. A for he convergence of he equence (β n ), ince (exp( βn u du)) = (E [M n ]) converge o E [M], i follow ha he equence ( βn u du) converge o log(e[m]), and (E[ βn u du]) i uniformly bounded. Hence, we can apply a compacne argumen, ee for inance [3, Theorem 1.4] applied on he produc pace, o obain a equence ( β n ) in he aympoic convex hull of (β n ) which converge P d o a poiive predicable proce β. In addiion, D β,t = E[M] ince he equence ( βn u du) and ( β n u du) converge o he ame limi. Now applying Faou lemma, convexiy and lower-emiconinuiy of g lead u o ε + lim inf α min(m n ) E n E lim inf M u qn D β n n lim inf M u qn D βn n,u g u(βu, n qu) n du,u g u( β u, n qu) n du = E Once again he reul i obained by leing ε end o. MuD q β,u g u(β u, q u ) du α min (M). 14

15 We recover he robu repreenaion of coheren (cah-ubaddiive) rik meaure. Corollary Under he aumpion of Theorem 3.1, if he generaor g i poiive homogeneou in he ene ha g(λy, λz) = λg(y, z) for all λ > and (y, z) R R d, hen E i alo poiive homogeneou and he dual repreenaion of E reduce o [ E (X) = up E Q q D β,t X], X H 1. (β,q) D + Q Proof. Le λ be ricly poiive, and (E (λx), Z) he minimal uperoluion in A(λX). By poiive homogeneiy of g, we have (E (λx)/λ, Z/λ) A(X), herefore E (λx) λe (X). Uing he ame reaoning on A(X) we have E (λx) λe (X), hence E i poiive homogeneou. The repreenaion (3.11) follow from Theorem 3.1 ince he convex conjugae of he poiive homogeneou funcion g i he indicaor of a cloed convex e (i.e. i i eiher or.) Le u conclude hi ecion wih an example. Example Le X be any random variable in H 1. Conider he BSDE dy = g(y, Z ), d + Z dw, Y T = X (3.23) wih generaor g defined on R R d by z 2 /y if y >, z R d g(y, z) := if y, z = + if y, z R d \ {}. The funcion g aifie he condiion of Theorem 3.1. Therefore, he minimal uperoluion E g (X) of Equaion (3.23) admi he dual repreenaion (3.19). Moreover, defining K := {(β, q) D + Q : β 14 } q 2, one can check ha g ake he value on K and + on he complemen of K. Thu, E g (X) = up [ E Q q D β,t X]. (β,q) K 4. Cah-Subaddiive Rik Meaure and BSDE The operaor E udied in he previou ecion can be een a a rik meaure. In fac, when he generaor doe no depend on y, he funcional ρ defined by ρ(x) := E ( X) i a convex rik meaure in he ene of Föllmer and Schied [1], and u(x) := E ( X) define a moneary uiliy funcion. If he generaor g doe depend on y and aifie (DEC), hen ρ i inead a cah-ubaddiive rik meaure a defined in [8]. In paricular, for all m hold ρ(x m) ρ(x) + m. In hi ecion we ar wih a cah-ubaddiive rik meaure aifying a given robu repreenaion and how, in Theorem 4.5, ha uch a rik meaure mu be he minimal uperoluion of a BSDE. Thu, we are given a dynamic cah-ubaddiive rik meaure 4 of he form φ (X) := e up E Q q D β,t X D,uf(β β u, q u ) du F, [, T ], (4.1) (β,q) D + Q 4 Acually, hi i only a rik meaure up o a ranformaion a explained above. 15

16 where X i a random variable in H 1 and f : R R d (, ] a given proper funcion. A funcion f i aid o be (NORM) null a he origin if, f(, ) =. Remark 4.1. Since D,D β β,u = D,u, β he penaly funcion α defined by Equaion (3.11) aifie he following cocycle propery inroduced in [2] for moneary convex rik meaure: α,u (β, q) = α, (β, q) + E Q q [D,α β,u (β, q) ] F for every (β, q) D + Q. (4.2) In he cah-addiive cae, he cocycle propery ake he form α,u (q) = α, (q) + E Q q [α,u (q) F ]. Hence, he characerizaion of ime-coniency in erm of he cocycle propery given by [2, Theorem 3.3] how ha when g doe no depend on y, E i ime-conien even if he normalizaion condiion g() = i no aumed, compare [6, Propoiion 3.6]. In wha follow we ue he noaion of he previou ecion. In paricular, for any q Q we denoe by M q he maringale deniy proce of he probabiliy meaure Q q wih repec o he reference meaure P. We follow a mehod already pu forh in Delbaen e al. [5] in he cah-addiive cae. The main idea i he following: Propoiion 4.2. For any X H 1 and for each (β, q) D + Q he proce i a Q q -upermaringale. ϕ(x) := D β, φ (X) D β,u f(β u, q u ) du [,T ] Proof. Le T. We ar by howing ha he e E Q D β q,t X D,uf(β β u, q u ) du F : (β, q) D + Q i direced upward. Le (β 1, q 1 ), (β 2, q 2 ) D + Q. Le u define he opping ime τ := inf { > : L 1 < L 2 }, wih L i := E Q q i [Dβi,T X D βi,uf(βu, i qu) i du F ], i = 1, 2, and pu ˆq := q 1 1 [,τ] + q 2 1 (τ,t ] and ˆβ := β 1 1 [,τ] + β 2 1 (τ,t ]. We have ( ˆβ, ˆq) D + Q and, by definiion, ˆL max{l 1, L 2 }, wih ˆL := E Qˆq[D ˆβ,T X D ˆβ,uf( ˆβ u, ˆq u ) du F ]. Therefore, by [1, Theorem A.32], here exi a equence (β n, q n ) D + Q uch ha φ (X) = lim n E Q qn D βn,t X D βn,uf(β n u, q n u) du F. 16

17 In addiion, hi convergence i monoone. Therefore, φ (X) i inegrable, and i i alo Q q -inegrable for every q Q ince dq q /dp L. Hence, for any (β, q) D + Q, i hold E Q q [ ϕ (X) F ] = EQ q = lim n Dβ, E Q q D β, lim E Q n qn E Q q D β n,d βn,t X E Q q D βn,t X D βn,uf(β n u, q n u) du D β,u f(β u, q u ) du F D β,d βn,uf(β n u, q n u) du F F D β,u f(β u, q u ) du D,uf(β β u, q u ) du F F D β,u f(β u, q u ) du, where he econd equaion follow by dominaed convergence heorem. We pu β n = β1 [,] + β n 1 (,T ] and q n = q1 [,] + q n 1 (,T ]. I follow ha E Q q [ ϕ (X) F ] = D β, lim n E Q qn D β, φ (X) D β n,t X,uf( β u, n q u) n du F D β n D β,u f(β u, q u ) du = ϕ (X), D β,u f(β u, q u ) du where he inequaliy follow by definiion of φ(x) and he fac ha ( β n, q n ) D + Q. Nex we give wo conequence of he previou reul. Corollary 4.3. Le X H 1, uppoe [ in addiion ha φ (X) admi a ubgradien (β, q) D + Q, i.e. (β, q) i uch ha φ (X) = E Q q D β,t X ] T Dβ,u f(β u, q u ) du. Then for each [, T ] we have φ (X) = E Q q D β,t X D,uf(β β u, q u ) du F, (4.3) ha i, (β, q) i a ubgradien of φ (X). Moreover, he proce i a Q q -maringale. D β, φ (X) D β,u f(β u, q u ) du [,T ] Proof. Le (β, q) D + Q be uch ha φ (X) = E Q q D β,t X D β,u f(β u, q u ) du. 17

18 By he previou propoiion and he choice of (β, q) we have for any [, T ] D β, φ (X) D β,u f(β u, q u ) du φ (X) = E Q q D β,t X D β,u f(β u, q u ) du, E Q q from which enue Since we have E Q q [ ] D β, φ (X) E Q q D β,t X = E Q q D β, D β,u f(β u, q u ) du D β,t X D,uf(β β u, q u ) du. φ (X) E Q q D β,t X D,uf(β β u, q u ) du F, and < D β, < we conclude ha φ (X) = E Q q D β,t X D,uf(β β u, q u ) du F Q q -a.. From Equaion (4.3) we have, for all [, T ], D β, φ (X) D β,u f(β u, q u ) du = E Q q D β,t X D β,u f(β u, q u ) du F Q q -a.. Corollary 4.4. Aume ha he funcion f aifie (NORM). Then, for every X H 1 he proce (φ (X)) [,T ] i a P -upermaringale and admi a Doob-Meyer decompoiion of he form φ(x) = φ (X) + M A where A i a càdlàg adaped and increaing proce wih A = and M a coninuou local maringale. Proof. The P -upermaringale propery of φ(x) follow from Propoiion 4.2 and he fac ha f(, ) =. Le u how ha φ(x) ha a càdlàg modificaion which i ill a P -upermaringale. Le [, T ], ince φ(x) i a P -upermaringale, for all [, T ] Q we have E[φ (X) F ] φ (X). Hence, by Faou lemma and due o he fac ha our filraion aifie he uual condiion we obain he inequaliy φ + (X) φ (X), where φ + (X) := lim φ (X)., Q On he oher hand by coninuiy of maringale we have, for all (β, q) D + Q, φ + (X) E Q q D β,t X D,uf(β β u, q u ) du F P -a.., o ha aking he upremum wih repec o β, q yield φ + (X) φ (X) P -a.., hu we have φ + (X) = φ(x) P -a.. We conclude by [12, Propoiion ] ha φ(x) ha a càdlàg modificaion which i again a upermaringale. Thi pah regulariy of φ(x) enure ha i admi a Doob-Meyer decompoiion. 18

19 Now we wan o link he dynamic rik meaure defined by Equaion (4.1) o a BSDE. In ha regard, we aume ha f i (CONV) and (LSC), and we denoe by g he funcion defined on R R d by g(y, z) := up { βy + qz f(β, q)}. β ;q R d The funcion g i (DEC) and if f i (NORM) hen g i (POS). Theorem 4.5. Aume ha he funcion f aifie (CONV), (LSC) and (NORM). For all X H 1, here exi a unique predicable d-dimenional proce Z uch ha (φ(x), Z) i he minimal uperoluion of he BSDE wih generaor g and erminal condiion X. Proof. Superoluion propery: Le X H 1. We ar by proving ha here exi Z uch ha (φ(x), Z) i a uperoluion of he BSDE wih generaor g and erminal condiion X. By Corollary 4.4 here exi procee A and M uch ha φ (X) = φ (X) + M A, and by maringale repreenaion here exi a proce Z L uch ha φ (X) = φ (X) + Z u dw u A. (4.4) By definiion of φ(x) and Equaion (4.4), Z u dw u E [X F ] φ (X). Thu, Z dw i a upermaringale a a local maringale bounded from below by a maringale. Le (β, q) D + Q. Applying Iô formula o D β, φ (X) lead u o ( ) d D β, φ (X) = β D β, φ (X) d + D β, dφ (X) Therefore, d D β, φ (X) = β D β, φ (X) d + D β, ( da + Z dw ) = β D β, φ (X) d + D β, ( da + Z q d) + D β, Z dw Qq. D β,u f(β u, q u ) du = D β, ( β φ (X) d da + Z q d f(β, q ) d) + D β, Z dw Qq. (4.5) By he Q q -upermaringale propery proved in Propoiion 4.2, we have Since β and q were aken arbirary, i hold Hence Equaion (4.4) give, for all T, da ( β φ (X) + q Z f(β, q )) d. φ (X) da g(φ (X), Z ) d. (4.6) g(φ u (X), Z u ) du + Z u dw u φ (X), which how ha (φ(x), Z) i an admiible uperoluion. Minimaliy: Showing ha he proce φ(x) i minimal i done uing exacly he ame argumen a hoe ued o prove Equaion (3.21) in he econd ep of he proof of Theorem 3.1 and he fir par of he proof of Theorem 3.4. Replacing by and he expecaion by he condiional expecaion in he proof of Equaion (3.21) doe no affec he reaoning. Recalling ha ince g i (CONV), (DEC) and (POS) he minimal uperoluion i unique conclude he proof. 19

20 Theorem 4.6. Aume ha he funcion f aifie (CONV), (LSC) and (NORM). Le X H 1, if φ (X) admi a ubgradien (β, q) D + Q hen he minimal uperoluion (φ(x), Z) i acually a oluion. In addiion, for P d-almo all (ω, ) Ω [, T ], (β, q ) g(ω,, φ (X), Z ), ubgradien of g wih repec o (φ (X), Z ). Proof. Le X H 1 and (β, q) D + Q be a ubgradien of φ (X). Then, by Corollary 4.3 and he decompoiion appearing in Equaion (4.5), we have Definiion of g and Equaion (4.6) give da = ( β φ (X) + q Z f(β, q )) d. g(φ (X), Z ) d ( β φ (X) + q Z f(β, q )) d = da g(φ (X), Z ) d. Then, da = g(φ (X), Z ) d, howing ha (β, q) g(φ(x), Z) P d-a.. Equaion (4.4) yield φ (X) = X g(φ u (X), Z u ) du + Z u dw u. Hence (φ(x), Z) i a oluion. We conclude by he following complee characerizaion of he minimal uperoluion uggeed by Corollary 3.5 and Theorem 4.5. Theorem 4.7. Aume ha he funcion g aifie (CONV), (DEC) and (POS), g aifie (NORM). If X L, hen he following are equivalen: (i) There exi a predicable d-dimenional proce Z uch ha (E(X), Z) i he minimal uperoluion of he BSDE wih erminal condiion X and driver g. (ii) The funcional E admi he repreenaion E (X) = e up E Q q D β,t X D,ug β (β u, q u ) du F, [, T ]. (β,q) D + Q A. Some Properie of he Minimal Superoluion Operaor The aim of hi appendix i o preen he proof of ome properie of he minimal uperoluion ued in he paper. Proof (Proof of Propoiion 2.2 ). See [6, Propoiion 3.2 and Theorem 4.9 and 4.12], bu for he ake of readabiliy we give he deail for he poin (iii), (iv) and (v). A for (iii), le m R wih m and X X. Since X + m X, if A(X) = hen A(X + m) =. In ha cae E (X + m) = = E (X). If A(X), le (Y, Z) A(X). For all T, ince g fulfill (DEC), we have Y +m g u (Y u + m, Z u ) du + Z u dw u m + Y g u (Y u, Z u ) du + Z u dw u m + Y. Thu, (Y + m, Z) A(X + m), which implie E (X + m) Y + m. Taking Y = E(X), we have E (X + m) E (X) + m howing he cah-ubaddiiviy. 2

21 A for (iv), if g doe no depend on y, one can how ha E i addiionally cah-uperaddiive, ha i, E (X + m) E (X) + m for m. Indeed, uing he ame argumen we have A(X) implie A(X + m) and (Y m, Z) A(X) for all (Y, Z) A(X + m). Then, if g doe no depend on y, i follow ha E (X + m) = E (X) + m for all m R +. Thu, E (X) + m = E (X) + m + m = E (X + m + ) m = E (X + m + m ) m = E (X + m) for all m R. A for (v), if g(y, ) =, we have (y, ) A(y), and herefore E (y) y. If g i (POS), for all (Y, Z) A(y), he upermaringale propery of Y and he erminal condiion yield Y E[Y T ] y. Hence, E (y) y. Nex, we recall he proof of he exience, uniquene and monoone abiliy of he minimal uperoluion wih repec o he generaor. Thee reul were already obained in [6]. Here we argue ha heir proof are alo valid, up o a ligh change, if we replace he aumpion (DEC) by (CONV) on he generaor. Recall ha for X X := {X L : X L 1 }, he condiion (POS) enure ha he value proce Y of a uperoluion (Y, Z) A(X) i a upermaringale uch ha ee [6, Lemma 3.3]. Y E[X F ] for all [, T ], (A.1) Proof (Skech of he Proof of Theorem 2.1). The uniquene of Z follow by he upermaringale propery of Ȳ and he maringale repreenaion heorem. The exience i proved by conrucing, hrough concaenaion, a equence of uperoluion (Y n, Z n ) whoe value procee (Y n ) decreae o he proce e inf{y : (Y, Z) A(X)}. By a compacne argumen, a ubequence in he aympoic convex hull of (Z n ) which converge rongly o a proce Z can be eleced. The proof i compleed by howing ha here i a modificaion Ȳ of e inf{y : (Y, Z) A(X)} uch he candidae (Ȳ, Z) i acually an admiible uperoluion. In he cae where g doe no aify (DEC) bu (CONV), hi i done a in he proof of he nex heorem. Theorem A.1. Le X X be a erminal condiion, and le (g n ) be an increaing equence of generaor, which converge poinwie o a generaor g. Suppoe ha each generaor i defined on R R d and fulfill (CONV), (LSC) and (POS) and denoe by Ȳ n he value proce of he minimal uperoluion of he BSDE wih generaor g n. Then lim n Ȳ n = E (X). If, in addiion, lim n Ȳ n <, hen for all [, T ] he e A(X) i nonempy and (Ȳ n ) converge P -a.. o E (X). Proof. By monooniciy, ee Propoiion 2.2, he equence (Ȳ n ) i increaing. Se Y = lim n Ȳ n, if Y = here i nohing o prove. Ele, we pu Y := lim n Ȳ n, [, T ]. I follow from he upermaringale propery of Ȳ n and he monoone convergence heorem ha Y i a càdlàg upermaringale. Uing he argumen of he proof of Theorem 2.1, we conruc a candidae conrol Z a poinwie limi of convex combinaion ( Z n ) of ( Z n ), where (Ȳ n, Z n ) i he minimal uperoluion of he BSDE wih generaor g n. I remain o verify ha (Y, Z) A(X). Faou lemma give Y g u (Y u, Z u ) du + Z u dw u lim up k Y gu(y k u, Z u ) du + Z u dw u. And for every k n, denoing by λ n i he convex weigh of he convex combinaion Z n, uing (CONV) 21

22 we have Y gu(y k u, Z u ) du + Z u dw u lim up n Ỹ n M n lim up n i=n M n lim up n Y. i=n λ n i λ n i Y i Y i gu(ỹ k u n, Z u n ) du A o he admiibiliy of Z, by mean of Equaion (A.1) and (A.2), we have Z u dw u E[X F ] Y g k (Y i u, Z i u) du + g i (Y i u, Z i u) du + Z n u dw u Zu i dw u Zu i dw u (A.2) o ha Z dw i a upermaringale a a local maringale bounded from below by a maringale. Thu, Z i admiible. Reference [1] P. Barrieu and N. El Karoui. Pricing, Hedging and Opimally Deigning Derivaive via Minimizaion of Rik Meaure. In e. Rene Carmona, edior, Indifference Princing, page , 27. [2] J. Bion-Nadal. Dynamic Rik Meaure: Time Coniency and Rik Meaure from BMO Maringale. Finance and Sochaic, 12(2): , April 28. [3] F. Delbaen and W. Schachermayer. A Compacne Principle for Bounded Sequence of Maringale wih Applicaion. In Proceeding of he Seminar of Sochaic Analyi, Random Field and Applicaion, Progre in Probabiliy, page Birkhauer, [4] F. Delbaen, S. Peng, and E. Roazza Gianin. Repreenaion of he Penaly Term of Dynamic Concave Uiliie. Finance and Sochaic, 14: , 21. [5] F. Delbaen, Y. Hu, and X. Bao. Backward SDE wih Superquadraic Growh. Probabiliy Theory and Relaed Field, 15(1-2): , 211. [6] S. Drapeau, G. Heyne, and M. Kupper. Minimal Superoluion of Convex BSDE. Annal of Probabiliy, 41 (6): , 213. [7] N. El Karoui and M.-C. Quenez. Dynamic Programming and Pricing of Coningen Claim in an Incomplee Marke. SIAM Journal on Conrol Opimizaion, 33(1):29 66, [8] N. El Karoui and C. Ravanelli. Cah Sub-addiive Rik Meaure and Inere Rae Ambiguiy. Mahemaical Finance, 19:561 59, 29. [9] N. El Karoui, S. Peng, and M. C. Quenez. Backward Sochaic Differenial Equaion in Finance. Mahemaical Finance, 1(1):1 71, January [1] H. Föllmer and A. Schied. Sochaic Finance. An Inroducion in Dicree Time. de Gruyer Sudie in Mahemaic. Waler de Gruyer, Berlin, New York, 2 ediion, 24. [11] Y. Hu, P. Imkeller, and M. Müller. Uiliy maximizaion in incomplee marke. Annal of Applied Probabiliy, 15(3): , 25. [12] I. Karaza and S. E. Shreve. Brownian Moion and Sochaic Calculu, volume 113 of Graduae Tex in Mahemaic. Springer-Verlag, New York, econd ediion,

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