Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE

Transcription

1 Feynman-Kac represenaion for Hamilon-Jacobi-Bellman IPDE Idris KHRROUBI 1), Huyên PHM 2) December 11, 2012 revised version: November 27, ) CEREMDE, CNRS, UMR ) Laboraoire de Probabiliés e Universié Paris Dauphine Modèles léaoires, CNRS, UMR 7599 kharroubi a ceremade.dauphine.fr Universié Paris 7 Didero, and CREST-ENSE pham a mah.univ-paris-didero.fr bsrac We aim o provide a Feynman-Kac ype represenaion for Hamilon-Jacobi-Bellman equaion, in erms of Forward Backward Sochasic Differenial Equaion (FBSDE) wih a simulaable forward process. For his purpose, we inroduce a class of BSDE where he jumps componen of he soluion is subjec o a parial nonposiive consrain. Exisence and approximaion of a unique minimal soluion is proved by a penalizaion mehod under mild assumpions. We hen show how minimal soluion o his BSDE class provides a new probabilisic represenaion for nonlinear inegro-parial differenial equaions (IPDEs) of Hamilon-Jacobi-Bellman (HJB) ype, when considering a regime swiching forward SDE in a Markovian framework, and imporanly we do no make any ellipiciy condiion. Moreover, we sae a dual formula of his BSDE minimal soluion involving equivalen change of probabiliy measures. This gives in paricular an original represenaion for value funcions of sochasic conrol problems including conrolled diffusion coefficien. Key words: BSDE wih jumps, consrained BSDE, regime-swiching jump-diffusion, Hamilon-Jacobi-Bellman equaion, nonlinear Inegral PDE, viscosiy soluions, inf-convoluion, semiconcave approximaion. MSC Classificaion: 60H10, 60H30, 35K55, 93E20. The auhors would like o hank Pierre Cardaliague for useful discussions. 1

2 1 Inroducion The classical Feynman-Kac heorem saes ha he soluion o he linear parabolic parial differenial equaion (PDE) of second order: v + b(x).d xv r(σσ (x)d 2 xv) + f(x) = 0, (, x) [0, T ) R d, v(t, x) = g(x), x R d, may be probabilisically represened under some general condiions as (see e.g. [11]): [ v(, x) = E f(xs,x )ds + g(x,x T ], ) (1.1) where X,x is he soluion o he sochasic differenial equaion (SDE) driven by a d- dimensional Brownian moion W on a filered probabiliy space (Ω, F, (F ), P): dx s = b(x s )ds + σ(x s )dw s, saring from x R d a [0, T ]. By considering he process Y = v(, X ), and from Iô s formula (when v is smooh) or in general from maringale represenaion heorem w.r.. he Brownian moion W, he Feynman-Kac formula (1.1) is formulaed equivalenly in erms of (linear) Backward Sochasic Equaion: Y = g(x T ) + f(x s )ds Z s dw s, T, wih Z an adaped process, which is idenified o: Z = σ (X )D x v(, X ) when v is smooh. Le us now consider he Hamilon-Jacobi-Bellman (HJB) equaion in he form: v + sup [ b(x, a).dx v + 1 a 2 r(σσ (x, a)dxv) 2 + f(x, a) ] = 0, on [0, T ) R d, (1.2) v(t, x) = g(x), x R d, where is a subse of R q. I is well-known (see e.g. [24]) ha such nonlinear PDE is he dynamic programming equaion associaed o he sochasic conrol problem wih value funcion defined by: [ v(, x) := sup E α where X,x,α is he soluion o he conrolled diffusion: ] f(xs,x,α, α s )ds + g(x,x,α T ), (1.3) dx α s = b(x α s, α s )ds + σ(x α s, α s )dw s, saring from x a, and given a predicable conrol process α valued in. Our main goal is o provide a probabilisic represenaion for he nonlinear HJB equaion using Backward Sochasic Differenial Equaion (BSDEs), namely he so-called nonlinear Feynman-Kac formula, which involves a simulaable forward process. One can hen hope 2

3 o use such represenaion for deriving a probabilisic numerical scheme for he soluion o HJB equaion, whence he sochasic conrol problem. Such issues have araced a lo of ineres and generaed an imporan lieraure over he recen years. cually, here is a crucial disincion beween he case where he diffusion coefficien is conrolled or no. Consider firs he case where σ(x) does no depend on a, and assume ha σσ (x) is of full rank. Denoing by θ(x, a) = σ (x)(σσ (x)) 1 b(x, a) a soluion o σ(x)θ(x, a) = b(x, a), we noice ha he HJB equaion reduces ino a semi-linear PDE: v r(σσ (x)d 2 xv) + F (x, σ D x v) = 0, (1.4) where F (x, z) = sup a [f(x, a)+θ(x, a).z] is he θ-fenchel-legendre ransform of f. In his case, we know from he seminal works by Pardoux and Peng [19], [20], ha he (viscosiy) soluion v o he semi-linear PDE (1.4) is conneced o he BSDE: Y = g(x 0 T ) + F (X 0 s, Z s )ds hrough he relaion Y = v(, X 0 ), wih a forward diffusion process dx 0 s = σ(x 0 s )dw s. Z s dw s, T, (1.5) This probabilisic represenaion leads o a probabilisic numerical scheme for he resoluion o (1.4) by discreizaion and simulaion of he BSDE (1.5), see [4]. lernaively, when he funcion F (x, z) is of polynomial ype on z, he semi-linear PDE (1.4) can be numerically solved by a forward Mone-Carlo scheme relying on marked branching diffusion, as recenly poined ou in [13]. Moreover, as showed in [9], he soluion o he BSDE (1.5) admis a dual represenaion in erms of equivalen change of probabiliy measures as: Y = ess sup E Pα[ f(xs 0, α s )ds + g(xt 0 ) F ], (1.6) α where for a conrol α, P α is he equivalen probabiliy measure o P under which dx 0 s = b(x 0 s, α s )ds + σ(x 0 s )dw α s, wih W α a P α -Brownian moion by Girsanov s heorem. In oher words, he process X 0 has he same dynamics under P α han he conrolled process X α under P, and he represenaion (1.6) can be viewed as a weak formulaion (see [8]) of he sochasic conrol problem (1.3) in he case of unconrolled diffusion coefficien. The general case wih conrolled diffusion coefficien σ(x, a) associaed o fully nonlinear PDE is challenging and led o recen heoreical advances. Consider he moivaing example from uncerain volailiy model in finance formulaed here in dimension 1 for simpliciy of noaions: dx α s = α s dw s, where he conrol process α is valued in = [a, ā] wih 0 a ā <, and define he value funcion of he sochasic conrol problem: v(, x) := sup E[g(X,x,α T )], (, x) [0, T ] R. α 3

4 The associaed HJB equaion akes he form: v + G(D2 xv) = 0, (, x) [0, T ) R, v(t, x) = g(x), x R, (1.7) where G(M) = 1 2 sup a [a 2 M] = ā 2 M + a 2 M. The unique (viscosiy) soluion o (1.7) is represened in erms of he so-called G-Brownian moion B, and G-expecaion E G, conceps inroduced in [22]: v(, x) = E G [ g(x + BT ) ]. Moreover, G-expecaion is closely relaed o second order BSDE sudied in [27], namely he process Y = v(, B ) saisfies a 2BSDE, which is formulaed under a nondominaed family of singular probabiliy measures given by he law of X α under P. This gives a nice heory and represenaion for nonlinear PDE, bu i requires a non degeneracy assumpion on he diffusion coefficien, and does no cover general HJB equaion (i.e. conrol boh on drif and diffusion arising for insance in porfolio opimizaion). On he oher hand, i is no clear how o simulae G-Brownian moion. We provide here an alernaive BSDE represenaion including general HJB equaion, formulaed under a single probabiliy measure (hus avoiding nondominaed singular measures), and under which he forward process can be simulaed. The idea, used in [16] for quasi variaional inequaliies arising in impulse conrol problems, is he following. We inroduce a Poisson random measure µ (d, da) on R + wih finie inensiy measure λ (da)d associaed o he marked poin process (τ i, ζ i ) i, independen of W, and consider he pure jump process (I ) equal o he mark ζ i valued in beween wo jump imes τ i and τ i+1. We nex consider he forward regime swiching diffusion process dx s = b(x s, I s )ds + σ(x s, I s )dw s, and observe ha he (unconrolled) pair process (X, I) is Markov. Le us hen consider he BSDE wih jumps w.r. he Brownian-Poisson filraion F = F W,µ : Y = g(x T ) + f(x s, I s )ds Z s dw s U s (a) µ (ds, da), (1.8) where µ is he compensaed measure of µ. This linear BSDE is he Feynman-Kac formula for he linear inegro-parial differenial equaion (IPDE): + v + b(x, a).d xv r(σσ (x, a)d 2 xv) (1.9) (v(, x, a ) v(, x, a))λ (da ) + f(x, a) = 0, (, x, a) [0, T ) R d, v(t, x, a) = g(x), (x, a) R d, (1.10) hrough he relaion: Y = v(, X, I ). Now, in order o pass from he above linear IPDE wih he addiional auxiliary variable a o he nonlinear HJB PDE (1.2), we consrain he jump componen o he BSDE (1.8) o be nonposiive, i.e. U (a) 0, (, a). (1.11) 4

5 Then, since U (a) represens he jump of Y = v(, X, I ) induced by a jump of he random measure µ, i.e of I, and assuming ha v is coninuous, he consrain (1.11) means ha U (a) = v(, X, a) v(, X, I ) 0 for all (, a). This formally implies ha v(, x) should no depend on a. Once we ge he non dependence of v in a, he equaion (1.9) becomes a PDE on [0, T ) R d wih a parameer a. By aking he supremum over a in (1.9), we hen obain he nonlinear HJB equaion (1.2). Inspired by he above discussion, we now inroduce he following general class of BSDE wih parially nonposiive jumps, which is a non Markovian exension of (1.8)-(1.11): wih Y = ξ + F (s, ω, Y s, Z s, U s )ds + K T K (1.12) Z s dw s U s (e) µ(ds, de), E 0 T, a.s. U (e) 0, dp d λ(de) a.e. on Ω [0, T ]. (1.13) Here µ is a Poisson random measure on R + E wih inensiy measure λ(de)d, a subse of E, ξ an F T measurable random variable, and F a generaor funcion. The soluion o his BSDE is a quadruple (Y, Z, U, K) where, besides he usual componen (Y, Z, U), he fourh componen K is a predicable nondecreasing process, which makes he -consrain (1.13) feasible. We hus look a he minimal soluion (Y, Z, U, K) in he sense ha for any oher soluion (Ȳ, Z, Ū, K) o (1.12)-(1.13), we mus have Y Ȳ. We use a penalizaion mehod for consrucing an approximaing sequence (Y n, Z n, U n, K n ) n of BSDEs wih jumps, and prove ha i converges o he minimal soluion ha we are looking for. The proof relies on comparison resuls, uniform esimaes and monoonic convergence heorem for BSDEs wih jumps. Noice ha compared o [16], we do no assume ha he inensiy measure λ of µ is finie on he whole se E, bu only on he subse on which he jump consrain is imposed. Moreover in [16], he process I does no influence direcly he coefficiens of he process X, which is Markov in iself. In conras, in his paper, we need o enlarge he sae variables by considering he addiional sae variable I, which makes Markov he forward regime swiching jump-diffusion process (X, I). Our main resul is hen o relae he minimal soluion o he BSDE wih -nonposiive jumps o a fully nonlinear IPDE of HJB ype: + E\ v [ + sup b(x, a).d x v(, x) + 1 a 2 r(σσ (x, a)dxv(, 2 x)) [ v(, x + β(x, a, e)) v(, x) β(x, a, e).dx v(, x) ] λ(de) + f ( ] x, a, v, σ (x, a)d x v) = 0, on [0, T ) R d. This equaion clearly exends HJB equaion (1.2) by incorporaing inegral erms, and wih a funcion f depending on v, D x v (acually, we may also allow f o depend on inegral 5

6 erms). By he Markov propery of he forward regime swiching jump-diffusion process, we easily see ha he minimal soluion o he he BSDE wih -nonposiive jumps is a deerminisic funcion v of (, x, a). The main ask is o derive he key propery ha v does no acually depend on a, as a consequence of he -nonposiive consrained jumps. This issue is a novely wih respec o he framework of [16] where here is a posiive cos a each change of he regime I, while in he curren paper, he cos is idenically degenerae o zero. The proof relies on sharp argumens from viscosiy soluions, inf-convoluion and semiconcave approximaion, as we don know a priori any coninuiy resuls on v. In he case where he generaor funcion F or f does no depend on y, z, u, which corresponds o he sochasic conrol framework, we provide a dual represenaion of he minimal soluion o he BSDE by means of a family of equivalen change of probabiliy measures in he spiri of (1.6). This gives in paricular an original represenaion for value funcions of sochasic conrol problems, and unifies he weak formulaion for boh unconrolled and conrolled diffusion coefficien. We conclude his inroducion by poining ou ha our resuls are saed wihou any ellipiciy assumpion on he diffusion coefficien, and includes he case of conrol affecing independenly drif and diffusion, in conras wih he heory of second order BSDE. Moreover, our probabilisic BSDE represenaion leads o a new numerical scheme for HJB equaion, based on he simulaion of he forward process (X, I) and empirical regression mehods, hence aking advanage of he high dimensional properies of Mone-Carlo mehod. Convergence analysis for he discree ime approximaion of he BSDE wih nonposiive jumps is sudied in [14], while numerous numerical ess illusrae he efficiency of he mehod in [15]. The res of he paper is organized as follows. In Secion 2, we give a deailed formulaion of BSDE wih parially nonposiive jumps. We develop he penalizaion approach for sudying he exisence and he approximaion of a unique minimal soluion o our BSDE class, and give a dual represenaion of he minimal BSDE soluion in he sochasic conrol case. We show in Secion 3 how he minimal BSDE soluion is relaed by means of viscosiy soluions o he nonlinear IPDE of HJB ype. Finally, we conclude in Secion 4 by indicaing exensions o our paper, and discussing probabilisic numerical scheme for he resoluion of HJB equaions. 2 BSDE wih parially nonposiive jumps 2.1 Formulaion and assumpions Le (Ω, F, P) be a complee probabiliy space on which are defined a d-dimensional Brownian moion W = (W ) 0, and an independen ineger valued Poisson random measure µ on R + E, where E is a Borelian subse of R q, endowed wih is Borel σ-field B(E). We assume ha he random measure µ has he inensiy measure λ(de)d for some σ-finie measure λ on (E, B(E)) saisfying ( 1 e 2 ) λ(de) <. E 6

7 We se µ(d, de) = µ(d, de) λ(de)d, he compensaed maringale measure associaed o µ, and denoe by F = (F ) 0 he compleion of he naural filraion generaed by W and µ. We fix a finie ime duraion T < and we denoe by P he σ-algebra of F-predicable subses of Ω [0, T ]. Le us inroduce some addiional noaions. We denoe by S 2 he se of real-valued càdlàg adaped processes Y = (Y ) 0 T such ha Y := S 2 ( [ ]) 1 E sup 0 T Y 2 2 <. [ ] L p T (0, T), p 1, he se of real-valued adaped processes (φ ) 0 T such ha E 0 φ p d <. L p (W), p 1, he se of R d -valued P-measurable processes Z = (Z ) 0 T such ha ( [ ]) 1 T Z L p (W) := E 0 Z p p d <. L p ( µ), p 1, he se of P B(E)-measurable maps U : Ω [0, T ] E R such ha ( U L p ( µ) := E [ ( 0 E U (e) 2 λ(de) ) ]) p 1 p 2 d <. ( ) 1 L 2 (λ) is he se of B(E)-measurable maps u : E R such ha u L 2 (λ) := E u(e) 2 2 λ(de) <. K 2 he closed subse of S 2 consising of nondecreasing processes K = (K ) 0 T wih K 0 = 0. We are hen given hree objecs: 1. erminal condiion ξ, which is an F T -measurable random variable. 2. generaor funcion F : Ω [0, T ] R R d L 2 (λ) R, which is a P B(R) B(R d ) B(L 2 (λ))-measurable map. 3. Borelian subse of E such ha λ() <. We shall impose he following assumpion on hese objecs: (H0) (i) The random variable ξ and he generaor funcion F saisfy he square inegrabiliy condiion: E [ ξ 2] [ ] + E F (, 0, 0, 0) 2 d <. 0 (ii) The generaor funcion F saisfies he uniform Lipschiz condiion: here exiss a consan C F such ha F (, y, z, u) F (, y, z, u ) C F ( y y + z z + u u L 2 (λ) ), for all [0, T ], y, y R, z, z R d and u, u L 2 (λ). 7

8 (iii) The generaor funcion F saisfies he monooniciy condiion: F (, y, z, u) F (, y, z, u ) γ(, e, y, z, u, u )(u(e) u (e))λ(de), E for all [0, T ], z R d, y R and u, u L 2 (λ), where γ : [0, T ] Ω E R R d L 2 (λ) L 2 (λ) R is a P B(E) B(R) B(R d ) B(L 2 (λ)) B(L 2 (λ))-measurable map saisfying: C 1 (1 e ) γ(, e, y, z, u, u ) C 2 (1 e ), for all e E, wih wo consans 1 < C 1 0 C 2. Le us now inroduce our class of Backward Sochasic Differenial Equaions (BSDE) wih parially nonposiive jumps wrien in he form: wih Y = ξ + F (s, Y s, Z s, U s )ds + K T K (2.1) Z s dw s U s (e) µ(ds, de), E 0 T, a.s. U (e) 0, dp d λ(de) a.e. on Ω [0, T ]. (2.2) Definiion 2.1 minimal soluion o he BSDE wih erminal daa/generaor (ξ, F ) and -nonposiive jumps is a quadruple of processes (Y, Z, U, K) S 2 L 2 (W) L 2 ( µ) K 2 saisfying (2.1)-(2.2) such ha for any oher quadruple (Ȳ, Z, Ū, K) S 2 L 2 (W) L 2 ( µ) K 2 saisfying (2.1)-(2.2), we have Y Ȳ, 0 T, a.s. Remark 2.1 Noice ha when i exiss, here is a unique minimal soluion. Indeed, by definiion, we clearly have uniqueness of he componen Y. The uniqueness of Z follows by idenifying he Brownian pars and he finie variaion pars, and hen he uniqueness of (U, K) is obained by idenifying he predicable pars and by recalling ha he jumps of µ are inaccessible. By misuse of language, we say someimes ha Y (insead of he quadruple (Y, Z, U, K)) is he minimal soluion o (2.1)-(2.2). In order o ensure ha he problem of geing a minimal soluion is well-posed, we shall need o assume: (H1) There exiss a quadruple (Ȳ, Z, K, Ū) S2 L 2 (W) L 2 ( µ) K 2 saisfying (2.1)-(2.2). We shall see laer in Lemma 3.1 how such condiion is saisfied in a Markovian framework. 8

9 2.2 Exisence and approximaion by penalizaion In his paragraph, we prove he exisence of a minimal soluion o (2.1)-(2.2), based on approximaion via penalizaion. For each n N, we inroduce he penalized BSDE wih jumps Y n = ξ + where K n is he nondecreasing process in K 2 defined by K n = n F (s, Ys n, Zs n, Us n )ds + KT n K n (2.3) Zs n dw s Us n (e) µ(ds, de), 0 T, E 0 [U n s (e)] + λ(de)ds, 0 T. Here [u] + = max(u, 0) denoes he posiive par of u. Noice ha his penalized BSDE can be rewrien as Y n = ξ + F n (s, Ys n, Zs n, Us n )ds Zs n dw s Us n (e) µ(ds, de), 0 T, E where he generaor F n is defined by F n (, y, z, u) = F (, y, z, u) + n [u(e)] + λ(de), for all (, y, z, u) [0, T ] R R d L 2 (λ). Under (H0)(ii)-(iii) and since λ() <, we see ha F n is Lipschiz coninuous w.r.. (y, z, u) for all n N. Therefore, we obain from Lemma 2.4 in [28], ha under (H0), BSDE (2.3) admis a unique soluion (Y n, Z n, U n ) S 2 L 2 (W) L 2 ( µ) for any n N. Lemma 2.1 Le ssumpion (H0) holds. The sequence (Y n ) n is nondecreasing, i.e. Y n Y n+1 for all [0, T ] and all n N. Proof. Fix n N, and observe ha F n (, e, y, z, u) F n+1 (, e, y, z, u), for all (, e, y, z, u) [0, T ] E R R d L 2 (λ). Under ssumpion (H0), we can apply he comparison Theorem 2.5 in [26], which shows ha Y n Y n+1, 0 T, a.s. The nex resul shows ha he sequence (Y n ) n is upper-bounded by any soluion o he consrained BSDE. Lemma 2.2 Le ssumpion (H0) holds. For any quadruple (Ȳ, Z, Ū, K) S 2 L 2 (W) L 2 ( µ) K 2 saisfying (2.1)-(2.2), we have Y n Ȳ, 0 T, n N. (2.4) 9

10 Proof. Fix n N, and consider a quadruple (Ȳ, Z, Ū, K) S 2 L 2 (W) L 2 ( µ) K 2 soluion o (2.1)-(2.2). Then, Ū clearly saisfies 0 [Ūs(e)] + λ(de)ds = 0 for all [0, T ], and so (Ȳ, Z, Ū, K) is a supersoluion o he penalized BSDE (2.3), i.e: Ȳ = ξ + F n (s, Ȳs, Z s, Ūs)ds + K T K Z s dw s Ū s (e) µ(ds, de), 0 T. E By a sligh adapaion of he comparison Theorem 2.5 in [26] under (H0), we obain he required inequaliy: Y n Ȳ, 0 T. We now esablish a priori uniform esimaes on he sequence (Y n, Z n, U n, K n ) n. Lemma 2.3 Under (H0) and (H1), here exiss some consan C depending only on T and he monooniciy condiion of F in (H0)(iii) such ha Y n 2 + S 2 Zn 2 + U n 2 + L 2 (W) L 2 Kn 2 ( µ) S 2 ( [ C E ξ 2 T ] + E F (, 0, 0, 0) 2 d + E [ sup Ȳ 2]), n N. (2.5) 0 0 T Proof. In wha follows we shall denoe by C > 0 a generic posiive consan depending only on T, and he linear growh condiion of F in (H0)(ii), which may vary from line o line. By applying Iô s formula o Y n 2, and observing ha K n is coninuous and Y n = E U n (e)µ({}, de), we have E ξ 2 = E Y n 2 2E + E E = E Y n 2 + E Ys n F (s, Ys n, Zs n, Us n )ds 2E Ys n dks n + E Zs n 2 ds { Y n s + U n s (e) 2 Y n Z n s 2 ds + E 2E Ys n F (s, Ys n, Zs n, Us n )ds 2E s 2 2Ys U n s n (e) } µ(de, ds) Us n (e) 2 λ(de)ds E Y n s dk n s, 0 T. From (H0)(iii), he inequaliy Y n Ȳ by Lemma 2.2 under (H1), and he inequaliy 2ab 1 α a2 + αb 2 for any consan α > 0, we have: E Y n 2 + E E ξ 2 + CE + 1 α E [ sup s [0,T ] Zs n 2 ds + E E U n s (e) 2 λ(de)ds ( ) Ys n F (s, 0, 0, 0) + Ys n + Zs n + Us n L 2 (λ) ds Ȳs 2] + αe K n T K n 2. 10

11 Using again he inequaliy ab a2 2 + b2 2, and (H0)(i), we ge E Y n E Zs n 2 ds E Us n (e) 2 λ(de)ds (2.6) E CE Ys n 2 ds + E ξ E F (s, 0, 0, 0) 2 ds + 1 [ 0 α E sup Ȳs 2] + αe KT n K n 2. s [0,T ] Now, from he relaion (2.3), we have: KT n K n = Y n ξ F (s, Ys n, Zs n, Us n )ds + Zs n dw s + Us n (e) µ(ds, de). Thus, here exiss some posiive consan C 1 depending only on he linear growh condiion of F in (H0)(ii) such ha E KT n K n 2 C 1 (E ξ 2 + E + E 0 E F (s, 0, 0, 0) 2 ds + E Y n 2 ( Y n s 2 + Z n s 2 + U n s 2 L 2 (λ) Hence, by choosing α > 0 s.. C 1 α 1 4, and plugging ino (2.6), we ge 3 4 E Y n E Zs n 2 ds E Us n (e) 2 λ(de)ds CE Y n s 2 ds E ξ E 0 E ) ds ), 0 T. (2.7) F (s, 0, 0, 0) 2 ds + 1 α E[ sup Ȳs 2], 0 T. s [0,T ] Thus applicaion of Gronwall s lemma o E Y n 2 yields: sup E Y n 2 + E Z n 2 d + E U n (e) 2 λ(de)d 0 T 0 0 E C ( E ξ 2 + E 0 F (, 0, 0, 0) 2 d + E [ sup Ȳ 2]), (2.8) [0,T ] which gives he required uniform esimaes (2.5) for (Z n, U n ) n and also (K n ) n by (2.7). Finally, by wriing from (2.3) ha sup Y n ξ + F (, Y n, Z n, U n ) d + KT n 0 T 0 + sup 0 T 0 Z n s dw s + sup 0 T 0 E Us n (e) µ(ds, de), we obain he required uniform esimae (2.5) for (Y n ) n by Burkholder-Davis-Gundy inequaliy, linear growh condiion in (H0)(ii), and he uniform esimaes for (Z n, U n, K n ) n. We can now sae he main resul of his paragraph. 11

12 Theorem 2.1 Under (H0) and (H1), here exiss a unique minimal soluion (Y, Z, U, K) S 2 L 2 (W) L 2 ( µ) K 2 wih K predicable, o (2.1)-(2.2). Y is he increasing limi of (Y n ) n and also in L 2 (0, T), K is he weak limi of (K n ) n in L 2 (Ω, F, P) for all [0, T ], and for any p [1, 2), as n goes o infiniy. Z n Z L p (W) + U n U L p ( µ) 0, Proof. By he Lemmaa 2.1 and 2.2, (Y n ) n converges increasingly o some adaped process Y, saisfying: Y < by he uniform esimae for (Y n S 2 ) n in Lemma 2.3 and Faou s lemma. Moreover by dominaed convergence heorem, he convergence of (Y n ) n o Y also holds in L 2 (0, T). Nex, by he uniform esimaes for (Z n, U n, K n ) n in Lemma 2.3, we can apply he monoonic convergence Theorem 3.1 in [10], which exends o he jump case he monoonic convergence heorem of Peng [21] for BSDE. This provides he exisence of (Z, U) L 2 (W) L 2 ( µ), and K predicable, nondecreasing wih E[KT 2 ] <, such ha he sequence (Z n, U n, K n ) n converges in he sense of Theorem 2.1 o (Z, U, K) saisfying: Y = ξ + F (s, Y s, Z s, U s )ds + K T K Z s dw s U s (e) µ(ds, de), 0 T. E Thus, he process Y is he difference of a càd-làg process and he nondecreasing process K, and by Lemma 2.2 in [21], his implies ha Y and K are also càd-làg, hence respecively in S 2 and K 2. Moreover, from he srong convergence in L 1 ( µ) of (U n ) n o U and since λ() <, we have E [Us n (e)] + λ(de)ds E [U s (e)] + λ(de)ds, 0 0 as n goes o infiniy. Since KT n = n 0 [U s n (e)] + λ(de)ds is bounded in L 2 (Ω, F T, P), his implies E [U s (e)] + λ(de)ds = 0, 0 which means ha he -nonposiive consrain (2.2) is saisfied. Hence, (Y, Z, K, U) is a soluion o he consrained BSDE (2.1)-(2.2), and by Lemma 2.2, Y = lim Y n is he minimal soluion. Finally, he uniqueness of he soluion (Y, Z, U, K) is given by Remark Dual represenaion In his subsecion, we consider he case where he generaor funcion F (, ω) does no depend on y, z, u. Our main goal is o provide a dual represenaion of he minimal soluion o he BSDE wih -nonposiive jumps in erms of a family of equivalen probabiliy measures. 12

13 Le V be he se of P B(E)-measurable processes valued in (0, ), and consider for any ν V, he Doléans-Dade exponenial local maringale ( L ν. ) := E (ν s (e) 1) µ(ds, de) 0 E ( ) = exp ln ν s (e)µ(ds, de) (ν s (e) 1)λ(de)ds, 0 T. (2.9) 0 E When L ν is a rue maringale, i.e. E[L ν T ] = 1, i defines a probabiliy measure Pν equivalen o P on (Ω, F T ) wih Radon-Nikodym densiy: dp ν dp 0 E = L ν, 0 T, (2.10) F and we denoe by E ν he expecaion operaor under P ν. Noice ha W remains a Brownian moion under P ν, and he effec of he probabiliy measure P ν, by Girsanov s Theorem, is o change he compensaor λ(de)d of µ under P o ν (e)λ(de)d under P ν. We denoe by µ ν (d, de) = µ(d, de) ν (e)λ(de)d he compensaed maringale measure of µ under P ν. We hen inroduce he subse V of V by: { V = ν V, valued in [1, ) and essenially bounded : } ν (e) = 1, e E \, dp d λ(de) a.e., and he subse V n as he elemens of ν V essenially bounded by n + 1, for n N. Lemma 2.4 For any ν V, L ν is a uniformly inegrable maringale, and L ν T inegrable. is square Proof. Several sufficien crieria for L ν o be a uniformly inegrable maringale are known. We refer for example o he recen paper [25], which shows ha if ( ST ν T ) := exp ν (e) 1 2 λ(de)d 0 E is inegrable, hen L ν is uniformly inegrable. By definiion of V, we see ha for ν V, ( ST ν T ) = exp ν (e) 1 2 λ(de)d, 0 which is essenially bounded since ν is essenially bounded and λ() <. Moreover, from he explici form (2.9) of L ν, we have L ν T 2 = L ν2 T Sν T, and so E Lν T 2 ST ν. We can hen associae o each ν V he probabiliy measure P ν hrough (2.10). We firs provide a dual represenaion of he penalized BSDEs in erms of such P ν. To his end, we need he following Lemma. Lemma 2.5 Le φ L 2 (W) and ψ L 2 ( µ). Then for every ν V, he processes. 0 φ dw and. 0 E ψ (e) µ ν (d, de) are P ν -maringales. 13

14 Proof. Fix φ L 2 (W) and ν V and denoe by M φ he process. 0 φ dw. Since W remains a P ν -Brownian moion, we know ha M φ is a P ν -local maringale. From Burkholder-Davis-Gundy and Cauchy Schwarz inequalies, we have E ν[ ] sup M φ CE ν[ ] [ ] M φ T = CE L ν T φ 2 d [0,T ] 0 [ ] [ C E L ν T T 2 E 0 ] φ 2 d <, since L ν T is square inegrable by Lemma 2.4, and φ L2 (W). This implies ha M φ is P ν - uniformly inegrable, and hence a rue P ν -maringale. The proof for. 0 E φ (e) µ ν (d, de) follows exacly he same lines and is herefore omied. Proposiion 2.1 For all n N, he soluion o he penalized BSDE (2.3) is explicily represened as Y n = ess sup E ν[ ξ + F (s)ds F ], 0 T. (2.11) ν V n Proof. Fix n N. For any ν V n, and by inroducing he compensaed maringale measure µ ν (d, de) = µ(d, de) (ν (e) 1)λ(de)d under P ν, we see ha he soluion (Y n, Z n, U n ) o he BSDE (2.3) saisfies: Y n = ξ + [ F (s) + E\ ( ) ] n[us n (e)] + (ν s (e) 1)Us n (e) λ(de) ds (2.12) (ν s (e) 1)U n s (e)λ(de)ds Zs n dw s Us n (e) µ ν (ds, de). E By definiion of V, we have (ν s (e) 1)Us n (e)λ(de)ds = 0, 0 T, a.s. E\ By aking expecaion in (2.12) under P ν ( P), we hen ge from Lemma 2.5: Y n = E ν[ ( ( ξ + F (s) + n[u n s (e)] + (ν s (e) 1)Us n (e) ) λ(de)) ds F ]. (2.13) Now, observe ha for any ν V n, hence valued in [1, n + 1], we have n[u n (e)] + (ν (e) 1)U n (e) 0, dp d λ(de) a.e. which yields by (2.13): Y n ess sup ν V n E ν[ ξ + F (s)ds F ]. (2.14) On he oher hand, le us consider he process ν V n defined by ( ) ν (e) = 1 e E\ + 1 U(e) 0 + (n + 1)1 U(e)>0 1 e, 0 T, e E. 14

15 By consrucion, we clearly have n[u n (e)] + (ν (e) 1)U n (e) = 0, 0 T, e, and hus for his choice of ν = ν in (2.13): Y n = E ν [ ξ + F (s)ds F ]. Togeher wih (2.14), his proves he required represenaion of Y n. Remark 2.2 rgumens in he proof of Proposiion 2.1 shows ha he relaion (2.11) holds for general generaor funcion F depending on (y, z, u), i.e. Y n = ess sup E ν[ ] ξ + F (s, Ys n, Zs n, Us n )ds F, ν V n which is in his case an implici relaion for Y n. Moreover, he essenial supremum in his dual represenaion is aained for some ν, which akes exreme values 1 or n+1 depending on he sign of U n, i.e. of bang-bang form. Le us hen focus on he limiing behavior of he above dual represenaion for Y n when n goes o infiniy. Theorem 2.2 Under (H1), he minimal soluion o (2.1)-(2.2) is explicily represened as Y = ess sup E ν[ ξ + F (s)ds F ], 0 T. (2.15) ν V Proof. Le (Y, Z, U, K) be he minimal soluion o (2.1)-(2.2). Le us denoe by Ỹ he process defined in he r.h.s of (2.15). Since V n V, i is clear from he represenaion (2.11) ha Y n Ỹ, for all n. Recalling from Theorem 2.1 ha Y is he poinwise limi of Y n, we deduce ha Y = lim n Y n Ỹ, 0 T. Conversely, for any ν V, le us consider he compensaed maringale measure µ ν (d, de) = µ(d, de) (ν (e) 1)λ(de)d under P ν, and observe ha (Y, Z, U, K) saisfies: [ ] Y = ξ + F (s) (ν s (e) 1)U s (e)λ(de) ds + K T K (2.16) E\ (ν s (e) 1)U s (e)λ(de)ds Z s dw s U s (e) µ ν (ds, de). E By definiion of ν V, we have: E\ (ν s(e) 1)U s (e)λ(de)ds = 0. Thus, by aking expecaion in (2.16) under P ν from Lemma 2.5, and recalling ha K is nondecreasing, we have: Y E ν[ ξ + E ν[ ξ + ( F (s) F (s)ds F ], ) ] (ν s (e) 1)U s (e)λ(de) ds F since ν is valued in [1, ), and U saisfies he nonposiive consrain (2.2). Since ν is arbirary in V, his proves he inequaliy Y Ỹ, and finally he required relaion Y = Ỹ. 15

16 3 Nonlinear IPDE and Feynman-Kac formula In his secion, we shall show how minimal soluions o our BSDE class wih parially nonposiive jumps provides acually a new probabilisic represenaion (or Feynman-Kac formula) o fully nonlinear inegro-parial differenial equaion (IPDE) of Hamilon-Jacobi- Bellman (HJB) ype, when dealing wih a suiable Markovian framework. 3.1 The Markovian framework We are given a compac se of R q, and a Borelian subse L R l \ {0}, equipped wih respecive Borel σ-fields B() and B(L). We assume ha (H) The inerior se Å of is connex, and = dh(å), he closure of is inerior. We consider he case where E = L and we may assume w.l.o.g. ha L = by idenifying and L respecively wih he ses {0} and {0} L in R q R l. We consider wo independen Poisson random measures ϑ and π defined respecively on R + L and R +. We suppose ha ϑ and π have respecive inensiy measures λ ϑ (dl)d and λ π (da)d where λ ϑ and λ π are wo σ-finie measures wih respecive suppors L and, and saisfying (1 l 2 )λ ϑ (dl) < and λ π(da) <, L and we denoe by ϑ(d, dl) = ϑ(d, dl) λ ϑ (dl)d and π(d, da) = π(d, da) λ π (da)d he compensaed maringale measures of ϑ and π respecively. We also assume ha (Hλ π ) (i) The measure λ π suppors he whole se Å: for any a Å and any open neighborhood O of a in R q we have λ π (O Å) > 0. (ii) The boundary of : = \ Å, is negligible w.r.. λ π, i.e. λ π ( ) = 0. In his conex, by aking a random measure µ on R + E in he form, µ = ϑ + π, we noice ha i remains a Poisson random measure wih inensiy measure λ(de)d given by ϕ(e)λ(de) = ϕ(l)λ ϑ (dl) + ϕ(a)λ π (da), E L for any measurable funcion ϕ from E o R, and we have he following idenificaions L 2 ( µ) = L 2 ( ϑ) L 2 ( π), L 2 (λ) = L 2 (λ ϑ ) L 2 (λ π ), (3.1) where L 2 ( ϑ) is he se of P B(L)-measurable maps U : Ω [0, T ] L R such ha U L 2 ( ϑ) := ( [ ]) 1 E U (l) 2 2 λ ϑ (dl)d 0 L <, 16

17 L 2 ( π) is he se of P B()-measurable maps R : Ω [0, T ] R such ha ( [ ]) 1 R L 2 ( π) := E R (a) 2 2 λ π (da)d <, 0 L 2 (λ ϑ ) is he se of B(L)-measurable maps u : L R such ha ( ]) 1 u L 2 (λ ϑ ) := u(l) 2 2 λ ϑ (dl) <, L 2 (λ π ) is he se of B()-measurable maps r : R such ha ( ) 1 r L 2 (λ π) := r(a) 2 2 λ π (da) <. L Given some measurable funcions b : R d R q R d, σ : R d R q R d d and β : R d R q L R d, we inroduce he forward Markov regime-swiching jump-diffusion process (X, I) governed by: dx s = b(x s, I s )ds + σ(x s, I s )dw s + β(x s, I s, l) ϑ(ds, dl), (3.2) L ( ) di s = a Is π(ds, da). (3.3) In oher words, I is he pure jump process valued in associaed o he Poisson random measure π, which changes he coefficiens of jump-diffusion process X. We make he usual assumpions on he forward jump-diffusion coefficiens: (HFC) (i) There exiss a consan C such ha b(x, a) b(x, a ) + σ(x, a) σ(x, a ) C ( x x + a a ), for all x, x R d and a, a R q. (ii) There exiss a consan C such ha β(x, a, l) C(1 + x ) ( 1 l ), β(x, a, l) β(x, a, l) C ( x x + a a )( 1 l ), for all x, x R d, a, a R q and l L. Remark 3.1 We do no make any ellipiciy assumpion on σ. In paricular, some lines and columns of σ may be equal o zero, and so here is no loss of generaliy by considering ha he dimension of X and W are equal. We can also make he coefficiens b, σ and β depend on ime wih he following sandard procedure: we inroduce he ime variable as a sae componen Θ =, and consider he forward Markov sysem: dx s = b(x s, Θ s, I s )ds + σ(x s, Θ s, I s )dw s + β(x s, Θ s, I s, l) ϑ(ds, dl), dθ s = ds di s = ( ) a Is π(ds, da). 17 L

18 which is of he form given above, bu wih an enlarged sae (X, Θ, I) (wih degenerae noise), and wih he resuling assumpions on b(x, θ, a), σ(x, θ, a) and β(x, θ, a, l). Under hese condiions, exisence and uniqueness of a soluion (X,x,a s, I,a s ) s T o (3.2)-(3.3) saring from (x, a) R d R q a ime s = [0, T ], is well-known, and we have he sandard esimae: for all p 2, here exiss some posiive consan C p s.. [ E sup s T for all (, x, a) [0, T ] R d R q. Xs,x,a p + Is,a p] C p (1 + x p + a p ), (3.4) In his Markovian framework, he erminal daa and generaor of our class of BSDE are given by wo coninuous funcions g: R d R q R and f : R d R q R R d L 2 (λ ϑ ) R. We make he following assumpions on he BSDE coefficiens: (HBC1) (i) The funcions g and f(., 0, 0, 0) saisfy a polynomial growh condiion: for some m 0. (ii) There exiss some consan C s.. g(x, a) + f(x, a, 0, 0, 0) sup x R d, a R q 1 + x m + a m <, f(x, a, y, z, u) f(x, a, y, z, u ) C ( x x + a a + y y + z z + u u L 2 (λ ϑ )), for all x, x R d, y, y R, z, z R d, a, a R q and u, u L 2 (λ ϑ ). (HBC2) The generaor funcion f saisfies he monooniciy condiion: f(x, a, y, z, u) f(x, a, y, z, u ) γ(x, a, l, y, z, u, u )(u(l) u (l))λ ϑ (dl), L for all x R d, a R q, z R d, y R and u, u L 2 (λ ϑ ), where γ : R d E R R d L 2 (λ ϑ ) L 2 (λ ϑ ) R is a B(R d ) B(E) B(R) B(R d ) B(L 2 (λ ϑ )) B(L 2 (λ ϑ ))-measurable map saisfying: C 1 (1 l ) γ(x, a, l, y, z, u, u ) C 2 (1 l ), for l L, wih wo consans 1 < C 1 0 C 2. Le us also consider an assumpion on he dependence of f w.r.. he jump componen used in [2], and sronger han (HBC2). (HBC2 ) The generaor funcion f is of he form f(x, a, y, z, u) = h ( x, a, y, z, u(l)δ(x, l)λ ϑ (dl) ) L for (x, a, y, z, u) R d R q R R d L 2 (λ), where 18

19 δ is a measurable funcion on R d L saisfying: 0 δ(x, l) C(1 l ), δ(x, l) δ(x, l) C x x (1 l 2 ), x, x R d, l L, for some posiive consan C. h is a coninuous funcion on R d R q R R d R such ha ρ h(x, a, y, z, ρ) is nondecreasing for all (x, a, y, z) R d R q R R d, and saisfying for some posiive consan C: h(x, a, y, z, ρ) h(x, a, y, z, ρ ) C ρ ρ, ρ, ρ R, for all (x, a, y, z) R d R q R R d. Now wih he idenificaion (3.1), he BSDE problem (2.1)-(2.2) akes he following form: find he minimal soluion (Y, Z, U, R, K) S 2 L 2 (W) L 2 ( ϑ) L 2 ( π) K 2 o wih Y = g(x T, I T ) + Z s.dw s f ( X s, I s, Y s, Z s, U s ) ds + KT K L U s (l) ϑ(ds, dl) R s (a) π(ds, da), (3.5) R (a) 0, dp d λ π (da) a.e. (3.6) The main goal of his paper is o relae he BSDE (3.5) wih -nonposiive jumps (3.6) o he following nonlinear IPDE of HJB ype: where w [ sup L a w + f ( ]., a, w, σ (., a)d x w, M a w) a = 0, on [0, T ) R d, (3.7) w(t, x) = sup g(x, a), x R d, (3.8) a L a w(, x) = b(x, a).d x w(, x) r(σσ (x, a)dxw(, 2 x)) [ + w(, x + β(x, a, l)) w(, x) β(x, a, l).dx w(, x) ] λ ϑ (dl), L M a w(, x) = ( w(, x + β(x, a, l)) w(, x) ) l L, for (, x, a) [0, T ] R d R q. Noice ha under (HBC1), (HBC2) and (3.4) (which follows from (HFC)), and wih he idenificaion (3.1), he generaor F (, ω, y, z, u, r) = f(x (ω), I (ω), y, z, u) and he erminal condiion ξ = g(x T, I T ) saisfy clearly ssumpion (H0). Le us now show ha ssumpion (H1) is saisfied. More precisely, we have he following resul. 19

20 Lemma 3.1 Le ssumpions (HFC), (HBC1) hold. Then, for any iniial condiion (, x, a) [0, T ] R d R q, here exiss a soluion {(Ȳ s,x,a,x,a, Z s, Ū s,x,a,x,a,x,a, R s, K s ), s T } o he BSDE (3.5)-(3.6) when (X, I) = {(Xs,x,a, Is,a ), s T }, wih Ȳ s,x,a = v(s, Xs,x,a ) for some deerminisic funcion v on [0, T ] R d saisfying a polynomial growh condiion: for some p 2, v(, x) sup (,x) [0,T ] R d 1 + x p <. (3.9) Proof. Under (HBC1) and since is compac, we observe ha here exiss some m 0 such ha g(x, a) + f(x, a, y, z, u) C f,g := sup x R d,a 1 + x m + y + z + u L 2 (λ ϑ ) <. (3.10) Le us hen consider he smooh funcion v(, x) = Ce ρ(t ) (1 + x p ) for some posiive consans C and ρ o be deermined laer, and wih p = max(2, m). We claim ha for C and ρ large enough, he funcion v is a classical supersoluion o (3.7)-(3.8). Indeed, observe firs ha from he growh condiion on g in (3.10), here exiss C > 0 s.. ĝ(x) := sup a g(x, a) C(1 + x p ) for all x R d. For such C, we hen have: v(t,.) ĝ. On he oher hand, we see afer sraighforward calculaion ha here exiss a posiive consan C depending only on C, C f,g, and he linear growh condiion in x on b, σ, β by (HFC) (recall ha is compac), such ha v [ sup L a v + f (., a, v, σ (., a)d x v, M a v) ] a 0, (ρ C) v by choosing ρ C. Le us now define he quinuple (Ȳ, Z, Ū, R, K) by: Ȳ = v(, X ) for < T, Ȳ T = g(x T, I T ), Z = σ (X, I )D x v(, X ), T, Ū = M I v(, X ), R = 0, T [ K = v 0 (s, X s) L Is v(s, X s ) f(x s, I s, Z ] s, Ūs) ds, K T = K T + v(t, X T ) g(x T, I T ). < T From he supersoluion propery of v o (3.7)-(3.8), he process K is nondecreasing. Moreover, from he polynomial growh condiion on v, linear growh condiion on b, σ, growh condiion (3.10) on f, g and he esimae (3.4), we see ha (Ȳ, Z, Ū, R, K) lies in S 2 L 2 (W) L 2 ( ϑ) L 2 ( π) K 2. Finally, by applying Iô s formula o v(, X ), we conclude ha (Ȳ, Z, Ū, R, K) is soluion a o (3.5), and he consrain (3.6) is rivially saisfied. Under (HFC), (HBC1) and (HBC2), we hen ge from Theorem 2.1 he exisence of a unique minimal soluion {(Ys,x,a, Zs,x,a, Us,x,a, Rs,x,a, Ks,x,a ), s T } o (3.5)-(3.6) when (X, I) = {(Xs,x,a, Is,a ), s T }. Moreover, as we shall see in he nex paragraph, 20

21 his minimal soluion is wrien in his Markovian conex as: Ys,x,a where v is he deerminisic funcion defined on [0, T ] R d R q R by: = v(s, X,x,a s, I,x,a s ) v(, x, a) := Y,x,a, (, x, a) [0, T ] R d R q. (3.11) We aim a proving ha he funcion v defined by (3.11) does no depend acually on is argumen a, and is a soluion in a sense o be precised o he parabolic IPDE (3.7)-(3.8). Noice ha we do no have a priori any smoohness or even coninuiy properies on v. To his end, we firs recall he definiion of (disconinuous) viscosiy soluions o (3.7)- (3.8). For a locally bounded funcion w on [0, T ) R d, we define is lower semiconinuous (lsc for shor) envelope w, and upper semiconinuous (usc for shor) envelope w by w (, x) = lim inf (, x ) (, x) < T w(, x ) and w (, x) = lim sup (, x ) (, x) < T w(, x ), for all (, x) [0, T ] R d. Definiion 3.1 (Viscosiy soluions o (3.7)-(3.8)) (i) funcion w, lsc (resp. usc) on [0, T ] R d, is called a viscosiy supersoluion (resp. subsoluion) o (3.7)-(3.8) if w(t, x) (resp. ) sup g(x, a), a for any x R d, and ( ϕ [ ]) sup L a ϕ + f(., a, w, σ (., a)d x ϕ, M a ϕ) (, x) a (resp. ) 0, for any (, x) [0, T ) R d and any ϕ C 1,2 ([0, T ] R d ) such ha (w ϕ)(, x) = min ϕ) (resp. [0,T ] Rd(w max ϕ)). [0,T ] Rd(w (ii) locally bounded funcion w on [0, T ) R d is called a viscosiy soluion o (3.7)-(3.8) if w is a viscosiy supersoluion and w is a viscosiy subsoluion o (3.7)-(3.8). We can now sae he main resul of his paper. Theorem 3.1 ssume ha condiions (H), (Hλ π ), (HFC), (HBC1), and (HBC2) hold. The funcion v in (3.11) does no depend on he variable a on [0, T ) R Å i.e. v(, x, a) = v(, x, a ), a, a Å, for all (, x) [0, T ) R d. [0, T ) R d by: Le us hen define by misuse of noaion he funcion v on v(, x) = v(, x, a), (, x) [0, T ) R d, (3.12) for any a Å. Then v is a viscosiy soluion o (3.7) and a viscosiy subsoluion o (3.8). Moreover, if (HBC2 ) holds, v is a viscosiy supersoluion o (3.8). 21

22 Remark Once we have a uniqueness resul for he fully nonlinear IPDE (3.7)-(3.8), Theorem 3.1 provides a Feynman-Kac represenaion of his unique soluion by means of he minimal soluion o he BSDE (3.5)-(3.6). This suggess consequenly an original probabilisic numerical approximaion of he nonlinear IPDE (3.7)-(3.8) by discreizaion and simulaion of he minimal soluion o he BSDE (3.5)-(3.6). This issue, especially he reamen of he nonposiive jump consrain, has been recenly invesigaed in [14] and [15], where he auhors analyze he convergence rae of he approximaion scheme, and illusrae heir resuls wih several numerical ess arising for insance in he superreplicaion of opions in uncerain volailiies and correlaions models. We menion here ha a nice feaure of our scheme is he fac ha he forward process (X, I) can be easily simulaed: indeed, noice ha he jump imes of I follow a Poisson disribuion of parameer λ π := λ π(da), and so he pure jump process I is perfecly simulaable once we know how o simulae he disribuion λ π (da)/ λ π of he jump marks. Then, we can use a sandard Euler scheme for simulaing he componen X. Our scheme does no suffer he curse of dimensionaliy encounered in finie difference mehods or conrolled Markov chains, and akes advanage of he high dimensional properies of Mone-Carlo mehods. 2. We do no address here comparison principles (and so uniqueness resuls) for he general parabolic nonlinear IPDE (3.7)-(3.8). In he case where he generaor funcion f(x, a) does no depend on (y, z, u) (see Remark 3.3 below), comparison principle is proved in [23], and he resul can be exended by same argumens when f(x, a, y, z) depends also on y, z under he Lipschiz condiion (HBC1)(ii). When f also depends on u, comparison principle is proved by [2] in he semilinear IPDE case, i.e. when is reduced o a singleon, under condiion (HBC2 ). We also menion recen resuls on comparison principles for IPDE in [3] and references herein. Remark 3.3 Sochasic conrol problem 1. Le us now consider he paricular and imporan case where he generaor f(x, a) does no depend on (y, z, u). We hen observe ha he nonlinear IPDE (3.7) is he Hamilon- Jacobi-Bellman (HJB) equaion associaed o he following sochasic conrol problem: le us inroduce he conrolled jump-diffusion process: dxs α = b(xs α, α s )ds + σ(xs α, α s )dw s + β(xs α, α s, l) ϑ(ds, dl), (3.13) where W is a Brownian moion independen of a random measure ϑ on a filered probabiliy space (Ω, F, F 0, P), he conrol α lies in F 0, he se of F 0 -predicable process valued in, and define he value funcion for he conrol problem: [ w(, x) := sup E α F 0 L ] f(xs,x,α, α s )ds + g(x,x,α T, α T ), (, x) [0, T ] R d, where {Xs,x,α, s T } denoes he soluion o (3.13) saring from x a s =, given a conrol α F 0. I is well-known (see e.g. [23] or [18]) ha he value funcion w is characerized as he unique viscosiy soluion o he dynamic programming HJB equaion (3.7)-(3.8), and herefore by Theorem 3.1, w = v. In oher words, we have provided a represenaion of fully nonlinear sochasic conrol problem, including especially conrol in 22

23 he diffusion erm, possibly degenerae, in erms of minimal soluion o he BSDE (3.5)- (3.6). 2. Combining he BSDE represenaion of Theorem 3.1 ogeher wih he dual represenaion in Theorem 2.2, we obain an original represenaion for he value funcion of sochasic conrol problem: [ ] sup E f(x α, α )d + g(xt α, α T ) α F 0 0 = sup E ν[ ] f(x, I )d + g(x T, I T ) ν V 0 The r.h.s. in he above relaion may be viewed as a weak formulaion of he sochasic conrol problem. Indeed, i is well-known ha when here is only conrol on he drif, he value funcion may be represened in erms of conrol on change of equivalen probabiliy measures via Girsanov s heorem for Brownian moion. Such represenaion is called weak formulaion for sochasic conrol problem, see [8]. In he general case, when here is conrol on he diffusion coefficien, such Brownian Girsanov s ransformaion can no be applied, and he idea here is o inroduce an exogenous process I valued in he conrol se, independen of W and ϑ governing he conrolled sae process X α, and hen o conrol he change of equivalen probabiliy measures hrough a Girsanov s ransformaion on his auxiliary process. 3. Non Markovian exension. n ineresing issue is o exend our BSDE represenaion of sochasic conrol problem o a non Markovian conex, ha is when he coefficiens b, σ and β of he conrolled process are pah-dependen. In his case, we know from he recen works by Ekren, Touzi, and Zhang [7] ha he value funcion o he pah-dependen sochasic conrol is a viscosiy soluion o a pah-dependen fully nonlinear HJB equaion. One possible approach for geing a BSDE represenaion o pah-dependen sochasic conrol, would be o prove ha our minimal soluion o he BSDE wih nonposiive jumps is a viscosiy soluion o he pah-dependen fully nonlinear HJB equaion, and hen o conclude wih a uniqueness resul for pah-dependen nonlinear PDE. However, o he bes of our knowledge, here is no ye such comparison resul for viscosiy supersoluion and subsoluion of pah-dependen nonlinear PDEs. Insead, we recenly proved in [12] by purely probabilisic argumens ha he minimal soluion o he BSDE wih nonposiive jumps is equal o he value funcion of a pah-dependen sochasic conrol problem, and our approach circumvens he delicae issue of dynamic programming principle and viscosiy soluion in he non Markovian conex. Our resul is also obained wihou assuming ha σ is non degenerae, in conras wih [7] (see heir ssumpion 4.7). The res of his paper is devoed o he proof of Theorem Viscosiy propery of he penalized BSDE Le us consider he Markov penalized BSDE associaed o (3.5)-(3.6): Y n = g(x T, I T ) + f(x s, I s, Ys n, Zs n, Us n )ds + n [Rs n (a)] + λ π (da)ds Zs n.dw s Us n (l) ϑ(ds, dl) Rs n (a) π(ds, da), (3.14) L 23

24 and denoe by {(Ys n,,x,a, Zs n,,x,a, Us n,,x,a, Rs n,,x,a ), s T } he unique soluion o (3.14) when (X, I) = {(Xs,x,a, Is,a ), s T } for any iniial condiion (, x, a) [0, T ] R d R q. From he Markov propery of he jump-diffusion process (X, I), we recall from [2] ha Ys n,,x,a = v n (s, Xs,x,a, Is,a ), s T, where v n is he deerminisic funcion defined on [0, T ] R d R q by: v n (, x, a) := Y n,,x,a, (, x, a) [0, T ] R d R q. (3.15) From he convergence resul (Theorem 2.1) of he penalized soluion, we deduce ha he minimal soluion of he consrained BSDE is acually in he form Ys,x,a = v(s, Xs,x,a, Is,a ), s T, wih a deerminisic funcion v defined in (3.11). Moreover, from he uniform esimae (2.5) and Lemma 3.1, here exiss some posiive consan C s.. for all n, ( v n (, x, a) 2 C E g(x,x,a T [ + E sup s T [, I,a T T ) 2 + E v(s, X,x,a s ) 2]), f(x,x,a s ], Is,a, 0, 0, 0) 2 ds for all (, x, a) [0, T ] R d R q. From he polynomial growh condiion in (HBC1)(i) for g and f, (3.9) for v, and he esimae (3.4) for (X, I), we obain ha v n, and hus also v by passing o he limi, saisfy a polynomial growh condiion: here exiss some posiive consan C v and some p 2, such ha for all n: v n (, x, a) + v(, x, a) C v ( 1 + x p + a p), (, x, a) [0, T ] R d R q. (3.16) We now consider he parabolic semi-linear penalized IPDE for any n: v n (, x, a) La v n (, x, a) f ( x, a, v n, σ (x, a)d x v n, M a v n ) (3.17) [v n (, x, a ) v n (, x, a)]λ π (da ) n [v n (, x, a ) v n (, x, a)] + λ π (da ) = 0, on [0, T ) R d R q, v n (T,.,.) = g, on R d R q. (3.18) From Theorem 3.4 in Barles e al. [2], we have he well-known propery ha he penalized BSDE wih jumps (2.3) provides a viscosiy soluion o he penalized IPDE (3.17)- (3.18). cually, he relaion in heir paper is obained under (HBC2 ), which allows he auhors o ge comparison heorem for BSDE, bu such comparison heorem also holds under he weaker condiion (HBC2) as shown in [26], and we hen ge he following resul. Proposiion 3.1 Le ssumpions (HFC), (HBC1), and (HBC2) hold. The funcion v n in (3.15) is a coninuous viscosiy soluion o (3.17)-(3.18), i.e. i is coninuous on [0, T ] R d R q, a viscosiy supersoluion (resp. subsoluion) o (3.18): v n (T, x, a) (resp. ) g(x, a), 24

25 for any (x, a) R d R q, and a viscosiy supersoluion (resp. subsoluion) o (3.17): ϕ (, x, a) La ϕ(, x, a) (3.19) f(x, a, v n (, x, a), σ (x, a)d x ϕ(, x, a), M a ϕ(, x, a)) [ϕ(, x, a ) ϕ(, x, a)] + λ π (da ) (resp. ) 0, [ϕ(, x, a ) ϕ(, x, a)]λ π (da ) n for any (, x, a) [0, T ) R d R q and any ϕ C 1,2 ([0, T ] (R d R q )) such ha (v n ϕ)(, x, a) = min [0,T ] R d R q(v n ϕ) (resp. max [0,T ] R d R q(v n ϕ)). (3.20) In conras o local PDEs wih no inegro-differenial erms, we canno resric in general he global minimum (resp. maximum) condiion on he es funcions for he definiion of viscosiy supersoluion (resp. subsoluion) o local minimum (resp. maximum) condiion. In our IPDE case, he nonlocal erms appearing in (3.17) involve he values w.r.. he variable a only on he se. Therefore, we are able o resric he global exremum condiion on he es funcions o exremum on [0, T ] R d. More precisely, we have he following equivalen definiion of viscosiy soluions, which will be used laer. Lemma 3.2 ssume ha (Hλ π ), (HFC), and (HBC1) hold. In he definiion of v n being a viscosiy supersoluion (resp. subsoluion) o (3.17) a a poin (, x, a) [0, T ) R d Å, we can replace condiion (3.20) by: 0 = (v n ϕ)(, x, a) = min (v n ϕ) (resp. max (v n ϕ)), [0,T ] R d Å [0,T ] R d Å and suppose ha he es funcion ϕ is in C 1,2,0 ([0, T ] R d R q ). Proof. We rea only he supersoluion case as he subsoluion case is proved by same argumens, and proceed in wo seps. Sep 1. Fix (, x, a) [0, T ) R d R q, and le us show ha he viscosiy supersoluion inequaliy (3.19) also holds for any es funcion ϕ in C 1,2,0 ([0, T ] R d R q ) s.. (v n ϕ)(, x, a) = min [0,T ] R d R q(v n ϕ). (3.21) We may assume w.l.o.g. ha he minimum for such es funcion ϕ is zero, and le us define for r > 0 he funcion ϕ r by ( ( x ϕ r (, x, a ) = ϕ(, x, a 2 + a 2 )) ( x 2 + a 2 ) (1 ) 1 Φ r 2 C v Φ + x r 2 p + a p), where C v > 0 and p 2 are he consan and degree appearing in he polynomial growh condiion (3.16) for v n, Φ : R + [0, 1] is a funcion in C (R + ) such ha Φ [0,1] 0 and Φ [2,+ ) 1. Noice ha ϕ r C 1,2,0 ([0, T ] R d R q ), (ϕ r, D x ϕ r, D 2 xϕ r ) (ϕ, D x ϕ, D 2 xϕ) as r (3.22) 25

26 locally uniformly on [0, T ] R d R q, and ha here exiss a consan C r > 0 such ha ϕ r (, x, a ) C r ( 1 + x p + a p) (3.23) for all (, x, a ) [0, T ] R q R d. Since Φ is valued in [0, 1], we deduce from he polynomial growh condiion (3.16) saisfied by v n and (3.21) ha ϕ r v n on [0, T ] R d R q for all r > 0. Moreover, we have ϕ r (, x, a) = ϕ(, x, a) (= v n (, x, a)) for r large enough. Therefore we ge (v n ϕ r )(, x, a) = min [0,T ] R d R q(v n ϕ r ), (3.24) for r large enough, and we may assume w.l.o.g. ha his minimum is sric. Le (ϕ r k ) k be a sequence of funcion in C 1,2 ([0, T ] (R d R q )) saisfying (3.23) and such ha (ϕ r k, D xϕ r k, D2 xϕ r k ) (ϕr, D x ϕ r, D 2 xϕ r ) as k, (3.25) locally uniformly on [0, T ] R d R q. From he growh condiions (3.16) and (3.23) on he coninuous funcions v n and ϕ r k, we can assume w.l.o.g. (up o an usual negaive perurbaion of he funcion ϕ k r for large (x, a )), ha here exiss a bounded sequence ( k, x k, a k ) k in [0, T ] R d R q such ha (v n ϕ r k )( k, x k, a k ) = min [0,T ] R d R q(v n ϕ r k ). (3.26) The sequence ( k, x k, a k ) k converges up o a subsequence, and hus, by (3.24), (3.25) and (3.26), we have ( k, x k, a k ) (, x, a), as k. (3.27) Now, from he viscosiy supersoluion propery of v n a ( k, x k, a k ) wih he es funcion, we have ϕ r k ϕr k ( k, x k, a k ) L a k ϕ r k ( k, x k, a k ) f(x k, a k, v n ( k, x k, a k ), σ (x k, a k )D x ϕ r k ( k, x k, a k ), M a k ϕ r k ( k, x k, a k )) [ϕ r k ( k, x k, a ) ϕ r k ( k, x k, a k )]λ π (da ) n [ϕ r k ( k, x k, a ) ϕ r k ( k, x k, a k )] + λ π (da ) 0, Sending k and r o infiniy, and using (3.22), (3.25) and (3.27), we obain he viscosiy supersoluion inequaliy a (, x, a) wih he es funcion ϕ. Sep 2. Fix (, x, a) [0, T ) R d Å, and le ϕ be a es funcion in C1,2 ([0, T ] (R d R q )) such ha 0 = (v n ϕ)(, x, a) = min (v n ϕ). (3.28) [0,T ] R d Å By same argumens as in (3.23), we can assume w.l.o.g. ha ϕ saisfies he polynomial growh condiion: ϕ(, x, a ) C(1 + x p + a p ), (, x, a ) [0, T ] R d R q, 26