Monique JEANBLANC 1, Thomas LIM 2 and Nacira AGRAM 3. Introduction

Transcription

1 ESAIM: PROCEEDINGS AND SURVEYS, June 217, Vol. 56, p S. Crépey, M. Jeanblanc and A. Nikeghbali Ediors SOME EXISENCE RESULS FOR ADVANCED BACKWARD SOCHASIC DIFFERENIAL EQUAIONS WIH A JUMP IME, Monique JEANBLANC 1, homas LIM 2 and Nacira AGRAM 3 Absrac. In his paper, we are ineresed by advanced backward sochasic differenial equaions ABSDEs, in a probabiliy space equipped wih a Brownian moion and a single jump process, wih a jump a ime τ. ABSDEs are BSDEs where he driver depends on he fuure pahs of he soluion. We show, ha under immersion hypohesis beween he Brownian filraion and is progressive enlargemen wih τ, assuming ha he condiional law of τ is equivalen o he uncondiional law of τ, and a Lipschiz condiion on he driver, he ABSDE has a soluion. Inroducion In he radiional approach, Backward Sochasic Differenial Equaions BSDEs are sudied for a driver which depends in a Markovian way of he parameers; in paricular, BSDEs wih a single jump and driven by a Brownian moion B have he form dy = f, Y, Z, U d Z db U dh where H is he process H = 1 {τ } associaed wih a given random ime τ. In his paper, we are ineresed by BSDEs in which he driver depends on he fuure of he soluion, in such a way ha he driver is adaped, for example if he driver depends on condiional expecaion of he fuure of he soluion, e.g., dy = ay be G Y 1 d Z dw U dh, we use he noaion E G [X := E[X G, where G = G is he progressive enlargemen filraion of he filraion of B by H. More precisely, we focus on BSDEs wrien in one of he following forms, called advanced backward sochasic differenial equaions in shor ABSDEs dy = f, Y, E G [Y δ, E G [Y s s δ, Z, E G [Z δ, E G [Z s s δ, U, E G [U δ, E G [U s s δ d Z db U dh,,.1 Y = ξ, δ, Z = P, U = Q 1 { τ}, < δ, his research was pored by Chaire Markes in ransiion, French Banking Federaion Insiu Louis Bachelier and Labex ANR 11-LABX-19. he research of N. Agram is also carried ou wih por of he Norwegian Research Council, wihin he research projec Challenges in Sochasic Conrol, Informaion and Applicaions SOCONINF, projec number 25768/F2. 1 LaMME, Universiy of Evry; monique.jeanblanc@univ-evry.fr 2 LaMME, Ecole Naionale Supérieure d Informaique pour l Indusrie e l Enreprise; lim@ensiie.fr 3 Deparmen of Mahemaics, Universiy of Oslo; naciraa@mah.uio.no c EDP Sciences, SMAI 217 Aricle published online by EDP Sciences and available a hp:// or hps://doi.org/1.151/proc/

2 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 89 and dy = E G[ f, Y, Y δ, Y s s δ, Z, Z δ, Z s s δ, U, U δ, U s s δ d Z db U dh,, Y = ξ, δ, Z = P, U = Q 1 { τ}, < δ. he soluion is he riple Y, Z, U, living in some spaces we shall define laer on. In his equaion, for a process V, he noaion V s s δ means ha we consider all he pah of he process beween and δ. he erminal condiions ξ, P and Q are given processes. We remark ha he drivers f of hese ABSDEs depend on he values of he processes Y, Z, U for presen ime as well as for fuure ime δ and also of he rajecory of he processes on he inerval [, δ. he ABSDE.1 was inroduced by Peng and Yang in [12 in a Brownian case seing roughly speaking, for τ. Øksendal e al. [11 have inroduced ABSDEs of he form.2 - in a simpler case where no all he pah of he soluion is involved - when dealing wih opimal conrol for delayed sysems, aking ino accoun a random Poisson measure, insead of a single jump process. Using he mehodology of BSDEs in an enlargemen of filraion seing as in Kharroubi and Lim [8, we give condiions such ha here exiss a unique soluion of.1 and of.2 under immersion hypohesis and in adequae spaces. his progressive enlargemen is ofen considered as progressive adding of informaion given in form of a random ime τ in a way which ransforms τ o a sopping ime wih respec o he filraion G. he opic of enlargemen of filraion was iniiaed by Jacod, Jeulin and Yor see [6, 7. Naurally, he enlargemen of filraion appears in credi risk and i has also been relaed recenly o sochasic opimal conrol by Pham [13 and o mean-variance hedging by Kharroubi e al. [9 where he opimal sraegy is described by non-sandard BSDEs driven by a Brownian moion and a jump maringale in he enlarged filraion. here are, in he lieraure, wo approaches of BSDEs wih a single jump. One of hem considers BSDEs driven by maringales and is based on he predicable represenaion propery PRP, as in Dumirescu e al. [3. hese auhors consider a Brownian moion B and he maringale M associaed o he jump process H. Under some condiions, in paricular ha he Brownian moion is a Brownian moion in he filraion generaed by he pair B, M which is a consequence of he assumed immersion propery, hey prove ha he pair B, M enjoys PRP and hey solve BSDEs of he form dy = f, Y, Z, U d Z dw U dm using usual mehodology. Kharroubi and Lim s mehod is differen: hey consider he case dy = g, Y, Z, U d Z dw U dh, assuming immersion propery, and hey show ha his BSDE is equivalen o a sysem of BSDE in he Brownian filraion. In fac, assuming he more general hypohesis of exisence of posiive condiional densiy α as in Hypohesis 1.2 below, hey can solve he problem wihou immersion propery as follows: he change of probabiliy dp = α τ 1 dp is such ha B is a P -Brownian moion independen of H so ha immersion holds under P and he soluion of he BSDE is he same under P and P. However, here, we can no use his mehodology, since a change of probabiliy will affec he condiional expecaion, and he Lipschiz condiion will be difficul o check. Neverheless, working wihou immersion hypohesis is doable as we explain in he Appendix of he paper. 1. Framework 1.1. Classical resuls abou progressive enlargemen Le Ω, G, P be a complee probabiliy space. We assume ha his space is equipped wih a one-dimensional sandard Brownian moion B and we denoe by F := F he righ-coninuous and complee filraion generaed by B. We consider on his space a random ime τ and we inroduce he righ-coninuous process H := 1 {τ.}. Since τ is no posed o be an F-sopping ime, we use he sandard approach of filraion enlargemen by considering he smalles righ-coninuous exension G of F ha urns τ ino a G-sopping ime. More precisely, he filraion G := G is defined by G := ε> G ε,.2

3 9 M. JEANBLANC,. LIM AND N. AGRAM for any, where G s := F s σh u, u [, s, for any s. We denoe by PF resp. PG he σ-algebra of F resp. G-predicable subses of Ω R, i.e., he σ- algebra generaed by he lef-coninuous F resp. G-adaped processes. We denoe by OF resp. OG he σ-algebra of F resp. G-opional subses of Ω R, i.e., he σ-algebra generaed by he righ-coninuous F resp. G-adaped processes. We impose he following hypohesis inroduced by Brémaud and Yor [2, which is classical in he filraion enlargemen heory and is called H-hypohesis or immersion propery. Hypohesis 1.1. he process B remains a G-Brownian moion. We observe ha, since he filraion F is generaed by he Brownian moion B, Hypohesis 1.1 is equivalen o all F-maringales are also G-maringales. In paricular, he sochasic inegral X sdb s is a well defined G-local maringale for all PG-measurable processes X such ha X s 2 ds < a.s., for all. We also inroduce anoher hypohesis, ofen called he Jacod equivalence hypohesis see, e.g., [1, chaper 4, ha he condiional law of τ is equivalen o he law of τ and ha τ admis a densiy w.r.. Lebesgue s measure, which will allow us o compue condiional expecaions w.r.. G in erms of condiional expecaions w.r.. F. Hypohesis 1.2. We assume ha here exiss a sricly posiive PF BR-measurable funcion ω,, u α ω, u coninuous in such ha a for any θ, he process α θ is an F-maringale, b for any, he measure α ω, θdθ is a version of Pτ dθ F ω, ha is for any Borel funcion f such ha fτ is inegrable, one has In paricular, he densiy of τ is α. E[fτ F = fθα θdθ, a.s. In all he paper, Hypoheses 1.1 and 1.2 are in force. We now recall some sandard resuls ha will be imporan for our purpose and we refer o [4 for heir proofs. We inroduce he F-ermaringale G called Azéma s ermaringale defined as G := Pτ > F = α θdθ,. he ermaringale G is sricly posiive, non-increasing and coninuous. he process M defined by τ α s s M := H ds,, G s is a G-maringale, wih a single jump a ime τ. he F-adaped process λ defined by λ := α G,, 1.1 is called he F-inensiy of τ. Under Hypoheses 1.1 and 1.2, we have, from [4, equaliy 11, α θ = α θ θ, θ, 1.2

4 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 91 which implies G = exp λ s ds, 1.3 since, by definiion of G and λ and he fac ha 1.2 holds, we have he following equaliies, G = α θdθ = 1 α θdθ = 1 α θ θdθ = 1 G θ λ θ dθ and G = 1. Noe ha, from immersion G = E Fs 1 {τ>}, s >. 1.4 Hypohesis 1.3. We assume ha he process λ is upper bounded by a consan k. Lemma 1.4. For any [,, he random variable G is lower bounded by e k, and for any θ [, we have < α θ k. Proof. he bound on G is obvious from 1.3. Le θ [,, hen for any θ, 1.1 and 1.2 lead o α θ = α θ θ = λ θ G θ k. Moreover, since αθ is a maringale, we ge α θ E F α θ θ k for any θ. Furhermore, if Y G is inegrable, hen we have E G [Y 1 {<τ} = 1 G 1 {<τ} E F Y 1 {<τ}. We recall a decomposiion resul for PG-measurable processes, proved in [7, Lemma 4.4 for bounded processes. I can be easily exended o he case of unbounded processes. Proposiion 1.5. Any PG-measurable process X = X can be represened as X = X b 1 { τ} X a τ1 {>τ}, for all, where X b is PF-measurable and X a is PF BR -measurable. Here, he erscrip b is for before τ and a for afer τ. In paricular, a G-predicable process is equal o an F-predicable process on he se { τ}. Song [14 has exended he previous resul o he class of opional processes under some hypoheses, which are saisfied under equivalence Jacod s hypohesis. Proposiion 1.6. Any OG-measurable process X = X can be represened as X = X b 1 {<τ} X a τ1 { τ}, for all, where X b is OF-measurable and X a is OF BR -measurable. If he process X is bounded by a consan K, hen he process X b is bounded by K and one can also choose he process X a θ bounded by K for any θ. We remark ha he uniqueness of X a θ is graned for θ. he process X b is uniquely deermined on [, by X b = 1 G E F [X 1 {<τ}, his quaniy will be called he pre-defaul par.

5 92 M. JEANBLANC,. LIM AND N. AGRAM Lemma 1.7. Le Y τ be a bounded F στ-measurable random variable. hen, for any, we have E G [Y τ = Y b 1 {<τ} Y a τ1 {τ } a.s. where [ = EF Y b which can be rewrien under he form Y uα udu G a.s. 1.5 Y a θ = E F[ Y θ a.s. for any θ Y a τ = E F[ Y θ θ=τ a.s. on τ. 1.6 Proof. he proof of his lemma is an applicaion of Proposiion 1.6 and ha Y a θ = EF [ Y θα θ α θ a.s. and α θ = α θ θ for any θ. herefore, if Y τ is bounded by a consan K hen he processes Y b and Y a θ are bounded by K for any θ. We now give a decomposiion resul for some Sochasic Differenial Equaions SDEs in G in erms of SDEs in F. Lemma 1.8. If he process X saisfies he following sochasic differenial equaion dx = µ, X, η d σ, X, η db ϕ, X, η dh, where µ, σ are OG BR BR-measurable maps, ϕ is a PG BR BR measurable map and η is a G-predicable process, hen X a and X b saisfy dx a τ = µ a, τ, X a τ, η a τd σ a, τ, X a τ, η a τdb, τ, dx b = µ b, X b, η b d σ b, X b, η b db,, X a X b = ϕ, X b, η b, τ. Proof. he proof of his lemma is an applicaion of Proposiion 1.6 and for he las equaliy, we have used ha if an F-predicable process K saisfies K τ =, hen K = on { τ} see [1, Lemma 3, Chaper Noaions o define soluions o ABSDEs, we inroduce he following spaces, where s, R wih s, and < is he erminal ime and δ is a sricly posiive consan. S 2 G [s, resp. S2 F [s, is he se of R-valued OG resp. OF-measurable processes Y u u [s, such ha Y 2 S 2 [s, := E[ Y u 2 <. u [s,

6 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 93 L 2 G [s, resp. L2 F [s, is he se of R-valued PG resp. PF-measurable processes Z u u [s, such ha [ Z 2 L 2 [s, := E s Z u 2 du <. L 2 F is he se of R-valued square inegrable F -measurable random variables. L 2 τ is he se of R-valued PF-measurable processes U such ha U = for > τ and U 2 L 2 τ [ := E U s 2 ds <. D[, δ is he se of càd-làg R-valued maps defined on [, δ. For Y D[, δ, we denoe Y := 1 δ δ Ys ds Exisence resuls for ABSDE in a Brownian filraion We exend he resuls of Peng and Yang [12 o more general drivers since we assume in our case ha he driver depends on he rajecory of he processes on he inerval [, δ. he proofs are based on sandard mehodologies, however hey require some care o check he needed Lipschiz condiions. o simplify he wriing we inroduce some new noaions for each Proposiion, and he same noaion y A is used in differen meanings which are clear from he conex. Proposiion 1.9. Le A := R 2 D[, δ R 2 D[, δ and, for any y = y, ŷ, Y, z, ẑ, Z A we define y by y = y ŷ Y z ẑ Z, where Y is defined in Secion 1.2. Le f be a map from Ω [, A valued in R. Le p and q be given bounded F-adaped processes. he following ABSDE dy = f, Y, E F [p δ Y δ, {E F [p s Y s } s δ, Z, E F [q δ Z δ, {E F [q s Z s } s δ d Z db,, Y = ξ, δ, Z = P, < δ, has a unique soluion in SF 2[, δ L2 F [, δ if a he map f, y is opional for any y A, b here exiss C > such ha, for any [,, any y A, we have f, y f, y C y y, c E[ fs, 2 ds <, d he erminal condiion ξ belongs o SF 2[, δ and P belongs o L2 F [, δ. Proof. In he driver, he map Y = Y s = E F [p s Y s, s δ is a family of F -measurable random variables. Le us firs inroduce a norm in he Banach space E := SF 2[, δ L2 F [, δ for β > : for Y, Z E [ δ Y, Z 2 β := E e β Y 2 Z 2 d, 1.7

7 94 M. JEANBLANC,. LIM AND N. AGRAM and define he mapping Φ : E E by Φy, z = Y, Z where Y, Z is defined by dy = f, y d Z db,, Y = ξ, δ, Z = P, < δ, where y = y, E F [p δ y δ, {E F [p s y s } s δ, z, E F [q δ z δ, {E F [q s z s } s δ. We now prove ha Φ is a conracion in E under he norm. β. For wo arbirary elemens y, z and y, z, we denoe heir difference by We can prove by using classical esimaes ha we have ỹ, z = y y, z z. [ E e β β 2 Ỹ 2 Z 2 d 2 β E [ e β f, y f, y 2 d. In he following inequaliies, K is a consan which does no depend on β and may change from line o line. By Lipschiz propery of he map f, he fac ha he square of a sum resp. inegral is bounded by he sum resp. inegral of he square up o a consan and he boundedness of p and q, i follows ha [ E e β β 2 Ỹ 2 Z 2 d K [ β E e β ỹ 2 z 2 ỹδ 2 z δ 2 1 δ By he change of variable u = s, we ge [ E e β β 2 Ỹ 2 Z 2 d Fubini s heorem leads o 1 δ K β E [ δ e β ỹ 2 z 2 1 d δ δ e β ỹu 2 z udu 2 d 1 δ u δ 1 e βδ where we have used ha 1 e βδ βδ. Combining 1.1 wih 1.9, we obain for β 2 βδ δ e β d u δ δ δ ỹs 2 z sds 2 d. 1.8 e β δ ỹu 2 z udu 2 e βu ỹ 2 u z 2 udu ỹu 2 z udu 2 d. 1.9 e βu ỹ 2 u z 2 udu 1.1 [ E e β Ỹ 2 Z 2 d K β E [ δ e β ỹ 2 z 2 d. Consequenly, since Ỹ = Z = for >, we ge Ỹ, Z 2 β K β ỹ, z 2 β,

8 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 95 and Φ is a conracion on SF 2[, δ L2 F [, δ for β large enough o ensure ha K/β < 1, and β > 2. As Φ is a conracion, using general resuls on BSDE as in [5 here exiss a unique soluion Y, Z in SF 2[, δ L2 F [, δ o ABSDE 1.7. We now give an esimaion of he soluion of he ABSDE. Proposiion 1.1. Suppose f saisfies he hypoheses of Proposiion 1.9. hen here exiss a sricly posiive consan K ha only depends on he Lipschiz consan C and on such ha for any ξ SF 2 [, δ and P L 2 F [, δ, he soluion Y, Z of he ABSDE 1.7 saisfies E F [ s Ys 2 Zs 2 ds K E F [ξ 2 δ ξs 2 Ps 2 ds fs, 2 ds, for any [,. Proof. he proof is obained wih sandard compuaions. For he sake of compleeness, we give deails in he Appendix. Using he same mehodology as in Proposiion 1.9, one obains he following resul, where D, [, δ is he family of maps Y from [, δ o R such ha Ys is F s -measurable, for any s [, δ. Proposiion For any [,, le A = R L 2 F δ D, [, δ R L 2 F δ D, [, δ and for any y = y, ζ, Y, z, η, Z A, we inroduce y = y z E F ζ η Y Z. For a map f such ha fω, : A R, he following ABSDE dy = f, Y, Y δ, Y s s δ, Z, Z δ, Z s s δ d Z db,, Y = ξ, δ, Z = P, < δ, has a unique soluion in SF 2[, δ L2 F [, δ if he map f saisfies: a for y A, f, y is F -measurable, b here exiss C such ha for any [,, any y, y in A, one has f, y f, y C y y, c E f, 2 d <, d he erminal condiion ξ belongs o SF 2[, δ and P belongs o L2 F [, δ. Moreover, here exiss a consan K such ha we have E F [ s Ys 2 Zs 2 ds K E F [ξ 2 δ ξs 2 Ps 2 ds fs, 2 ds, for any [,. Proof. We use similar argumens o he proofs of Proposiion 1.9 and 1.1.

9 96 M. JEANBLANC,. LIM AND N. AGRAM 2. ABSDE wih jump of ype.1 We assume ha Hypoheses 1.1, 1.2 and 1.3 hold and ha f., y is opional. We consider in his secion an ABSDE of he following form: find a riple Y, Z, U S 2 G [, δ L2 G [, δ L2 τ saisfying dy = f, Y, E G [Y δ, {E G [Y s } s δ, Z, E G [Z δ, {E G [Z s } s δ, U, E G [U δ, {E G [U s } s δ d Z db U dh,, Y = ξ, δ, Z = P, U = Q 1 { τ}, < δ. From Proposiions 1.5 and 1.6, all he involved processes can be decomposed in wo pars, before and afer τ. In paricular, since ξ will be given as a G-opional process and P as a G-predicable process, we have for any [, and we have for any [, δ f, y = f b, y1 {<τ} f a, τ, y1 { τ} opional decomposiion, { ξ = ξ b 1 {<τ} ξ a τ1 { τ} opional decomposiion We work under he following hypoheses: P = P b 1 { τ} P a τ1 {>τ} predicable decomposiion. Hypoheses 2.1. Le A := R 2 D[, δ R 2 D[, δ R 2 D[, δ and, for any y A, we define y by y = y ŷ Y z ẑ Z u û U. a he erminal condiions saisfy ξ SG 2[, δ, P L2 G [, δ, Q L2 F [, δ, here exiss a consan K such ha E[ ξuθ a 2 K and E[ Pu a θ 2 K for any θ, u [, [, δ. b he driver f : Ω [, A R of he ABSDE is Lipschiz, i.e., here exiss a consan C such ha, for any [,, any y and y in A, we have c For any y A, he process f, y is G-opional. d here exiss a consan C such ha fs, < C. From Proposiions 1.5 and 1.6, we can wrie f, y f, y C y y. { Y = Y b 1 {<τ} Y a τ1 { τ} Z = Z b 1 { τ} Z a τ1 {>τ} opional decomposiion predicable decomposiion I follows, from Lemma 1.8, ha dy a τ = f a, τ, Y a τ, E G [Y a δτ, {E G [Y a sτ} s δ, Z a τ, E G [Z a δτ, {E G [Z a sτ} s δ,,, d Z a τdb, τ, Y a τ = ξ a τ, δ, Z a τ = P a τ, < δ, 2.1

10 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 97 and dy b = f b, Y b, E F [Y δ, {E F [Y s } s δ, Z b, E F [Z δ, {E F [Z s } s δ, Y a Y b, E F [U δ, {E F [U s } s δ d Z b db,, 2.2 Y b = ξ b, δ, Z b = P b, U b = Q, < δ. On he righ-hand side of his equaion, we sill have o make precise if Y δ is par of he soluion before τ or afer τ, ha is o separae he case δ < τ and he case δ τ. Furhermore, U = [ Y a Y b 1 { } Q 1 { < δ} 1{ τ} Sudy of he Equaion 2.1 Our aim is o wrie 2.1 as a family of ABSDEs in he filraion F. For ha purpose, we noe ha, on he se { τ}, we have from 1.6 E G [Y a δτ = E F [Y a δθ θ=τ. he same equaliy holds for he par involving f, y and Zδ a τ. herefore, we sudy he family of ABSDE dy a θ = f a, θ, Y a θ, E F [Y a δθ, {E F [Y a sθ} s δ, Z a θ, E F [Z a δθ, {E F [Z a sθ} s δ,,, d Z a θdb,, Y a θ = ξ a θ, δ, Z a θ = P a θ, < δ. For any fixed θ [,, he map F := f a θ defined as F, y = f a, θ, y inheris he Lipschiz condiions of Proposiion 1.9 from he one of f. Due o he boundedness of f,, he map F, is also bounded, and saisfies [ E f a, θ, 2 d θ and he exisence of a soluion follows from Proposiion 1.9. Using Proposiion 1.1, here exiss a consan K such ha E F s Y a s θ Sudy of he Equaion 2.2 Zs a θ 2 ds <, K E F ξ a θ 2 δ ξ a s θ 2 P a s θ 2 ds 2.3 f a s, θ, 2 ds. 2.4 Our aim is o wrie 2.2 as an ABSDE in he filraion F, ha is o ge rid of he quaniies involving processes afer ime τ as, e.g., Y δ on { δ > τ} and working only wih condiional expecaion w.r.. F. Obviously, for any u δ, we have Furhermore, from 1.5, we have E G [Y u = E G [Y u 1 {u<τ} E G [Y u 1 {u τ}. E G [Y u 1 {u<τ} 1 {<τ} = E G [Y b u 1 {u<τ} 1 {<τ} = 1 G E F [Y b u G u 1 {<τ}, 2.5

11 98 M. JEANBLANC,. LIM AND N. AGRAM and E G [Y u 1 {u τ} 1 {<τ} = E G [Yu a τ1 {u τ} 1 {<τ} = 1 [ u E F Yu a θα u θdθ 1 {<τ} =: J Y a u. 2.6 G he same equaliies hold for he par involving Z a. hen, we inroduce, relying on he uniqueness of pre-defaul pars, he following BSDE which is a ransformaion of 2.2 dy b = g, Y b, Y b δ, {Y b s} s δ, Z b, Z b δ, {Z b s} s δ d Z b db,, Y b = ξ b, δ, Z b = P b, < δ. Here, due o he equaliies 2.5 and 2.6, g is he map Ω [, R L 2 F.δ D, [, δ R L 2 F.δ D, [, δ R defined, for y and z in R, ζ and η in L 2 F.δ, and Y and Z in D, [, δ, in erms of soluion of he equaion 2.3 by, for y = y, ζ, Y, z, η, Z g, y = f b, I 1, I 2, I 3 where, recalling ha he quaniies J are defined in 2.6 I 1 = y, I 2 = z, 1 E F [ζg δ J Y a δ, { 1 E F Y sg s J Y a s} s δ, G G 1 E F [ηg δ J Z a δ, { 1 E F Z sg s J Za s} s δ, G G 1 E F [1 {δ } Y G δ a δ ζg δ 1 {δ> } Q δ G δ, 1 { } E F [Y a G s s Y sg s 1 {s } Q s G s 1 {s> }. s δ I 3 = Y a y, I is sraighforward ha g is F-opional. We now show ha g saisfies Lipschiz condiions recalled in Proposiion 1.9. Since we have f b, y = 1 G E F [f, y1 {<τ}, 2.7 we obain ha, using he Lipschiz condiion for f and ha G is bounded, here exiss a consan K such ha g, y g, y K y y z z E F 1 {<τ} G E F [ ζ ζ η η G δ 1 {<τ} E F[ YG Y G ZG Z G 1 {<τ}. Since, for X F s and s >, one has from 1.4 E F [X1 {<τ} = E F [XE Fs 1 {<τ} = E F [XG = G E F [X,

12 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 99 we deduce g, y g, y K y y z z E F [ ζ ζ η η G δ E F [ YG Y G ZG Z G. Noing G is upper bounded by 1, he Lipschiz propery of Proposiion 1.9 for g holds. We now check he inegrabiliy condiion on g, 2. We noice, using noaion 2.6, we have g, = f b } },, J Y a δ, {J Y a s,, J Za δ, {J Za s, Y a, s δ s δ 1 E F [Y G δ a δg δ 1 {δ } Q δ G δ 1 {δ> }, { 1 } E F [Y a G s sg s 1 {s } Q s G s 1 {s> }. s δ From Lipschiz propery of f, since f, is bounded and G = E F 1 {<τ}, we have f b, y 1 G E F [f, C y 1 {<τ} C 1 C y. Using again ha he square of a sum is bounded up o a consan by he sum of he squares, and using again he fac ha G is lower bounded, he inegrabiliy condiion of g, 2 will follow from he boundedness of he quaniies E J Y δ δ 2 d, E J Y s 2 ds d 2.8 and similar expressions wih J Z, as well as E E F Yδ a δ 2 d E Y a 2 d δ E E F Q 2 d [ δ E Y a s s 2 1 s Q 2 s1 s> ds d. he quaniies in 2.8 are bounded since α is bounded and is bounded since δ E Ys a θ 2 K θ s dθ 1 {<θ<δ} EYδθ a 2 d θ E ξ a θ 2 f a s, θ, 2 ds δ ξ a s θ 2 P a s θ 2 ds and he assumed boundedness of P and ξ. he oher quaniies are sudied using he same mehodology and ha Q L 2 [, δ.

13 1 M. JEANBLANC,. LIM AND N. AGRAM he exisence of a unique soluion Y b, Z b of he ABSDE 2.7 follows from Proposiion Moreover we have E F s Y b s 2 Zs b 2 ds KE F ξ b 2 δ ξ b s 2 P b s 2 du f b s, 2 ds Inegrabiliy of he soluions In his par we consider he inegrabiliy of he soluions Y, Z, U where Y = Y b 1 {<τ} Y a τ1 { τ}, Z = Z b 1 { τ} Z a τ1 {>τ}, U = Y a Y b 1 { τ}. From Subsecions 2.1 and 2.2 we know Y, Z, U saisfy he ABSDE.1. Proposiion 2.2. he process U belongs o L 2 τ. Proof. We have [ δ τ E Us 2 ds [ τ = E 2 [ 2E [ δ τ Ys a s Ys b 2 ds Ys a s 2 ds 2E [ τ [ E Ys a s 2 ds 2 E [ Ys b 2 ds Q 2 sds [ δ Y b 2 E [ δ Q 2 sds Q 2 sds and he quaniies on he righ-hand side are finie. Proposiion 2.3. here exiss a sricly posiive consan K such ha he soluion Y, Z, U of he ABSDE.1 saisfies E G [ s KE F [ξ b 2 for any [,. Ys 2 δ K [ α τ EF ξ a θ 2 K1 { τ} E F [ Zs 2 ds ξ b s 2 P b s 2 ds δ { ξ a θ 2 f b s, 2 ds ξ a s θ 2 P a s θ 2 ds δ ξ a s θ 2 P a s θ 2 ds f a s, θ, 2 ds 1 {τ<} θ=τ } f a s, θ, 2 ds dθ

14 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 11 Proof. In he proof, he consan K can vary from line o line. We remark 1 E G [ = E G [ E G [ s s s Ys 2 Y 2 s Y 2 s τ Zs 2 ds Zs b 2 ds Z b s 2 ds τ On he se {τ < }, since λ is bounded Hypohesis 1.3, we use ha E G [ s Ys 2 On he se { τ}, we remark E G [ s = E G [ s ke k E F [ Ys 2 E F [ τ Y a s τ 2 = 1 G E F [ s s Zs a τ 2 ds Zs a τ 2 ds. s Y a s θ 2 KE F [ Y b s 2 E G [ τ s [ [ From E G τ s Ys a τ 2 = 1 α τ EF θ s Ys a θ 2 α θ E G [ τ s Ys a τ 2 K [ α τ EF θ s Ys a θ 2 α θ s θ=τ Y a s θ 2. Y a s τ 2. and ha α is bounded, we have Ys a θ 2. θ=τ We proceed in he same way for he par τ Za s τ 2 ds. Using we can conclude Uniqueness of he soluion In his par we are concerned wih he uniqueness of he soluion of ABSDE.1. Suppose his AB- SDE has wo soluions Y, Z, U and Ȳ, Z, Ū. Each process admis a unique decomposiion under he form Y b, Z b, U b -Y a τ, Z a τ and Ȳ b, Z b, Ū b -Ȳ a τ, Z a τ. Moreover we know Y b, Z b and Ȳ b, Z b are soluion of ABSDE 2.2, hus by uniqueness of he soluion of ABSDE 2.2 from Proposiion 1.11 we ge ha Y b = Ȳ b and Z b = Z b. We have wih he same argumens Y a τ = Ȳ a τ and Z a τ = Z a τ. Moreover we have U = Y a Y b 1 { τ}, hus U = Ū. Finally we ge he uniqueness of he soluion of ABSDE ABSDE wih jump of ype.2 We assume ha Hypoheses 1.1, 1.2 and 1.3 hold. We define, for any [,, A = R L 2 F δ D, [, δ R L 2 F δ D, [, δ R L 2 F δ D, [, δ. We consider in his secion an ABSDE of he following form: find a riple Y, Z, U S 2 G [, δ L2 G [, δ L2 τ saisfying dy = E G[ f, Y, Y δ, {Y s } s δ, Z, Z δ, {Z s } s δ, U, U δ, {U s } s δ d Z db U dh,, Y s = ξ s, s δ Z s = P s, U s = Q s 1 { τ}, s δ, 1 wih he convenion b a.ds = if b < a 3.1

15 12 M. JEANBLANC,. LIM AND N. AGRAM wih he following hypoheses Hypoheses 3.1. Suppose ha a he erminal condiions saisfy ξ SG 2[, δ, P L2 G [, δ and Q L2 F [, δ, and θ ξ a θ SF 2[, δ and θ P a θ L 2 F [, δ. b he driver f : Ω [, A. R is Lipschiz, ha means here exiss a consan C such ha for any [,, for any y and y in A, one has f, y f, y C y y. c here exiss a consan C such ha fs, C. Proceeding as before, we consider, on he se {τ }, he ABSDE dy a τ = E G[ f a, τ, Y a τ, Y a δτ, {Y a sτ} s δ, Z a τ, Z a δτ, {Z a sτ} s δ,,, d Z a τdb, τ, Y a sτ = ξ a sτ, s δ, Z a sτ = P a sτ, < s δ, whereas, due o he uniqueness of pre-defaul pars we consider he ABSDE dy b = E G[ f b, Y b, Y δ, {Y s } s δ, Z b, Z δ, {Z s } s δ, U b, U δ, {U s } s δ d Z b db,, Y b s = ξ b s, s δ, Z b s = P s, U b s = Q s, < s δ Sudy of he Equaion 3.2 Using he same argumens as in Subsecion 2.1 we sudy he family of ABSDEs dy a θ = E F[ f a, θ, Y a θ, Y a δθ, {Y a sθ} s δ, Z a θ, Z a δθ, {Z a sθ} s δ,,, d Z a θdb, θ, Y a θ = ξ a θ, δ, Z a θ = P a θ, < δ. his ABSDE can be wrien under he following form dy a θ = g, θ, Y a θ, Y a δθ, {Y a sθ} s δ, Z a θ, Z a δθ, {Z a sθ} s δ d Z a θdb, θ, Y a θ = ξ a θ, δ, Z a θ = P a θ, < δ, 3.4 which is on he form of Proposiion 1.11, where g is given by g, θ, y, ŷ, Y, z, ẑ, Z = E F[ f a, θ, y, ŷ, Y, z, ẑ, Z,,,.

16 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 13 he [ Lipschiz condiion on g follows from he hypohesis on f. he square inegrabiliy of g, = E F f a, θ, follows as in Subsecion 2.1 from he boundedness hypohesis of f,. hus from Proposiion 1.11 we ge he exisence of a unique soluion o his ABSDE saisfying E F s Y a s θ 2 Zs a θ 2 ds CE F ξ a θ 2 δ ξ a s θ 2 P a s θ 2 ds gs, θ, 2 ds Sudy of he Equaion 3.3 Using he same argumens as in Subsecion 2.2, we are lead o consider dy b = g, Y b, Yδ, b {Ys} b s δ, Z b, Zδ, b {Zs} b s δ d Z b db, Y b s = ξ b s, s δ, Z b s = P b s, < s δ, 3.6 where g, y, ζ, Y, z, η, Z = 1 G δ E F[ f b, K 1, K 2, K 3 αδ θ dθ 1 G E F[ f b, y, ζ, Y, z, η, Z, K 4. wih K 1 = y, Yδθ, a {Y s 1 s<θ Ysθ1 a s θ } s δ K 2 = z, Z a δθ, {Z s 1 s θ Z a sθ1 s>θ } s δ K 3 = Y a y,, { Y a s s Y s 1 s< Q s 1 s 1 s θ } K 4 = Y a y, Y a δ δ ζ1 {δ } Q δ 1 {δ> } G δ, { Y a s s Y s 1 {s } Q s 1 {s> } G s } We show ha he hypoheses of Proposiion 1.11 are saisfied. Firs, we show ha he driver is Lipschiz. Using ha f is Lipschiz we ge s δ f b, y f b, y 1 E F [ f, y f, y 1 {<τ} C y y. G. s δ I follows ha, seing Y = y, Yδθ, a {Y s 1 s<θ Ysθ1 a s θ } s δ, z, Zδθ, a {Z s 1 s θ Zsθ1 a s>θ } s δ, { } Y a y,, Ys a s Y s 1 s< Q s 1 s 1 s θ, s δ

17 14 M. JEANBLANC,. LIM AND N. AGRAM here exiss a consan C such ha E F [f b, Yα δ θ E F [f b, Y α δ θ C y y z z E F α δ θ Yαθ Y αθ Zαθ Z αθ C y y z z Y Y Z Z α θ, where we use ha αθ is a maringale. Hence, using ha α θ dθ = 1, we ge δ E F [ f b, Y f b, Y α δ θ dθ C y y z z Y Y Z Z. In he oher hand, using he Lipschiz propery of f b, ha G is upper bounded and denoing ϕ, y := E F [f b, y { for y = y, ζ, Y, z, η, Z, Y a y, Yδ a δ ζ1 {δ }Q δ 1 {δ> } G δ, Yss Y a s 1 {s } } Q s 1 {s> } G s here exiss a consan K such ha one has s δ ϕ, y ϕ, y K y y. I follows using one more ime ha G is lower bounded ha here exiss a consan K such ha and he Lipschiz propery holds. here exiss an unique soluion of Inegrabiliy he inegrabiliy condiion of g, = 1 G δ g, y g, y K y y E F [f b,, Y a δθ, {Y a sθ1 s θ } s δ,, Z a δθ, {Z a sθ1 s θ } s δ, Y a,, {Y a s s1 {s } Q s 1 {s> } 1 s θ } s δ α δ θdθ 1 G E F [f b,,,,, Y a, Y a δ δ G δ 1 {δ< } Q δ G δ 1 {δ } follows wih he same argumens as in Secion 2.2. We also consider he inegrabiliy of he soluions Y, Z, U for ABSDE 3.1, where Y = Y b 1 {<τ} Y a τ1 { τ}, Z = Z b 1 { τ} Z a τ1 {>τ}, U = Y a Y b 1 { τ}. One can apply he same mehodology han he one in he previous secion, since Proposiion 1.1 is valid in he case of ABSDE 3.1 and we obain similar resuls.

18 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME Paricular cases 4.1. Moving average erm We consider he case dy = E G[ δ f, Y, a s Y s ηds, Z, U d Z db U dh,, Y = ξ, δ, Z = P, U = Q 1 { τ} < δ, where a is a bounded F-adaped process and η is a measure of he form ηds = lsds i k i ε si ds where l is a bounded Borel funcion, ε a he Dirac measure a a, and f, y, ŷ, z, u is Lipschiz. Since δ a s Y s ηds δ K Y s ds i 1 s i [,δy si, he driver saisfies he Lipschiz condiion, and he above ABSDE has a soluion Linear ABSDE In his par we give a closed formula for he soluion of linear ABSDEs. ha means he driver f is linear w.r.. Y, Z and U. We firs give a resul abou he form of Y, par of he soluion of a linear ABSDEs in he Brownian case. Proposiion 4.1. Consider he following ABSDE [ dy = < a, Y > l d Z db,, Y = ξ, δ, Z = P, < δ, 4.1 where <, > is he scalar produc, a and Y are defined by a = µ, µ, µ, σ, σ, σ, δ Y = Y, E F [p δ Y δ, E F p u Y u du, δ Z, E F [q δ Z δ, E F q u Z u du, and where he processes µ, µ, µ, σ, σ, σ L 2 F [ δ, δ are assumed o be uniformly bounded and l L2 F [,. hen, for any [,, he soluion Y is given by Y = E [X F ξ δ δ l s X sds δ {µ s u p s ξ s σ s u q s P s } X s udu ds { µs δ p s ξ s σ s δ q s P s } X s δ ds,

19 16 M. JEANBLANC,. LIM AND N. AGRAM where [ δ dxs = µ s Xs p s µ s δ Xs δ p s µ s u Xs udu ds [ δ σ s Xs q s σ s δ Xs δ q s σ s u Xs udu db s, s δ, X = 1, Xs =, δ s. Proof. he proof of his resul is similar o he proof of heorem 2.1 in [12. We give some deails in Appendix. We now exend he previous resul o he case of an ABSDE wih jump. Proposiion 4.2. Consider he following ABSDE dy = [ < a, Y > l d Z db U dh,, Y = ξ, δ, Z = P, U = Q 1 { τ}, < δ, where <, > is he scalar produc, and a and Y are defined by a = µ, µ, µ, σ, σ, σ, ρ, ρ, ρ, Y = Y, E G [Y δ, E G [ δ Y udu, Z, E G [Z δ, E G [ δ Z udu, U, E G [U δ, E G [ δ U udu. We assume he funcions µ, µ, µ, σ, σ, σ, ρ, ρ, ρ L 2 G [ δ, δ are assumed o be uniformly bounded and l L 2 G [,. hen, for any [,, he soluion Y is given by Y = Y b 1 <τ Y a τ1 τ where Y b and Y a θ, for any θ [,, are defined by [ Y a θ = E F X,a θξa θ X,a s θl a s θds P a s θσ a s δθx,a s δ θds δ [ Y b = E F X,b ξb Xs,b L s ds δ δ δ δ δ Gs ξ b sµ b s u G s P b s σ b s u X,b s udu ds ξ a s θµ a s δθ ξ s a θµ as u θ P s a θσ a s u θ Xs uθdu,a ds Gs ξ b sµ b s δ G s P b s σ b s δ X,b s δ ds, wih L = l b µ b J Y a δ µ b δ J Y a udu σ b J Za δ σ b ρ b Y a ρb G E F [ G δ Y a δ1 {δ } G δ Q δ 1 {δ> } ρb G E F [ δ δ Y a u u1 {u } Q u 1 {u> } Gu du J Za udu,

20 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 17 where X,a θ is he soluion of he following linear sochasic delayed differenial equaion SDDE dx,a s θ = [ µ a sxs,a θ µ s δ θx,a X,a θ = 1, [ s δ θ δ σs δθx a s,a θ σ a s δx,a X,a s θ =, s [ δ,, and X,b is he soluion of he following linear SDDE dx,b s = [ µ b s ρ b s X,b s X,b = 1, δ [ s δ θ δ µ a s u θx,a s uθdu µ b s δ G s δ ρb s δ µ b s u G s u ρb s u G s u 1 {s u } σ b sx,b s X,b s =, s [ δ,. δ σb s δ X,b s δ G s δ ds σ a s uθx,a s uθdu 1 {s } G s δ σ b s u X,b s udu G s u X,b s udu X,b s δ ds db s, s [, δ, db s, s [, δ, Proof. he resul is an applicaion of he resuls of Secion 2 and Proposiion Proof of Proposiion Appendix Applying Iô s formula o e βs Y 2 s, we obain for any s [, e β ξ 2 e β Y 2 = e βs βys 2 Zs 2 ds 2 e βs Y s fs, Y s ds 2 e βs Y s Z s db s 5.1 where Y s = Y s, E Fs [p sδ Y sδ, {E Fs [p su Y su } u δ, Z s, E Fs [q sδ Z sδ, {E Fs [q su Z su } u δ. By he Lipschiz assumpion on f, using similar esimaions as in he previous proofs and he boundedness of p and q, we obain here exiss K such ha f, Y f, Y f, f, 5.2 K Y Z E F[ Y δ Z δ [ δ E F Y u Z u du f,. Combining 5.1 and 5.2, and using several imes ha 2 ab a2 c for example cb2 for differen posiive consans c, as 2 Y s E Fs Y sδ Y 2 s c 1 c 1 E Fs Y sδ 2,

21 18 M. JEANBLANC,. LIM AND N. AGRAM we ge afer some compuaions e β Y 2 [β 2 c 1 c 2 c 3 c 4 c 5 K e βs Ys 2 ds [1 K e βs Zs 2 ds c 3 e β ξ 2 2 e βs Y s Z s db s 2 e βs Y s fs, ds K e βs E Fs Y c sδds 2 1 K δ e βs e βs E Fs Z c sδds 2 K δ e βs 2 c 5 E Fs Y 2 sudu ds K c 4 E Fs Z 2 sudu ds. aking condiional expecaion w.r.. F, noing ha from Fubini e βs δ Y 2 sudu ds δ δ e βs Y 2 s ds and ha e βs Y 2 sδds δ e βs Y 2 s ds we obain, afer some simplificaions, and using again 2 ab a2 c cb2 wih a new consan c 6! e β Y 2 [β 2 c 1 c 2 c 3 c 4 c 5 K K δk K c 6 E F c 1 c 2 [1 K K δk E F e βs Zs 2 ds c 3 c 4 c 5 e β E F ξ 2 1 E F e βs fs, 2 ds [ δk K E F c 6 c 1 [ δk c 5 K c 4 E F δ P 2 s ds. c 2 δ ξ 2 sds e βs Y 2 s ds I remains o choose he consans c i such ha β = 2 c 1 c 2 c 3 c 4 c 5 K K c 1 K c 3 K c 4 δk c 5 1 o obain ha here exiss a consan K such ha δk c 2 K c 6 and E F [ Zs 2 ds K E F [ξ 2 δ ξ 2 s P 2 s ds fs, 2 ds, for any [,. he proof for he erm wih he remum on Y is obained using he same mehodology and Burkholder- Davis-Gundy inequaliy.

22 5.2. Proof of Proposiion 4.1 SOME EXISENCE RESULS FOR ABSDE WIH A JUMP IME 19 Applying Iô s formula o X Y and aking he condiional expecaion we ge X Y = E [X F Y l s Xsds σ s X sq sδ Z sδ ds µ s X s σ s X s δ δ p su Y su du ds q su Z su du ds µ s Xsp sδ Y sδ ds µ s δ Xs δp s Y s ds σ s δ X s δq s Z s ds his can be rewrien, by using X u = for u, under he form X Y = E [X F Y δ δ δ Since X = 1, we conclude l s X sds σ s δ X s δq s Z s ds δ µ s u X s up s Y s du ds σ s u Xs uq s Z s du ds. Y = E [X F ξ δ δ 5.3. Wihou Immersion propery l s X sds δ p s Y s µ s u Xs udu ds q s Z s δ µ s δ Xs δp s Y s ds σ s δ X s δq s Z s ds δ δ δ σ s u Xs udu ds. δ δ σ s u X s uq s Z s du ds {µ s u p s ξ s σ s u q s P s } X s udu ds µ s δ X s δp s Y s ds { µs δ p s ξ s σ s δ q s P s } X s δ ds. µ s u X s up s Y s du ds We presen how o solve he sudy of ABSDE in a general seing, wihou immersion hypohesis and explain ha i will only require more aenion and more complex compuaions. A firs mehod is o noe ha, under densiy Hypohesis 1.2, i is known ha B is a G-semimaringale of he form B = B G τ d B, G F s d B, αu F s u=τ G s τ α s τ =: B G ν s ds, where G = Pτ > F is now a ermaringale and B G is a G-Brownian moion. herefore, he BSDE.1 has he form dy = f, Y, E G [Y δ, E G [Y s s δ, Z, E G [Z δ, E G [Z s s δ, U, E G [U δ, E G [U s s δ Z ν d Z db G U dh,

23 11 M. JEANBLANC,. LIM AND N. AGRAM and where Y is given, and he same analysis can be done, under a square inegrabiliy assumpion on ν, which can be jusified in a financial meaning o avoid arbirages in he enlarged filraion. A second mehod 2 is based on he change of probabiliy on he iniial enlarged filraion suggesed in he Inroducion, i.e., has o be used on G and is where l is given by dp F στ = l dp F στ, dp G = l dp G, l = Eα τ 1 G = 1 <τ G G 1 τ 1 α τ, wih G = Pτ >. Under P, he random ime τ is independen of F and B is a P -Brownian moion. Under ha probabiliy, he ABSDE.1 has he form dy = f, Y, l E G [l δ 1 Y δ, l E G [l s 1 Y s s δ, Z, l E G [l δ 1 Z δ, l E G [l s 1 Z s s δ, U, l E G [l δ 1 U δ, l E G [l s 1 U s s δ d Z db U dh, and where Y is given. he same mehodology works, providing condiions on l such ha he required Lipschiz condiions are saisfied. Aknowledgemen: We hank he anonymous referee for his her commens. References [1 A. Aksami, and M. Jeanblanc, Enlargemen of filraion wih finance in view, Springer, forhcoming. [2 P. Brémaud, and M. Yor, Changes of filraion and of probabiliy measures, Z. Wahr. Verw. Gebiee, 45, , [3 R. Dumirescu, M-C. Quenez, and A. Sulem, BSDEs wih defaul jump, Preprin, 216. [4 N. El Karoui, M. Jeanblanc, and Y. Jiao, Wha happens afer a defaul: he condiional densiy approach, Sochasic Processes and heir Appl., 127, , 21. [5 N. El Karoui, S. Peng, and M-C. Quenez, Backward sochasic differenial equaions in finance, Mahemaical Finance, 7 1, 1-71, [6. Jeulin and M. Yor, Grossissemen de filraion: exemples e applicaions, Lecure Noes in Mahs, 1118, Springer, [7. Jeulin, Semimaringales e grossissemens d une filraion, Lecure Noes in Mahs, 833, Springer, 198. [8 I. Kharroubi and. Lim, Progressive enlargemen of filraions and backward sochasic differenial equaions wih jumps, Journal of heoreical Probabiliy, 27 3, , 214. [9 I. Kharroubi,. Lim and A. Ngoupeyou, Mean-variance hedging on uncerain ime horizon in a Marke wih a Jump, Applied Mahemaics and Opimizaion, 68 3, , 213. [1 A. Ngoupeyou, Opimisaion des porefeuilles d acifs soumis au risque de défau, PHD hesis Evry universiy 21. [11 B. Øksendal, A. Sulem, and. Zhang, Opimal conrol of sochasic delay equaions and ime-advanced backward sochasic differenial equaions, Adv. Appl. Prob., 43, , 211. [12 S. Peng and Z. Yang, Anicipaed backward sochasic differenial equaions, he Annals of Probabiliy, 37 3, , 29. [13 H. Pham, Sochasic conrol under progressive enlargemen of filraions and applicaions o muliple defauls risk managemen, Sochasic Processes and heir Applicaions, 129, , 21. [14 S. Song, Opional spliing formula in a progressively enlarged filraion, ESAIM Probabiliy and Saisics, 18, , suggesed by he anonymous referee