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Available online a www.sciencedirec.com Sochasic Processes and heir Applicaions 123 (213) 3328 3357 www.elsevier.com/locae/spa BSDEs wih jumps, opimizaion and applicaions o dynamic risk measures Marie-Claire Quenez a,b, Agnès Sulem b,c, a LPMA, Universié Paris 7 Denis Didero, Boie courrier 712, 75251 Paris cedex 5, France b INRIA Paris-Rocquencour, Domaine de Voluceau, Rocquencour, BP 15, Le Chesnay Cedex, 78153, France c Universié Paris-Es, F-77455 Marne-la-Vallée, France Received 11 June 212; received in revised form 8 December 212; acceped 22 February 213 Available online 14 March 213 Absrac In he Brownian case, he links beween dynamic risk measures and BSDEs have been widely sudied. In his paper, we consider he case wih jumps. We firs sudy he properies of BSDEs driven by a Brownian moion and a Poisson random measure. In paricular, we provide a comparison heorem under quie weak assumpions, exending ha of Royer [21]. We hen give some properies of dynamic risk measures induced by BSDEs wih jumps. We provide a represenaion propery of such dynamic risk measures in he convex case as well as some resuls on a robus opimizaion problem in he case of model ambiguiy. c 213 Published by Elsevier B.V. MSC: 93E2; 6J6; 47N1 Keywords: Backward sochasic differenial equaions wih jumps; Comparison heorems; Risk measures; Dual represenaion; Robus opimizaion 1. Inroducion Linear backward sochasic differenial equaions (BSDEs) were inroduced by Bismu (1976) [4] as he adjoin equaions associaed wih sochasic Ponryagin maximum principles in sochasic conrol heory. The general case of non-linear BSDEs was hen sudied by Pardoux and Peng (199) (see [16] and [17] in he Brownian framework). In [17], hey provided Feynman Kac Corresponding auhor a: INRIA Paris-Rocquencour, Domaine de Voluceau, Rocquencour, BP 15, Le Chesnay Cedex, 78153, France. Tel.: +33 3963 55 69. E-mail addresses: quenez@mah.jussieu.fr (M.-C. Quenez), agnes.sulem@inria.fr (A. Sulem). 34-4149/$ - see fron maer c 213 Published by Elsevier B.V. hp://dx.doi.org/1.116/j.spa.213.2.16
M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3329 represenaions of soluions of non-linear parabolic parial differenial equaions. In he paper by El Karoui e al. (1997) [9], some addiional properies are given and several applicaions o opion pricing and recursive uiliies are sudied. The case of a disconinuous framework is more involved, especially concerning he comparison heorem, which requires an addiional assumpion. In 1994, Tang and Li [22] provided an exisence and uniqueness resul in he case of a naural filraion associaed wih a Brownian moion and a Poisson random measure. In 1995, Barles, Buckdahn, Pardoux [1] provided a comparison heorem as well as some links beween BSDEs and non-linear parabolic inegral parial differenial equaions, generalizing some resuls of [16] o he case of jumps. In 26, Royer [21] proved a comparison heorem under weaker assumpions, and inroduced he noion of non-linear expecaions in his framework. Furhermore, in 24 25, various auhors have inroduced dynamic risk measures in a Brownian framework, defined as he soluions of BSDEs (see [18,2,11,19]). More precisely, given a Lipschiz driver f (, x, π) and a erminal ime T, he risk measure ρ a ime of a posiion ξ is given by X, where X is he soluion of he BSDE driven by a Brownian moion, associaed wih f and erminal condiion ξ. By he comparison heorem, ρ saisfies he monooniciy propery, which is usually required for a risk measure (see [1]). Many sudies have been recenly done on such dynamic risk measures, especially concerning robus opimizaion problems and opimal sopping problems, in he case of a Brownian filraion and a concave driver (see, among ohers, Bayrakar and coauhors in [3]). In his paper, we are concerned wih dynamic risk measures induced by BSDEs wih jumps. We sudy heir properies as well as some relaed opimizaion problems. We begin by sudying BSDEs wih jumps and heir properies. We firs focus on linear BSDEs which play an imporan role in he comparison heorems as well as in he applicaions o finance. We show ha he soluion is given by a condiional expecaion via an exponenial semimaringale, usually called he adjoin process. We also provide some addiional properies of he soluion and is adjoin process, which are specific o he jump case. Using hese properies, we provide a comparison heorem as well as a sric comparison heorem, under mild assumpions, which generalize hose saed in [21]. We also prove some opimizaion principles for BSDEs wih jumps. More precisely, we consider a family of conrolled drivers f α, α A and show ha, under some hypohesis, he infimum of he associaed soluions X α can be characerized as he soluion of a BSDE. Moreover, he driver of his BSDE is given by he infimum of he drivers f α, α A. We provide a sufficien condiion of opimaliy. Also, from he sric comparison heorem, we derive a necessary opimaliy condiion. We hen sae some properies of dynamic risk measures induced by BSDEs wih jumps. Noe ha conrary o he Brownian case, he monooniciy propery does no generally holds, and requires an addiional assumpion. In he case of a concave driver f, we provide a dual represenaion propery of he associaed convex risk measure via a se of probabiliy measures which are absoluely coninuous wih respec o he iniial probabiliy P. A las, we sudy he case of ambiguiy on he model. More precisely, we consider a model parameerized by a conrol α as follows. Wih each coefficien α, is associaed a probabiliy measure Q α, equivalen o P, called prior, as well as a monoone risk measure ρ α induced, under Q α, by a BSDE wih jumps. We consider an agen who is averse o ambiguiy and define her risk measure as he supremum over α of he risk measures ρ α. We show ha his dynamic risk measure is induced, under P, by a BSDE. The paper is organized as follows. In Secion 2, we inroduce he noaion and he basic definiions. Secion 3 is dedicaed o linear BSDEs wih jumps. In Secion 4, comparison
333 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 heorems for BSDEs wih jumps are provided. We also prove wo opimizaion principles which allow us o characerize he value funcion of an opimizaion problem wrien in erms of BSDEs. Secion 5 is dedicaed o dynamic risk measures induced by BSDE wih jumps and relaed robus opimizaion problems. In Secion 5.1, we give properies of dynamic risk measures induced by BSDEs wih jumps. In he case of a concave driver, we provide a dual represenaion of he associaed convex risk measure (Secion 5.2). The problem of dynamic risk measures under model ambiguiy is addressed in Secion 5.3. Finally, in Secion 5.4, we inerpre he dependence of he driver wih respec o x in erms of he insananeous ineres rae. In he Appendix, we provide some useful addiional properies on exponenial local maringales, and BSDEs wih jumps. 2. BSDEs wih jumps: noaion and definiions Le (Ω, F, P) be a probabiliy space. Le W be a one-dimensional Brownian moion and N(d, du) be a Poisson random measure wih compensaor ν(du)d such ha ν is a σ -finie measure on R, equipped wih is Borel field B(R ). Le Ñ(d, du) be is compensaed process. Le F = {F, } be he naural filraion associaed wih W and N. The resuls of his paper can be generalized o muli-dimensional Brownian moions and Poisson random measures wihou difficuly. Noaion. Le P be he predicable σ -algebra on [, T ] Ω. For each T > and p > 1, we use he following noaion. L p (F T ) is he se of random variables ξ which are F T -measurable and p-inegrable. H p,t is he se of real-valued predicable processes φ such ha T p φ p := E φ 2 2 H p,t d <. For β > and φ H 2,T, we inroduce he norm φ 2 β,t := E T eβs φs 2ds. Lν p is he se of borelian funcions l : R R such ha l p p,ν := R l(u) p ν(du) < +. The se L 2 ν is a Hilber space equipped wih he scalar produc δ, l ν := δ(u)l(u)ν(du) for all δ, l L 2 R ν L2 ν, and he norm l 2 2,ν = R l(u) 2 ν(du) < +, also denoed by l 2 ν. Hν p,t is he se of processes l which are predicable, ha is, measurable such ha l : ([, T ] Ω R, P B(R )) (R, B(R)); (ω,, u) l (ω, u) l p H p,t ν T p := E l 2 2 ν d <. For β > and l H 2,T ν, we se l 2 ν,β,t := E T eβs l s 2 ν ds. S p,t is he se of real-valued RCLL adaped processes φ wih φ p S p := E(sup T φ p ) <.
M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3331 When T is fixed and here is no ambiguiy, we denoe H p insead of H p,t, H p ν insead of H p,t ν, and S p insead of S p,t. T denoes he se of sopping imes τ such ha τ [, T ] a.s. Definiion 2.1 (Driver, Lipschiz Driver). A funcion f is said o be a driver if f : [, T ] Ω R 2 L 2 ν R (ω,, x, π, l( )) f (ω,, x, π, l( )) is P B(R 2 ) B(L 2 ν )-measurable, f (.,,, ) H 2. A driver f is called a Lipschiz driver if moreover here exiss a consan C such ha d P d-a.s., for each (x 1, π 1, l 1 ), (x 2, π 2, l 2 ), f (ω,, x 1, π 1, l 1 ) f (ω,, x 2, π 2, l 2 ) C( x 1 x 2 + π 1 π 2 + l 1 l 2 ν ). Definiion 2.2 (BSDE wih Jumps). A soluion of a BSDE wih jumps wih erminal ime T, erminal condiion ξ and driver f consiss of a riple of processes (X, π, l) saisfying d X = f (, X, π, l ( ))d π dw l (u)ñ(d, du); X T = ξ (2.1) R where X is a RCLL opional process, and π (resp. l) is an R-valued predicable process defined on Ω [, T ] (resp. Ω [, T ] R ) such ha he sochasic inegral wih respec o W (resp. Ñ) is well defined. This soluion is denoed by (X (ξ, T ), π(ξ, T ), l (ξ, T )). Noe ha he process f (, X, π, l ( )) is predicable and saisfies f (, X, π, l ( )) = f (, X, π, l ( ))d P d-a.s. We recall he exisence and uniqueness resul for BSDEs wih jumps esablished by Tang and Li (1994) in [22]. Theorem 2.3 (Exisence and Uniqueness). Le T >. For each Lipschiz driver f, and each erminal condiion ξ L 2 (F T ), here exiss a unique soluion (X, π, l) S 2,T H 2,T H 2,T ν of he BSDE wih jumps (2.1). 3. Linear BSDEs wih jumps We now focus on linear BSDEs wih jumps which play a crucial role in he sudy of properies of general BSDEs. We firs provide some useful properies of exponenial local maringales driven by a Brownian moion and a Poisson random measure. 3.1. Some properies of exponenial local maringales Le (β ) be an R-valued predicable process, a.s. inegrable wih respec o dw. Le (γ (.)) be an R-valued predicable process defined on [, T ] Ω R, ha is, P B(R )-measurable, and a.s. inegrable wih respec o Ñ(ds, du). Le M = (M ) T be a local maringale given by M := β s dw s + γ s (u)ñ(ds, du). (3.2) R
3332 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 Le Z = (Z ) T be he soluion of d Z s = Z s d M s ; Z = 1. The process Z is he so-called exponenial local maringale associaed wih he local maringale M, denoed by E(M). I is given by he Doléans-Dade formula (see (A.6) in he Appendix): s E(M) s = exp β u dw u 1 s s βu 2 2 du γ r (u)ν(du)dr R (1 + γ r ( Y r )) (3.3) <r s where Y := R un([, ], du). Classically, if γ ( Y ) 1, T a.s. hen we have E(M), T a.s. Noe ha his propery sill holds for general exponenial local maringales (see Appendix). Since here M is driven by a Brownian moion and a Poisson random measure, we have more precisely he following propery. Proposiion 3.1. Le (β ) and (γ (.)) be predicable R-valued processes and le M be he local maringale defined by (3.2). The following asserions are equivalen. (i) For each n N, γ Tn ( Y Tn ) 1P-a.s., where (T n ) n N is he increasing sequence of sopping imes corresponding o he jump imes of Y. (ii) γ (u) 1d P d dν(u)-a.s. Moreover, if one of his condiion is saisfied, hen we have E(M), T a.s. Similarly, if γ (u) > 1d P d dν(u)-a.s., hen, for each, E(M) > a.s. These precisions will be useful in he sequel, in paricular o prove Theorem 5.2. Proof. For each s >, we have <r s (1 + γ r ( Y r )) = n N,<T n s (1 + γ T n ( Y Tn )). Hence, by formula (3.3), condiion (i) implies ha for each s, E(M) s a.s. I remains o show ha (i) is equivalen o (ii). Now, we have E 1 {γt n ( Y Tn )< 1} = E 1 {γr (u)< 1}N(du, dr) n N R R + = E 1 {γr (u)< 1}ν(du)dr, R R + because ν(du)d is he predicable compensaor of he Poisson random measure N(du, d). The resul follows. We now provide a sufficien condiion for he square inegrabiliy propery of E(M). Proposiion 3.2. Le (β ) and (γ (.)) be predicable R-valued processes and le M be he local maringale defined by (3.2). Suppose ha T β 2 s ds + T is bounded. Then, we have E[E(M) 2 T ] < +. γ s 2 νds (3.4) Noe ha in his case, by maringale inequaliies, (E(M) s ) T S 2,T.
M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3333 Proof. By he produc formula (or by using he Doléans-Dade formula (3.3)), we ge E(M) 2 = E(2M + [M, M]), where [M, M] = β2 s ds + s γ s 2( Y s). Now, γs 2 ( Y s) = γ 2 R s (u)n(ds, du) = γ s 2 ν ds + s I follows ha E(M) 2 = E. N +. βs 2 ds + γ s 2 ν ds.. = E(N) exp βs 2 ds + γ 2 R s (u)ñ(ds, du). γ s 2 ν ds, (3.5) where N := 2M + R γs 2 (u)ñ(ds, du). Noe ha N is a local maringale. Now, by assumpion, here exiss K > such ha exp{ T β2 s ds + T γ s 2 ν ds} K a.s. Also, by (3.5), E(N) is non negaive. Since i is also a local maringale, i follows ha i is a supermaringale. Hence, we have E[E(M) 2 T ] E[E(N) T ] K K, which ends he proof. Remark 3.3. For example, if he processes β and γ ν are bounded, he random variable (3.4) is hen bounded, and hence, by he above proposiion, E(M) T L 2. This propery will be used in he sudy of linear BSDEs as well as in he comparison heorem (Theorem 4.2). For example, condiion is saisfied when here exiss ψ L 2 ν such ha d d P dν(u)-a.s. γ (u) ψ(u). More generally, we have he following propery: if β and γ p,ν are bounded, for all p 2, hen E(M) T is p-inegrable for all p 2. This propery, as well as addiional ones, is shown in he Appendix (see Proposiion A.1). I will be used in Secion 5.3, o solve a robus opimizaion problem, where some p-inegrabiliy condiions, wih p > 2, are required. 3.2. Properies of linear BSDEs wih jumps We now show ha he soluion of a linear BSDE wih jumps can be wrien as a condiional expecaion via an exponenial semimaringale. Le (δ ) and (β ) be R-valued predicable processes, supposed o be a.s. inegrable wih respec o d and dw. Le (γ (.)) be a predicable R-valued process defined on [, T ] Ω R, supposed o be a.s. inegrable wih respec o Ñ(ds, du). For each [, T ], le (Γ,s ) s [,T ] be he unique soluion of he following forward SDE dγ,s = Γ,s δ s ds + β s dw s + γ s (u)ñ(ds, du) ; Γ, = 1. (3.6) R The process Γ,. can be wrien as Γ,s = e s δ u du Z,s, where (Z,s ) s [,T ] is he soluion of he following SDE d Z,s = Z,s β s dw s + γ s (u)ñ(ds, du) ; Z, = 1. R Theorem 3.4. Le (δ, β, γ ) be a bounded predicable process. Le Γ be he so-called adjoin process defined as he soluion of SDE (3.6). Suppose ha Γ S 2.
3334 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 Le (X, π, l ) be he soluion in S 2,T H 2,T H 2,T ν of he linear BSDE d X = (ϕ + δ X + β π + γ, l ν )d π dw l (u)ñ(d, du); R X T = ξ. The process (X ) saisfies X = E Γ,T ξ + T (3.7) Γ,s ϕ(s)ds F, T, a.s. (3.8) Proof. Fix [, T ]. To simplify noaion, le us denoe Γ,s by Γ s for s [, T ]. By he Iô produc formula, and denoing Γ,s by Γ s for s [, T ], we have: d(x s Γ s ) = X s dγ s Γ s d X s d[x, Γ ] s = X s Γ s δ s ds + Γ s [ϕ s + δ s X s + β s π s + γ s, l s ν ] ds β s π s Γ s ds Γ s γ s, l s ν ds Γ s (X s β s + π s )dw s Γ s l s (u)(1 + γ s (u))ñ(du, ds) R = Γ s ϕ s ds d M s, wih d M s = Γ s (X s β s + π s )dw s Γ s and T, we ge X ξγ,t = T R l s(u)(1 + γ s (u))ñ(du, ds). By inegraing beween Γ,s ϕ s ds M T + M a.s. (3.9) Recall ha Γ,. S 2 and ha X S 2, π H 2 and l H 2 ν. Moreover, he processes δ, β and γ are bounded. I follows ha he local maringale M is a maringale. Hence, by aking he condiional expecaion in equaliy (3.9), we ge equaliy (3.8). This propery ogeher wih Proposiion 3.1 yields he following corollary, which will be used o prove he comparison heorems. Corollary 3.5. Suppose ha he assumpions of Theorem 3.4 are saisfied. Suppose ha he inequaliy γ (u) 1 holds d P d dν(u)-a.s. If ϕ, [, T ], d P d a.s. and ξ a.s., hen X a.s. for all [, T ]. Suppose ha he inequaliy γ (u) > 1 holds d P d dν(u)-a.s. If ϕ, [, T ], d P d a.s. and ξ a.s., and if X = a.s. for some [, T ], hen ϕ = d P d-a.s. on [, T ], and ξ = a.s on A. Proof. Le us prove he firs asserion. Since γ (u) 1d P d dν(u)-a.s., by Proposiion 3.1, we ge Γ,T a.s. and he resul follows from he represenaion formula for linear BSDEs (3.8). The second asserion follows from similar argumens and he fac ha if γ (u) > 1d P d dν(u)-a.s., hen Γ,T > a.s. Noe ha when ξ and ϕ, if he process γ can ake values < 1 wih sricly posiive probabiliy, hen he soluion X of he linear BSDE may ake sricly negaive values. Moreover,
M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3335 suppose ha ξ, ϕ a.s. and ha he process γ 1 a.s., bu can ake he value 1 wih sricly posiive probabiliy. Then, in general, he equaliy X = does no imply ha ξ = and ϕ = a.s. This is illusraed in he example below. Example 3.1. Suppose ha γ is a real consan, δ = β =, ϕ =, ν(du) := δ 1 (du), where δ 1 denoes Dirac measure a 1. The process N := N([, ] {1}) is hen a Poisson process wih parameer 1, and we have Ñ := Ñ([, ] {1}) = N. Suppose ha he driver f is given by f (l) := γ, l ν = γ l(1). (3.1) One can show ha he associaed adjoin process Γ,s, denoed here by Γ s, saisfies Γ T = (1 + γ ) N T e γ T. (3.11) Consider now he special case when he erminal condiion is given by ξ := N T. Le X be he soluion of he BSDE associaed wih driver f and his erminal condiion. By he represenaion propery of linear BSDEs wih jumps (see equaliy (3.8)) and classical compuaions, we ge X = E[Γ T N T ] = (1 + γ )T. Consequenly, if γ < 1, hen X < alhough ξ = N T a.s. Consider now he special case when f (l) = l(1) and ξ = 1 {T1 T }. (3.12) From equaliy (3.11) wih γ = 1, i follows ha Γ T. a.s. and Γ T = a.s. on {N T 1} = {T 1 T }. The soluion X of he associaed BSDE saisfies X = E[Γ T 1 {T1 T }] = alhough ξ a.s. and P(ξ > ) = P(T 1 T ) >. 4. Comparison heorems and opimizaion principles for BSDEs wih jumps 4.1. Comparison heorems The comparison heorems are key properies of BSDEs and play a crucial role in he sudy of opimizaion problems expressed in erms of BSDEs. In [21], Royer esablished a comparison heorem and a sric comparison heorem for BSDEs wih jumps. Here, we prove hese heorems under less resricive hypoheses and provide some opimizaion principles for BSDEs wih jumps. We begin by a preliminary resul which will be used o prove he comparison heorems. Lemma 4.1 (Comparison Resul wih Respec o a Linear BSDE). Le (δ, β, γ ) be a bounded predicable process and for each, le Γ,. be he exponenial semimaringale soluion of SDE (3.6). Suppose ha Γ,. S 2 and γ (u) 1 d P d ν(du)-a.s. Le ξ L 2 (F T ) and h be a driver (non necessarily Lipschiz). Le (X, π, l ) be a soluion in S 2,T H 2,T H 2,T ν of he BSDE d X = h(, X, π, l ( ))d π dw l (u)ñ(d, du); R X T = ξ. (4.13) Le ϕ H 2,T. Suppose ha h(, X, π, l ) ϕ + δ X + β π + γ, l ν, T, d P d-a.s. (4.14)
3336 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 Then, X is a.s. greaer or equal o he soluion given by (3.8) of he linear BSDE (3.7). In oher erms, T X E Γ,T ξ + Γ,s ϕ(s)ds F, T, a.s. (4.15) Proof. Fix [, T ]. Since γ (u) 1d P d ν(du)-a.s., i follows ha Γ,. a.s. To simplify noaion, le us denoe Γ,s by Γ s for s [, T ]. By he Iô produc formula, and denoing Γ,s by Γ s for s [, T ], we have: d(x s Γ s ) = X s Γ s δ s ds + β s dw s + Γ s π s dw s + l s (u)ñ(du, ds) R Γ s γ s (u)l s (u)n(du, ds). R R γ s (u)ñ(du, ds) π s Γ s β s ds + Γ s h(s, X s, π s, l s )ds Using inequaliy (4.14) ogeher wih he non negaiviy of Γ, and doing he same compuaions as in he proof of Theorem 3.4, we derive ha d(x s Γ s ) Γ s ϕ s ds d M s, where M is a maringale (since Γ,. S 2 and since (δ ) and (β ) are bounded). By inegraing beween and T and by aking he condiional expecaion, we derive inequaliy (4.15). Now, by Theorem 3.4, he second member of his inequaliy corresponds o he soluion of he linear BSDE (3.7). The proof is hus complee. The commens made in he linear case (see in paricular Example 3.1, wih γ < 1 and ξ = N T ) show he relevance of he assumpion γ (u) 1 in he above lemma. Noe also ha if δ, β, γ are bounded and if γ ψ, where ψ L 2 ν, Proposiion 3.2 yields ha, for each, Γ,. S 2. Using his remark ogeher wih he above lemma, we now show he general comparison heorems for BSDEs wih jumps. Theorem 4.2 (Comparison Theorem for BSDEs wih Jumps). Le ξ 1 and ξ 2 L 2 (F T ). Le f 1 be a Lipschiz driver. Le f 2 be a driver. For i = 1, 2, le (X i, πi, li ) be a soluion in S 2,T H 2,T H 2,T ν of he BSDE d X i = f i (, X i, πi, li )d πi d B l i R (u)ñ(d, du); X T i = ξ i. (4.16) Assume ha here exiss a bounded predicable process (γ ) such ha d d P ν(du)-a.s., γ (u) 1 and γ (u) ψ(u), (4.17) where ψ L 2 ν, and such ha f 1 (, X 2, π 2, l1 ) f 1(, X 2, π 2, l2 ) γ, l 1 l 2 ν, [, T ], d d P a.s. Assume ha ξ 1 ξ 2 a.s. and f 1 (, X 2, π 2, l2 ) f 2(, X 2, π 2, l2 ), [, T ], d d P a.s. (4.18) (4.19)
Then, we have M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3337 X 1 X 2 a.s. for all [, T ]. (4.2) Moreover, if inequaliy (4.19) is saisfied for (X 1, π 1, l1 ) insead of (X 2, π 2, l2 ) and if f 2 (insead of f 1 ) is Lipschiz and saisfies (4.18), hen inequaliy (4.2) sill holds. Proof. Pu X = X 1 X 2; π = π 1 π 2; l (u) = l 1(u) l2 (u). Then d X = h d π dw l (u)ñ(d, du); X T = ξ 1 ξ 2, R where h := f 1 (, X 1, π 1, l1 ) f 2(, X 2, π 2, l2 ). The proof now consiss o show ha here exiss δ and β such ha h saisfies inequaliy (4.14) and hen o apply he comparison resul wih respec o a linear BSDE (Lemma 4.1). We have h = f 1 (, X 1, π 1, l1 ) f 1(, X 2, π 1, l1 ) + f 1(, X 2, π 1, l1 ) f 1(, X 2, π 2, l1 ) + f 1 (, X 2, π 2, l1 ) f 1(, X 2, π 2, l2 ) + f 1 (, X 2, π 2, l2 ) f 2(, X 2, π 2, l2 ). (4.21) Le ϕ := f 1 (, X 2, π 2, l2 ) f 2(, X 2, π 2, l2 ) and δ := f 1(, X 1, π 1, l1 ) f 1(, X 2, π 1, l1 ) X 1 { X } β := f 1(, X 2, π 1, l1 ) f 1(, X 2, π 2, l1 ) 1 { π }. π By he assumpion (4.18) on f 1, we ge h ϕ + δ X + β π + γ l ν d d P-a.s. Since f 1 is Lipschiz, he predicable processes (δ ) and (β ) are bounded. By assumpion (4.17), i follows from Proposiion 3.2 ha for each, Γ,. S 2, where he process Γ,. is defined by (3.6). Since γ (u) 1, i follows ha Γ,. a.s. By he comparison resul wih respec o a linear BSDE (see Lemma 4.1), we hus derive ha X E Γ,T (ξ 1 ξ 2 ) + T Γ,s ϕ(s)ds F, T, a.s. (4.22) Now, by assumpion, ϕ d P d-a.s. and ξ 1 ξ 2 a.s. Hence X 1. X 2.. The second asserion follows from he same argumens bu linearizing f 2 insead of f 1. Remark 4.3. Noe ha he presence of jumps as well as inequaliy (4.18), which is a relaively weak assumpion, do no really allow us o proceed wih a linearizaion mehod as in he Brownian case (see [9]). Indeed, here is somehow an asymmery beween he role of negaive jumps and ha of posiive ones of X. The above lemma hus appears as a preliminary sep before proving he general comparison heorem in he case of jumps. We now provide a sric comparison heorem, which holds under an addiional assumpion. This propery is an imporan ool for he sudy of opimizaion problems expressed in erms of BSDEs since i allows us o obain a necessary condiion of opimaliy (see Proposiion 4.9 asserion 2). Theorem 4.4 (Sric Comparison Theorem). Suppose ha he assumpions of Theorem 4.2 hold and ha he inequaliy γ (u) > 1 holds d d P dν(u)-a.s.
3338 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 If X 1 = X 2 a.s. on A for some [, T ] and A F, hen X 1 = X 2 a.s. on [, T ] A, ξ 1 = ξ 2 a.s. on A and (4.19) holds as an equaliy in [, T ] A. Proof. The resul follows from inequaliy (4.22) and he second asserion of Corollary 3.5. Remark 4.5. Example 3.1 wih (3.12) shows he relevance of he assumpion γ (u) > 1 in he sric comparison heorem. Noe ha he condiions under which he above comparison heorems hold, are weaker ha hose made in [21] (see some more deails in Secion 4.3). 4.2. Opimizaion principles From he comparison heorem, we derive opimizaion principles for minima of BSDEs which generalize hose of El Karoui e al. (1997) o he case of jumps. Theorem 4.6 (Opimizaion Principle I). Le ξ in L 2 (F T ) and le ( f, f α ; α A T ) be a family of Lipschiz drivers. Le (X, π, l) (resp. (X α, π α, l α )) be he soluion of he BSDE associaed wih erminal condiion ξ and driver f (resp. f α ). Suppose ha f (, X, π, l ) = ess inf α f α (, X, π, l ) = f ᾱ(, X, π, l ), T, d P d-a.s. for some parameer ᾱ A T (4.23) and ha for each α A, here exiss a predicable process γ α saisfying (4.17) and Then, f α (, X, π, l α ) f α (, X, π, l ) γ α, l α l ν, [, T ], d d P a.s. (4.24) X = ess inf X α α = X ᾱ, T, a.s. (4.25) Proof. For each α, since f (, X, π, l ) f α (, X, π, l )d P d-a.s., he comparison Theorem 4.2 gives ha X X α, T, P-a.s. I follows ha X ess inf α X α T, P-a.s. (4.26) By assumpion, X is he soluion of he BSDE associaed wih f ᾱ. By uniqueness of he soluion of he Lipschiz BSDE associaed wih f ᾱ, we derive ha X = X ᾱ, T, a.s. which implies ha inequaliy in (4.26) is an equaliy. Theorem 4.7 (Opimizaion Principle II). Le ξ in L 2 (F T ) and le ( f, f α ; α A) be a family of Lipschiz drivers. Suppose ha he drivers f α, α A are equi-lipschiz wih common Lipschiz consan C. Le (X, π, l) be a soluion of he BSDE associaed wih erminal condiion ξ and driver f and (X α, π α, l α ) be he soluion of he BSDE associaed wih erminal condiion ξ and driver f α. Suppose ha for each α A, f (, X, π, l ) f α (, X, π, l ), T, d P d-a.s. (4.27) and ha here exiss γ α and δ α saisfying (4.17) and γ α, l α l ν f α (, X, π, l α ) f α (, X, π, l ) δ α, lα l ν, [, T ], d d Pa.s. (4.28)
Then, M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3339 Suppose also ha for each ε >, here exiss α ε A such ha f αε (, X, π, l ) ϵ f (, X, π, l ), T, d P d-a.s. (4.29) X = ess inf X α α, T, a.s. (4.3) Proof. By he comparison heorem, X X α, T, a.s. Hence, X ess inf α X α a.s. for each [, T ]. Le us now show he inverse inequaliy. By esimaion (A.65) in he Appendix, wih η = 1 and β = 3C 2 + 2C, we ge X C 2 X αε 2 1 e βt T ε a.s., which yields ha C 2 X X αε εk C,T ess inf X α α εk C,T, T, a.s., (4.31) where K C,T = C 1 e βt 2 T. Since his inequaliy holds for each ε >, we obain X ess inf α X α a.s. Hence, his inequaliy is an equaliy. Remark 4.8. Noe ha α ε is ε -opimal for (4.3) wih ε = εk C,T since by (4.31), we have X X αε ε a.s. By he sric comparison heorem, we derive he following necessary opimaliy condiion. Proposiion 4.9. Suppose ha Assumpions of Theorem 4.6 (resp. Theorem 4.7) are saisfied. Le ˆα A and le S T. Suppose ˆα is S-opimal, ha is, ess inf α X α S = X ˆα S a.s. and ha he associaed process γ ˆα γ ˆα > 1 d P d dν-a.s. Then, we have saisfies he sric inequaliy f (, X, π, l ) = f ˆα (, X, π, l ), S T, d P d-a.s. 4.3. Remarks on he assumpions of he comparison heorem Le us inroduce he following condiion. Le T >. Assumpion 4.1. A driver f is said o saisfy Assumpion 4.1 if he following holds: d P d-a.s. for each (x, π, l 1, l 2 ) R 2 (L 2 ν )2, wih f (, x, π, l 1 ) f (, x, π, l 2 ) θ x,π,l 1,l 2, l 1 l 2 ν, θ : [, T ] Ω R 2 (L 2 ν )2 L 2 ν ; (ω,, x, π, l 1, l 2 ) θ x,π,l 1,l 2 (ω,.) P B(R 2 ) B((L 2 ν )2 )-measurable, bounded, and saisfying d P d dν(u)-a.s., for each (x, π, l 1, l 2 ) R 2 (L 2 ν )2, θ x,π,l 1,l 2 (u) 1 and θ x,π,l 1,l 2 (u) ψ(u), (4.32) where ψ L 2 ν.
334 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 Assumpion 4.1 is sronger han he one made in he comparison heorem (Theorem 4.2). Indeed, if he driver f 1 saisfies Assumpion 4.1, hen i saisfies condiion (4.18) wih γ =, bu he converse does no hold. Noe also ha condiion (4.18) is only required along (X 2, π 2, l2 ) (he soluion of he second BSDE) as well as l1 (he hird coordinae of he soluion of he firs BSDE) bu no necessarily for all (x, π, l). Also, if f α saisfies Assumpion 4.1, hen i saisfies he weaker condiion (4.28) assumed in he opimizaion principle II. An imporan poin is ha Assumpion 4.1 ensures a monoony propery wih respec o he erminal condiion, in he following sense: for all ξ 1, ξ 2 L 2 (F T ) wih ξ 1 ξ 2 a.s., we have X (ξ 1 ) X (ξ 2 ) a.s., where X (ξ 1 ) (resp. X (ξ 2 )) denoes he soluion of he BSDE associaed wih f and ξ 1 (resp. ξ 2 ). This clearly follows from he comparison heorem applied o f 1 = f 2 = f. As we will see in he nex secion, his assumpion will be appropriae o ensure he monooniciy propery of a dynamic risk measure induced by a BSDE. θ X 2,π 2,l1,l2 Remark 4.1. Assumpion 4.1 implies ha for each (x, π, l 1, l 2 ), f (, x, π, l 1 ) f (, x, π, l 2 ) γ x,π,l 1,l 2, l 1 l 2 ν (4.33) where γ x,π,l 1,l 2 (u) = θ x,π,l 2,l 1 (u). Noe ha Assumpion 4.1 is weaker han he assumpion made by Royer [21], where, in paricular, i is moreover required ha θ x,π,l 1,l 2 C 1 (or equivalenly γ x,π,l 1,l 2 C 1 ) wih C 1 > 1. 5. Dynamic risk measures induced by BSDEs wih jumps, robus opimizaion problems 5.1. Definiions and firs properies Le T > be a ime horizon. Le f be a Lipschiz driver such ha f (,,, ) H 2,T. We define he following funcional: for each T [, T ] and ξ L 2 (F T ), se ρ f (ξ, T ) = ρ (ξ, T ) := X (ξ, T ), T, (5.34) where X (ξ, T ) denoes he soluion of he BSDE (2.1) wih erminal condiion ξ and erminal ime T. If T represens a given mauriy and ξ a financial posiion a ime T, hen ρ (ξ, T ) will be inerpreed as he risk measure of ξ a ime. The funcional ρ : (ξ, T ) ρ (ξ, T ) defines hen a dynamic risk measure induced by he BSDE wih driver f. We now provide properies of such a dynamic risk measure. We poin ou ha, conrary o he Brownian case, he monooniciy propery of ρ, which is naurally required for risk measures, is no auomaically saisfied and needs Assumpion 4.1, inroduced in Secion 4.3. Consisency. By he flow propery (see (A.69) in he Appendix), ρ is consisen: more precisely, le T [, T ] and le S T,T be a sopping ime, hen for each ime smaller han S, he riskmeasure associaed wih posiion ξ and mauriy T coincides wih he risk-measure associaed wih mauriy S and posiion ρ S (ξ, T ) = X S (ξ, T ), ha is, (CS) S, ρ (ξ, T ) = ρ ( ρ S (ξ, T ), S) a.s. Coninuiy. Le T [, T ]. Le {θ α, α R} be a family of sopping imes in T,T, converging a.s. o a sopping ime θ T,T as α ends o α. Le (ξ α, α R) be a family of random variables such ha E[ess sup α (ξ α ) 2 ] < +, and for each α, ξ α is F θ α-measurable. Suppose also ha ξ α converges a.s. o an F θ -measurable random variable ξ as α ends o α. Then, for each S T,T, he random variable ρ S (ξ α, θ α ) ρ S (ξ, θ) a.s. and he process ρ(ξ α, θ α ) ρ(ξ, θ) in S 2,T when α α (see Proposiion A.6 in he Appendix).
M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3341 zero one law. If f (,, ) =, hen he risk-measure associaed wih he null posiion is equal o. More precisely, he risk-measure saisfies he zero one law propery: (ZO) ρ (1 A ξ, T ) = 1 A ρ (ξ, T )a.s for T, A F, and ξ L 2 (F T ). Translaion invariance. If f does no depend on x, hen he associaed risk-measure saisfies he ranslaion invariance propery: (TI) ρ (ξ + ξ, T ) = ρ (ξ, T ) ξ, for any ξ L 2 (F T ) and ξ L 2 (F ). This siuaion can be inerpreed as a marke wih an ineres rae r equal o zero. The case r corresponds o a BSDE wih a driver of he form r x + g(, π, l) and can be reformulaed as a problem wih a driver independen of x by discouning he posiions ξ (see Secion 5.4.1). The general case when f depends on x in a non-linear way may be inerpreed in erms of ambiguiy on he ineres rae (see Secions 5.4.2 and 5.4.3). Homogeneous propery. If f is posiively homogeneous wih respec o (x, π, l), hen he riskmeasure ρ is posiively homogeneous wih respec o ξ, ha is, for λ, T [, T ] and ξ L 2 (F T ), ρ. (λξ, T ) = λρ. (ξ, T ). From now on, we assume ha he driver f saisfies Assumpion 4.1 wih T = T. The comparison heorem for BSDEs wih jumps (see Theorem 4.2) can hen be applied, and yields he monooniciy of he risk measure ρ. Monooniciy. ρ is nonincreasing wih respec o ξ, ha is : for each T [, T ]. (MO) For each ξ 1, ξ 2 L 2 (F T ), if ξ 1 ξ 2 a.s., hen ρ (ξ 1, T ) ρ (ξ 2, T ), T a.s. Noe ha he dynamic risk measure ρ f associaed wih driver f (l) = γ l(1) wih γ < 1, in he case of a Poisson process wih parameer 1, is no monoone (see Example 3.1). The comparison heorem also yields he following propery. Convexiy. If f is concave wih respec o (x, π, l), hen he dynamic risk-measure ρ is convex, ha is, for any λ [, 1], T [, T ], ξ 1, ξ 2 L 2 (F T ), ρ(λξ 1 + (1 λ)ξ 2, T ) λρ(ξ 1, T ) + (1 λ)ρ(ξ 2, T ). (5.35) Suppose now ha in Assumpion 4.1, we have θ x,π,l 1,l 2 > 1. The sric comparison heorem (see Theorem 4.4) can hen be applied and yields he no arbirage propery. No Arbirage. The dynamic risk measure ρ saisfies he no arbirage propery: for each T [, T ], and ξ 1, ξ 2 L 2 (F T ) (NA) If ξ 1 ξ 2 a.s. and if ρ (ξ 1, T ) = ρ (ξ 2, T ) a.s. on an even A F, hen ξ 1 = ξ 2 a.s. on A. Conrary o he monooniciy propery, he no arbirage propery is generally no required for risk-measures. Noe ha he dynamic risk measure ρ f associaed driver f (l) = l(1), in he case of a Poisson process wih parameer 1, is monoone bu does no saisfy he no arbirage propery (see Example 3.1).
3342 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 The inverse problem. We now look a he inverse problem: when can a dynamic risk-measure be represened by a BSDE wih jumps? The following proposiion gives an answer. Proposiion 5.1 (Royer M.). Suppose ha he inensiy measure ν of he Poisson random measure saisfies R (1 u 2 )ν(du) < +. Le ρ be a dynamic risk measure, ha is, a map which, o each ξ L 2 (F T ) and T, associaes an adaped RCLL process (ρ (ξ, T )) { T }. Suppose ha ρ is nonincreasing, consisen, ranslaion invarian and saisfies he zero one law as well as he no arbirage propery. Moreover, suppose ha ρ saisfies he so-called E C,C1 - dominaion propery: here exiss C > and 1 < C 1 such ha ρ (ξ + ξ, T ) ρ (ξ, T ) X C,C 1 (ξ, T ), (5.36) for any ξ, ξ L 2 (F T ), where X C,C 1 (ξ, T ) is he soluion of he BSDE associaed wih erminal condiion ξ and driver f C,C1 (, π, l) := C π + C R (1 u )l + (u)ν(du) C 1 R (1 u )l (u)ν(du). Then, here exiss a Lipschiz driver f (, π, l) such ha ρ = ρ f, ha is, ρ is he dynamic risk measure induced by a BSDE wih jumps wih driver f (, π, l). This proposiion corresponds o Theorem 4.6 in [21], here wrien in erms of risk measures, which generalizes he resul shown in he Brownian case by [5] o he case of jumps. For he proof, we refer he reader o [21]. 5.2. Represenaion of convex dynamic risk measures We now provide a represenaion for dynamic risk measures induced by concave BSDEs wih jumps (which hus are convex risk measures). This dual represenaion is given via a se of probabiliy measures which are absoluely coninuous wih respec o P. Le f be a given driver independen of x. For each (ω, ), le F(ω,,,, ) be he polar funcion of f wih respec o (π, l), defined for each (α 1, α 2 ) in R L 2 ν by F(ω,, α 1, α 2 ) := sup (π,l) R 2 L 2 ν [ f (ω,, π, l) α 1 π α 2, l ν ]. (5.37) Theorem 5.2. Suppose ha he Hilber space L 2 ν is separable. Le f be a Lipschiz driver wih Lipschiz consan C, which does no depend on x. Suppose also ha f saisfies Assumpion 4.1 and is concave wih respec o (π, l). Le T [, T ]. Le A T be he se of predicable processes α = (α 1, α 2 (.)) such ha F(, α 1, α2 ) belongs o H2 T, where F is defined by (5.37). For each α A T, le Q α be he probabiliy absoluely coninuous wih respec o P which admis ZT α as densiy wih respec o P on F T, where Z α is he soluion of d Z α = Z α α 1 dw + (u)d Ñ(d, du) ; Z α = 1. (5.38) R α 2 The convex risk-measure ρ(., T ) has he following represenaion: for each ξ L 2 (F T ), ρ (ξ, T ) = sup α A T EQ α( ξ) ζ(α, T ), (5.39)
M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3343 where he funcion ζ, called penaly funcion, is defined, for each T and α A T, by T ζ(α, T ) := E Q α F(s, αs 1, α2 s )ds. Moreover, for each ξ L 2 (F T ), here exiss ᾱ = (ᾱ 1, ᾱ 2 ) A T such ha F(, ᾱ 1, ᾱ2 ) = f (, π, l ) ᾱ 1 π ᾱ 2, l ν, T, d P d-a.s., where (X, π, l) is he soluion o he BSDE wih driver f, erminal ime T and erminal condiion ξ. Also, he process ᾱ is opimal for (5.39). Remark 5.3. In he paricular case of a Brownian filraion, his represenaion corresponds o ha provided in [9,6] by BSDEs argumens. In his case, conrary o he case wih jumps, all he probabiliy measures Q α, α A, are equivalen o P. In our framework, due o he presence of jumps, he conrols α are no valued in a finie dimensional space as R p bu in he Hilber space R L 2 ν. Noe ha he separabiliy assumpion made on his Hilber space is used in he proof o solve some measurabiliy problems. In paricular, i allows us o apply he secion heorem of [7], which requires ha he space is lusinian, ha is, isomorph o a borelian par of a polish space. Noe ha he above represenaion is relaed o some recen sudies on robus porfolio opimizaion, wih a specific quadraic driver (see [13]). Proof. Since, by assumpion, R L 2 ν is separable, i admis a dense counable subse I. Since f is coninuous wih respec o (π, l), he supremum in (5.37) hus coincides wih he supremum over I, which implies he measurabiliy of F. By resuls of convex analysis in Hilber spaces (see e.g. Ekeland and Temam (1976) [8]), he polar funcion F is convex. I is also lower semiconinuous wih respec o α 1, α 2 as supremum of coninuous funcions. Also, since f is concave and coninuous, f and F saisfy he conjugacy relaion, ha is, f (ω,, π, l) = inf {F(ω,, α 1, α 2 ) + α 1 π + α 2, l ν }, α D (ω) where for each (, ω), D (ω) is he non empy se of α = (α 1, α 2 ) R L 2 ν F(ω,, α 1, α 2 ) >. Now, he following lemma holds. such ha Lemma 5.4. For each (, ω), D (ω) U, where U is he closed subse of he Hilber space R L 2 ν of he elemens α = (α 1, α 2 ) such ha α 1 is bounded by C and ν(du)-a.s α 2 (u) 1 and α 2 (u) ψ(u) C, (5.4) where C is he Lipschiz consan of f. For each process α = (α 1, α2 ) A T, le f α be he associaed linear driver defined by f α (ω,, π, l) := F(ω,, α 1 (ω), α2 (ω)) + α1 (ω)π + α2 (ω), l ν. Noe firs ha for each α A T, f α f. Le T [, T ] and ξ L 2 (F T ). Le (X (ξ, T ), π(ξ, T ), l(ξ, T )) (also denoed (X, π, l)) be he soluion in S 2 H 2 H 2 ν of he BSDE associaed wih driver f, erminal ime T and erminal condiion ξ. The following lemma holds.
3344 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 Lemma 5.5. There exiss a process ᾱ = (ᾱ 1, ᾱ 2 (.)) A T such ha f (, π, l ) = ess inf α A T { f α (, π, l )} = f ᾱ(, π, l ), T, d P d-a.s. By he opimizaion principle for BSDEs wih jumps (see Theorem 4.6), we hus derive ha X (ξ, T ) = inf X α (ξ, T ) = X ᾱ (ξ, T ) (5.41) α A T where for each process α A T, X α (ξ, T ) is he soluion of he linear BSDE associaed wih driver f α, erminal ime T and erminal condiion ξ. Le α = (α 1, α 2 ) A T. By Lemma 5.4, α 2 (u) ψ(u) Cd P d ν(du) a.s. Hence, by Proposiion 3.2, he process Z α, defined by (5.38), belongs o S 2. Consequenly, by he represenaion formula of linear BSDEs (see (3.8)), we have X α (ξ, T ) = E Z α T ξ + T Z α s F(s, α1 s, α2 s )ds. Now, by Lemma 5.4, we also have ha α 2 1d d P dν-a.s. Hence, (Z α) T is a non negaive maringale and he probabiliy Q α which admis ZT α as densiy wih respec o P on F T is well defined. We hus have T X α (ξ, T ) = E Q ξ α + F(s, αs 1, α2 s )ds, which complees he proof of he heorem. Proof of Lemma 5.4. Wihou loss of generaliy, we can suppose ha Assumpion 4.1 is saisfied for each (ω, ). Le (, ω) [, T ] Ω and le α = (α 1, α 2 ) D (ω). Le us firs show ha α 2 1ν-a.s. Suppose by conradicion ha ν({u R, α 2 (u) < 1}) >. Since f saisfies Assumpion 4.1, he following inequaliy f (ω,,, l) f (ω,,, ) + θ,l, (ω), l ν holds for each l L 2 ν. I follows ha, using he definiion of F (see (5.37)), F(ω,, α 1, α 2 ) f (ω,,, l) α 2, l ν f (ω,,, ) + θ,l, (ω) α 2, l ν. By applying his inequaliy o l := n1 {α2 < 1}, where n N, we hus derive ha, F(ω,, α 1, α 2 ) f (ω,,, ) + n (ω, u) α 2 (u) ν(du), {α 2 < 1} θ,l, and his holds for each n N. Now, θ,l, (ω, u) 1. By leing n end o + in his inequaliy, we ge F(ω,, α 1, α 2 ) = +, which provides he expeced conradicion since (α 1, α 2 ) D (ω). We hus have proven ha α 2 1ν-a.s. By similar argumens, one can show ha α 1 is bounded by C and ha α 2 (u) ψ(u) Cν(du)-a.s., which ends he proof. Proof of Lemma 5.5. The proof is divided in wo seps.
M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3345 Sep 1: Le us firs prove ha for each (ω, ), here exiss ᾱ = (ᾱ 1, ᾱ 2 ) U such ha inf {F(ω,, α 1, α 2 ) + α 1 π (ω) + α 2, l (ω) ν } α U = F(ω,, ᾱ 1, ᾱ 2 ) + ᾱ 1 π (ω) + ᾱ 2, l (ω) ν. (5.42) The proof is based on classical argumens of convex analysis. Fix (ω, ). The se U is srongly closed and convex in R L 2 ν. Hence, U is closed for he weak opology. Moreover, since U is bounded, i is a compac se for he weak opology. Le φ be he funcion defined for each α = (α 1, α 2 ) R L 2 ν by φ(α) := F(ω,, α 1, α 2 ) + α 1 π (ω) + α 2, l (ω) ν. This funcion is convex and lower semi-coninuous (l.s.c.) for he srong opology in R L 2 ν. By classical resuls of convex analysis, i is l.s.c. for he weak opology. Now, here exiss a sequence α n = (α n 1, αn 2 ) n N of U such ha φ(α n ) inf α U φ(α) as n. Since U is weakly compac, here exiss an exraced sequence sill denoed by (α n ) which converges for he weak opology o ᾱ = (ᾱ 1, ᾱ 2 ) for some α U. Since φ is l.s.c. for he weak opology, i follows ha φ(ᾱ) lim inf φ(α n ) = inf α U φ(α). Therefore, φ(ᾱ) = inf α U φ(α). Hence ᾱ saisfies (5.42), which ends he proof of sep 1. Sep 2: Le us now inroduce he se U of processes α: [, T ] Ω R L 2 ν ; (, ω) α (ω,.) which are measurable wih respec o σ -algebras P and B(R) B(L 2 ν ) and which ake heir values in Ud P d-a.s. Since he Hilber space L 2 ν is supposed o be separable, i is a polish space. Hence, he secion heorem Secion 81 in he Appendix of Ch. III in Dellacherie and Meyer (1975) [7] can be applied. I follows ha here exiss a process ᾱ = (ᾱ 1, ᾱ 2 (.)) which belongs o U such ha, d P d-a.s., f (, π, l ) = ess inf α U { f α (, π, l )} = f ᾱ(, π, l ), d P d-a.s. T (5.43) Le us show ha he process ᾱ 2(.) is predicable. Since L2 ν is a separable Hilber space, here exiss a counable orhonormal basis (e i ) i N of L 2 ν. For each i N, define λi (ω) = ᾱ2 (ω), e i ν. Since he map., e i ν is coninuous on L 2 ν, he process (λi ) is P-measurable. As ᾱ2 (u) = i λi (ω)e i(u), i follows ha ᾱ 2 : [, T ] Ω R R; (, ω, u) ᾱ 2(ω, u) is P B(R )- measurable. I is hus predicable. Equaliy (5.43) ogeher wih he definiion of f ᾱ yields ha F(, ᾱ 1, ᾱ2 ) = f (, π, l ) ᾱ 1π ᾱ 2, l ν, which implies ha he process F(, ᾱ 1, ᾱ2 ) belongs o H 2 T as a sum of processes in H2 T. Hence, (ᾱ ) A T, which ensures ha equaliy (5.43) sill holds wih U replaced by A T. Remark 5.6. In he dual represenaion, he supremum canno be generally aken over he probabiliy measures Q α equivalen o P. For insance, consider Example 3.1 wih γ = 1, ha is wih driver f (l) := l(1). Then, D = { 1} and A = { 1}. Also, Q 1 is he probabiliy measure wih densiy wih respec o P given by Γ T = N T e T, and is hus no equivalen o P. Noe also ha if f is posiively homogeneous, hen F =. The penaly funcion ζ is hus equal o zero, and for all T, he se A T coincides wih he se of predicable processes (α ) valued in D.
3346 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 5.3. Dynamic risk-measures under model ambiguiy We consider now dynamic risk-measures in he case of model ambiguiy, parameerized by a conrol α as follows. Le A be a polish space (or a borelian subse of a polish space) and le A he se of A-valued predicable processes α. To each coefficien α A, is associaed a model via a probabiliy measure Q α called prior as well as a dynamic risk measure ρ α. More precisely, for each α A, le Z α be he soluion of he SDE: d Z α = Z α β 1 (, α )dw + β 2 (, α, u)d Ñ(d, du) ; Z α R = 1, where β 1 : (, ω, α) β 1 (, ω, α), is a P B(A)-measurable funcion defined on [, T ] Ω A and valued in [ C, C], wih C >, and β 2 : (, ω, α, u) β 2 (, ω, α, u) is a P B(A) B(R )-measurable funcion defined on [, T ] Ω A R which saisfies d d P dν(u)-a.s. β 2 (, α, u) C 1 and β 2 (, α, u) ψ(u), (5.44) wih C 1 > 1 and ψ is a bounded funcion Lν p for all p 1. Hence, ZT α > a.s. and, by Proposiion A.1, ZT α L p (F T ) for all p 1. For each α A, le Q α be he probabiliy measure equivalen o P which admis ZT α as densiy wih respec o P on F T. By Girsanov s heorem (see [14]), he process W α := W β1 (s, α s )ds is a Brownian moion under Q α and N is a Poisson random measure independen of W α under Q α wih compensaed process Ñ α (d, du) = Ñ(d, du) β 2 (, α, u)ν(du)d. Even if he filraion F is no generaed by W α and Ñ α, we have a represenaion heorem for Q α -maringales wih respec o W α and Ñ α. More precisely, we have he following. Lemma 5.7. Le (M ) be a maringale under Q α, and p-inegrable under Q α, for some p > 2. Then, here exiss a unique pair of predicable processes (π, l (.)) such ha M = M + π s dws α + l s (u)ñ α (ds, du) T a.s. (5.45) R Proof. Suppose firs ha (M ) is p-inegrable under Q α. Since (M ) is a Q α -maringale, he process N := Z α M is a maringale under P. By assumpion (5.44), i follows from Proposiion A.1 ha ZT α L q (F T ) for all q 1. By Hölder s inequaliy, N T is hus square inegrable under P. By he maringale represenaion heorem of Tang and Li [22], here exiss a unique pair of predicable processes (ψ, k (.)) H 2 H 2 ν such ha N = N + ψ s dw s + k s (u)ñ(ds, du) T a.s. R Then, by applying Iô s formula o M = N (Z α) 1 and by classical compuaions, one can derive he exisence of (π, l (.)) saisfying (5.45). Remark 5.8. By Proposiion A.2, (π, l (.)) H p α H p ν,α, where he spaces H p α and H p ν,α are defined as H p and H p ν, bu under probabiliy Q α insead of P. For each conrol α, he associaed dynamic risk measure will be induced by a BSDE under Q α and driven by W α and Ñ α, which makes sense by he above Q α -maringale represenaion propery. Le us firs inroduce a funcion
M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 3347 F : [, T ] Ω R L 2 ν A R; (, ω, π, l, α) F(, ω, π, l, α) which is P B(R) B(L 2 ν ) B(A)-measurable. Suppose F is uniformly Lipschiz wih respec o (π, l), coninuous wih respec o α, and such ha ess sup α A F(,,,,, α) H p,t, for each p 2. Suppose also ha F(, π, l 1, α) F(, π, l 2, α) θ π,l 1,l 2,α, l 1 l 2 ν, (5.46) for some adaped process θ π,l 1,l 2,α ( ) wih θ : Ω [, T ] R (L 2 ν )2 A L 2 ν being P B(R) B(L 2 ν )2 B(A)-measurable and saisfying θ π,l 1,l 2,α (u) ψ(u), where ψ is bounded and in Lν p, for all p 1, and θ π,l 1,l 2,α 1 C 1. For each α A, he associaed driver is given by F(, ω, π, l, α (ω)). (5.47) Noe ha hese drivers are equi-lipschiz. For each α A, le ρ α be he dynamic risk-measure induced by he BSDE associaed wih driver F(., α ) and driven by W α and Ñ α. More precisely, fix T [, T ] and ξ L p (F T ) wih p > 2. By Proposiion A.1, ZT α L q (F T ) for all q 1. Hence, by Hölder s inequaliy, ξ L 2 α, where L2 α denoes he space of random variables which are square inegrable under Q α. Similarly, F(,,,,, α ) H 2,T α. Hence, here exiss a unique soluion (X α, π α, l α ) in Sα 2 H2 α H2 α,ν of he Qα -BSDE d X α = F(, π α, lα, α )d π α dw α l α R (u)ñ α (d, du); XT α = ξ, (5.48) driven by W α and Ñ α. The dynamic risk-measure ρ α (ξ, T ) of posiion ξ is hus well defined by ρ α (ξ, T ) := X α (ξ, T ), T, (5.49) wih X α (ξ, T ) = X α. Assumpion (5.46) yields he monooniciy propery of ρ α. The agen is supposed o be averse o ambiguiy. Her risk measure a ime is hus given, for each T [, T ] and ξ L p (F T ), p > 2, by ess sup ρ α (ξ, T ) = ess inf X α (ξ, T ). (5.5) α A α A Noe ha i defines a monoonous dynamic risk measure. We now show ha his dynamic risk measure is induced by a BSDE driven by W and Ñ under probabiliy P. Theorem 5.9. Le f be he funcion defined for each (, ω, π, l) by f (, ω, π, l) := inf α A {F(, ω, π, l, α) + β1 (, ω, α)π + β 2 (, ω, α), l ν }. (5.51) Le ρ be he dynamic risk measure associaed wih driver f, defined for each T [, T ] and ξ L p (F T ) (p > 2), by ρ (ξ, T ) := X (ξ, T ), T, (5.52) wih X (ξ, T ) = X, where (X, π, l) is he unique soluion in S 2,T H 2,T H 2,T ν of he P-BSDE associaed wih driver f, ha is, d X = f (, π, l )d π dw l (u)ñ(d, du); X T = ξ. (5.53) R
3348 M.-C. Quenez, A. Sulem / Sochasic Processes and heir Applicaions 123 (213) 3328 3357 For each T [, T ] and ξ L p (F T ) wih p > 2, we have for each [, T ], ρ (ξ, T ) = ess sup ρ α (ξ, T ) a.s. (5.54) α A Proof. In order o prove his resul, we will express he problem in erms of BSDEs under probabiliy P and hen apply he second opimizaion principle. Fix now ξ L p (F T ) wih p > 2. Since (X α, π α, l α ) is a soluion of BSDE (5.48), i clearly saisfies d X α = f α (, π α, lα )d π α dw l α R (u)ñ(d, du); X T α = ξ, (5.55) which is a P-BSDE driven by W and Ñ, and where he driver is given by f α (, π, l) := F(, π, l, α ) + β 1 (, α )π + β 2 (, α ), l ν. (5.56) The drivers f α are clearly equi-lipschiz. Le p be a real number such ha 2 < p < p. Now, ZT α is q-inegrable, for all q 1. Hence, by Hölder s inequaliy, ξ Lα p, where Lα p denoes he space of random variables which are p -inegrable under Q α. Similarly, F(,,, α ) Hα p. By Proposiion A.2 in he Appendix, here exiss a unique soluion (X α, π α, l α ) in Sα p Hα p Hα,ν p of he Q α -BSDE (5.48). Now, suppose we have shown ha (Z α ) 1 S q,t for all q 1. Since p > 2, by Hölder s inequaliy, we derive ha (X α, π α, l α ) belongs o S 2 H 2 H 2 ν and is hus he unique soluion of P-BSDE (5.55) in S 2 H 2 H 2 ν. Moreover, for each α, f α saisfies Assumpion 4.1. Indeed, we have f α (, π, l 1 ) f α (, π, l 2 ) = F(, π, l 1, α ) F(, π, l 2, α ) + β 2 (, α ), l 1 l 2 ν θ π,l 1,l 2 + β 2 (, α ), l 1 l 2 ν, wih θ π,l 1,l 2 + β 2 (, α ) ( 1 C 1 ) + C 1 1 and θ π,l 1,l 2 + β 2 (, α ) ψ + ψ. Le us show ha f, defined by (5.51), is a Lipschiz driver. Since A is a polish space, here exiss a counable subse D of A which is dense in A. As F is coninuous wih respec o α, i follows ha he above equaliy sill holds wih A replaced by D, which gives ha f is P B(R) B(L 2 ν )-measurable. Also, f is Lipschiz and f (,, ) H2,T, which yields ha f is a Lipschiz driver. By he definiions of f (see (5.51)) and f α (see (5.56)), we ge ha for each α A, f f α. Also, for each ϵ > and each (, ω, π, l) Ω [, T ] R L 2 ν, here exiss αϵ A such ha f (, ω, π, l) + ϵ F(, ω, π, l, α ϵ ) + β 1 (, ω, α ϵ )π + β 2 (, ω, α ϵ ), l ν. By he secion heorem of [7], for each ϵ >, here exiss an A-valued predicable process (α ϵ) such ha f (, π, l ) + ϵ f αϵ (, π, l )d P d-a.s. Consequenly, by he second opimizaion principle for BSDEs (Theorem 4.7), equaliy (5.54) holds, which is he desired resul. I remains o show ha (Z α ) 1 S q,t α for all q > 1. Now, by classical compuaions, we derive ha (Z α ) 1 saisfies he following SDE: d(z α ) 1 = (Z α ) 1 β 1 (, α )dw α β 2 (, α, u)d Ñ α (d, du) ; (Z α R ) 1 = 1.