Numerical approximation of Backward Stochastic Differential Equations with Jumps

Transcription

1 Numerical approximaio of Bacward Sochasic Differeial Equaios wih Jumps Aoie Lejay, Ereso Mordeci, Soledad Torres To cie his versio: Aoie Lejay, Ereso Mordeci, Soledad Torres. Numerical approximaio of Bacward Sochasic Differeial Equaios wih Jumps. Research Repor 214, pp.32. <iria v3> HAL Id: iria hps://hal.iria.fr/iria v3 Submied o 9 Sep 213 (v3), las revised 18 Sep 214 (v4) HAL is a muli-discipliary ope access archive for he deposi ad dissemiaio of scieific research documes, wheher hey are published or o. The documes may come from eachig ad research isiuios i Frace or abroad, or from public or privae research ceers. L archive ouvere pluridiscipliaire HAL, es desiée au dépô e à la diffusio de documes scieifiques de iveau recherche, publiés ou o, émaa des éablissemes d eseigeme e de recherche fraçais ou éragers, des laboraoires publics ou privés.

2 Numerical approximaio of Bacward Sochasic Differeial Equaios wih Jumps Aoie Lejay Mordeci Soledad Torres Absrac. I his oe we propose a umerical mehod o approximae he soluio of a Bacward Sochasic Differeial Equaios wih Jumps (BS- DEJ). This mehod is based o he cosrucio of a discree BSDEJ drive by a complee sysem of hree orhogoal discree ime-space marigales, he firs a radom wal covergig o a Browia moio; he secod, aoher radom wal, idepede of he firs oe, covergig o a Poisso process. The soluio of his discree BSDEJ is show o wealy coverge o he soluio of he coiuous ime BSDEJ. A applicaio o parial iegro-differeial equaios is give. Keywords: Bacward SDEs wih jumps, Sorohod opology, Poisso Process, Moe Carlo mehod. 2 AMS MSC. Primary: 6H1, Secodary: 34F5, 93E3 Uiversi de Lorraie, IECN, UMR 752, Vadœuvre-lès-Nacy, F-545, Frace; CNRS, IECL, UMR 752, Vadœuvre-lès-Nacy, F-545, Frace; Iria, Villers-ls-Nacy, F-546, Frace Aoie.Lejay@uiv-lorraie.fr. Parially suppored by he MahAmSud Worshop Sochasic Aalysis Research Newor. Cero de Maemáica, Faculad de Ciecias, Uiversidad de la República. Iguá 4225, 114, Moevideo, Uruguay; mordeci@cma.edu.uy. This auhor acowledges suppor of Proyeco Fodo Clemee Esable 111. DEUV-CIMFAV, Uiversidad de Valparaíso, Casilla 53 Valparaíso, Chile; soledad.orres@uv.cl. Parially suppored by Laboraorio de Aálisis Esocásico, PBCT-ACT13 ad Fodecy Gra N

3 1 Iroducio Cosider a sochasic process {Y, Z, U : T } ha is he soluio of a Bacward Sochasic Differeial Equaio wih Jumps (i shor BSDEJ) of he form T T Y = ξ + f(s, Y s, Z s, U s )ds Z s db s U s (x)ñ(ds, dx), (,T R T. (1.1) Here {B : T } is a oe dimesioal sadard Browia moio; Ñ(d, dx) = N(d, dx) ν(d, dx) is a compesaed Poisso radom measure defied i, T (R \ {}). Boh processes are defied o a sochasic basis B = (Ω, F T, F = {F } T, P), ad, as usual i his framewor, we assume ha hey are idepede. The ermial codiio ξ is a F T -measurable radom variable i L q (P), q > 2 (see codiio (B) i Secio 2); he coefficie f is a o-aicipaive (w.r.. (F ) ), coiuous, bouded ad Lipschiz fucio f : Ω, T R 3 R (see codiio (A) i Secio 2). The problem of solvig a BSDEJ give he ermial codiio ξ, he coefficie f, ad he drivig processes B ad Ñ, defied o a sochasic basis B, is o fid hree adaped processes {Y, Z, U : T } such ha (1.1) holds. I his oe oce he exisece ad uiqueess of he soluio of a BSDEJ as described above is ow we propose a umerical mehod ha approximaes he soluio of he equaio (1.1) i he case whe Ñ is a compesaed Poisso process. Noliear bacward sochasic differeial equaios of he form Y = ξ + T f(s, Y s, Z s )ds T Z s dw s, T, were firs iroduced by E. Pardoux ad S. Peg 15 i 199. Uder some Lipschiz codiios o he geeraor f, he auhors saed he firs exisece ad uiqueess resul. Laer o, he same auhors i 1992 developed he BSDE heory relaxig he hypoheses ha esure he exisece ad uiqueess o his ype of equaios i 16. May subseque effors have bee made i order o relax furher he assumpios o he coefficie f(s, y, z), ad may applicaios i mahemaical fiace have bee proposed. The bacward sochasic differeial equaios wih jumps heory begis wih a exisece resul obaied by S. Tag ad X. Li 12. The auhors 2

4 saed such a heorem whe he geeraor saisfies some Lipschiz codiio. Aoher releva coribuio o BSDEJ is he paper by R. Siu 21. I 2, G. Barles, R. Bucdah ad E. Pardoux cosider a BSDEJ whe he drivig oises are a Browia moio ad a idepede Poisso radom measure. They show he exisece ad uiqueess of he soluio, ad i addiio, hey esablish a li wih a parial iegro-differeial equaio (i shor PIDE). A releva problem i he heory of BSDEs is o propose implemeable umerical mehods o approximae he soluio of such equaios. Several effors have bee made i his direcio as well. For example, i he Marovia case, J. Douglas, J. Ma ad P. Proer 9 proposed a umerical mehod for a class of forward-bacward SDEs, based o a four sep scheme developed by J. Ma, P. Proer ad J. Yog 14. O he oher had, D. Chevace 6 proposed a umerical mehod for BSDEs. I 23, J. Zhag proposed a umerical scheme for a class of bacward sochasic differeial equaios wih possible pah-depede ermial values. See also 4, 11, amog ohers, where umerical mehods for decoupled forward-bacward differeial equaios are proposed, ad 1, 3, 13 for bacward differeial equaios. I he prese wor we propose o approximae he soluio of a BSDEJ drive by a Browia Moio ad a idepede compesaed Poisso process, hrough he soluio of a discree bacward equaio, followig he approach proposed for BSDE by P. Briad, B. Delyo ad J. Mémi i 5. The algorihm o compue his approximaio is simple. I he case wihou jumps, umerical examples of implemeaios of his scheme may be foud i 17 for example. Noe ha whaever he mehod, he compuaio of a codiioal expecaio is umerically cosly. However, he rae of covergece is difficul o esablish. The difficuly comes from he represeaio heorem. No surprisigly, sudyig closely his erm requires sophisicaed ools such as Mallavia calculus, o which he wor of B. Bouchard ad R. Élie relies 3. Here, we prefer o use relaively simple ools a he price of o sudyig he rae of covergece. The res of he oe is orgaized as follows. I secio 2 we prese he problem, propose a approximaio process ad a simple algorihm o compue i, ad prese he mai covergece resul of he oe. Secio 3 is devoed o he proof of he previous mai resul, i he adequae opology. I secio 4 we prese a applicaio of he mai resul o a decoupled sysem of a sochasic differeial equaio ad a bacward sochasic differeial equaio, resulig i a umerical approximaio of he soluio of a associaed parial iegro-differeial equaio. 3

5 2 Mai resul I order o propose a implemeable umerical scheme we cosider ha he Poisso radom measure is simply geeraed by he jumps of a Poisso process For he sae of simpliciy of oaio we wor i he ime ierval, 1. We he cosider a Poisso process {N : 1} wih iesiy λ ad jump epochs {τ : =, 1,... }. The radom measure is he N 1 Ñ(d, dx) = δ (τ,1)(d, dx) λdδ 1 (dx), =1 where δ a deoes he Dirac dela a he poi a. We also deoe Ñ = N λ. As a cosequece, he uow fucio U (x) ha i priciple depeds o he jump magiude x becomes U = U (1), ad our BSDEJ i (1.1) becomes Y = ξ + = ξ + f(s, Y s, Z s, U s )ds f(s, Y s, Z s, U s )ds Z s db s Z s db s N 1 i=n +1 (,1 U Ti + λ U s ds, U s dñs, (2.2) for 1. Here, we resric ourselves wihou loss of geeraliies o a ime horizo T = 1. I order o esure exisece ad uiqueess of he soluio of his equaio we cosider he followig codiios o he coefficie: (A) The fucio f : Ω, 1 R 3 R is o-aicipaive wih respec o (F ) ad here exiss K ad a bouded, o-decreasig coiuous fucio Λ wih Λ() = such ha f(ω, s 1, y 1, z 1, u 1 ) f(ω, s 2, y 2, z 2, u 2 ) Λ(s 2 s 1 ) + K ( y 1 y 2 + z 1 z 2 + u 1 u 2 ), a.s. (2.3) I wha respecs he ermial codiio, we assume: (B) The radom variable ξ is F 1 -measurable ad E ξ q < for some q > 2. 4

6 2.1 Discree ime BSDE wih jumps We propose o approximae he soluio of he BSDEJ i (2.2) by he soluio of a discree bacward sochasic differeial equaio wih jumps i a discree sochasic basis wih a filraio geeraed by wo idepede, ceered radom wals. I order o obai he represeaio propery a hird marigale is cosidered. The covergece of his approximaio relies o he fac ha he firs radom wal coverges o he drivig Browia moio, ad he secod o he drivig compesaed Poisso process. Le us defie hese wo radom wals. For N we iroduce he firs radom wal {W : =,..., } by W =, W = 1 ɛ i ( = 1,..., ), (2.4) where ɛ 1,..., ɛ are idepede symmeric Beroulli radom variables: i=1 P ( ɛ = 1 ) = P ( ɛ = 1 ) = 1/2, ( = 1,..., ). The secod radom wal {Ñ : =,..., } is o symmeric, ad give by Ñ =, Ñ = ηi ( = 1,..., ), (2.5) i=1 where η1,..., η are idepede ad ideically disribued radom variables wih probabiliies, for each, give by P(η = κ 1) = 1 P(η = κ ) = κ ( = 1,..., ), (2.6) where κ = e λ/. We assume ha boh sequeces ɛ 1,..., ɛ ad η1,..., η are defied o he origial probabiliy space (Ω, F, P) (ha ca be elarged if ecessary), ad ha hey are muually idepede. The (discree) filraio i he probabiliy space is F = {F : =,..., } wih F = {Ω, } ad F = σ{ɛ 1,..., ɛ, η 1,..., η } for = 1,...,. I his discree sochasic basis, give a F+1 -measurable radom variable y +1, o represe he marigale differece m +1 := y +1 E ( ) y +1 F we eed a se of hree srogly orhogoal marigales. This is a moivaio o iroduce a hird marigale icremes sequece {µ = ɛ η : = 5

7 ,..., }. I his coex here exis uique F -measurable radom variables z, u, v such ha m +1 = y +1 E ( y +1 F ha ca be compued as ) = 1 z ɛ +1 + u η +1 + v µ +1, (2.7) z = E ( ) ( ) y +1 ɛ +1 F = E y+1 ɛ +1 F, (2.8) u = E( y +1 η+1 F ) E ( 1 ) = (η+1 )2 F κ (1 κ ) E( ) y +1 η+1 F, (2.9) v = E( y +1 µ +1 F ) E ( 1 ) = (µ +1 )2 F κ (1 κ ) E( ) y +1 µ +1 F. (Observe ha he marigales are orhogoal bu o orhoormal, hece he ormalizaio.) Wih his iformaio we proceed o formulae he discree BSDEJ. Le us iroduce a supplemeary codiio o f: (A ) There exiss a sequece of fucios f : Ω, T R 3 R oaicipaive w.r.. (F ) =,...,, saisfyig (2.3) ad such ha f (ω,,, ) coverges uiformly o f(ω,,, ) almos surely. Remar 1. The prooypal example of such a sequece f is whe here is a uderlyig sochasic process X drive by B ad N ad f(ω, s, y, z) = g(x s (ω), s, y, z). If g is uiformly coiuous wih respec o is firs variable ad for =,...,, X is a approximaio of X / cosruced from {ɛ i } i=,..., ad {ηi } i=,...,, he se f (ω, s, y, z) = g(x (ω), /, y, z) for s /, ( + 1)/). This allows oe o cosider a sysem of decoupled Forward-Bacward Sochasic Differeial Equaios (See Secio 4). From ow, we drop he referece o he radomess ω i f ad f. Cosider he a square iegrable ad F -measurable ermial codiio ξ ad deoe h = 1/. Cosider he discree ime bacward sochasic differeial equaio wih ermial codiio y := y = ξ, ad for = 1,..., give by 1 y = ξ + i= 1 hf ( i, yi, zi, u i ) i= ( ) hz i ɛ i+1 + u i ηi+1 + vi µ i+1, (2.1) 6

8 where yi := y, ad i i = i/ (i =,..., ). A soluio o (2.1) is a F -adaped sequece {y, z, u, v : =,..., } such ha y = ξ ad (2.1) holds. Equaio (2.1) is equivale o a bacward recursive sysem, begiig by y = ξ, followed by y = y +1 + hf (, y, z, u ) hz ɛ +1 u η +1 v µ +1 (2.11) = y +1 + hf (, y, z, u ) m +1, for = 1,...,. I view of he represeaio propery (2.7), his las equaio (2.11) is equivale o y = E(y +1 F ) + hf (, y, z, u ). (2.12) The soluio ca be compued by he followig simple algorihm: (I) Se (y, z, u, v ) = (ξ,,, ). (II) For from 1 dow o, compue E(y+1 F ). Compue z ad u as i (2.8) ad (2.9) ad solve (2.12) o fid y, usig a fixed poi priciple. This algorihm shows he exisece ad uiqueess of he soluio of he discree BSDEJ defied i (2.1), as i is always possible o solve i usig he fixed poi priciple for large eough, such ha Kh = K/ < 1. Bouds o he soluio are give below i Secio 3.1. Remar 2 (Compuig codiioal expecaios). Our scheme is very simple o impleme. As for ay scheme ha solves BSDEs umerically, i requires o compue codiioal expecaios. Various mehods have he bee proposed: rees 5, quaizaio 1, regressio 11. Ay mehod faces he explosio of is compuaioal cos as he dimesio icreases. I our case, for a fucio Φ: R 2+2 R we use he formula EΦ(ɛ 1,..., ɛ +1, η 1,..., η +1) F = κ 2 Φ(ɛ 1,..., ɛ, 1, η 1,..., η, κ 1) + κ 2 Φ(ɛ 1,..., ɛ, 1, η 1,..., η, κ 1) + 1 κ Φ(ɛ 2 1,..., ɛ, 1, η1,..., η, κ ) + 1 κ Φ(ɛ 2 1,..., ɛ, 1, η1,..., η, κ ). (2.13) 7

9 Remar 3. The simpliciy of his mehod is also is drawbac. Theoreically, his wor for Browia moio i space dimesio ad ay Poisso process of he form Π(dz) = i=1,...,l π iµ xi (dz). However, i he pah-depeda cases, we should cosider compuig some values for all he possible oucomes of he {ɛ } = ad he {η } = for each of he Browia compoe ad each of he Poisso measure µ xi. This leads o a very high compuaioal cos as soo as he dimesio icreases. I he case where f does o deped o a forward par, or oly o he Browia moio, ad ξ depeds oly o he ermial value of W ad N, he he compuaioal cos may be reduced sice oe eed oly o compue some values a he ode of a muli-dimesioal ree represeig he possibles values of he marigales { i= ɛ i } = ad { i= η i } =. Ayway, he compuaio cos remais high as soo as he Poisso radom measures ivolves more ha oe Dirac mass or he Browia moio have several dimesios. 2.2 The mai covergece resul Cosider he coiuous ime versio {Y, Z, U, V : 1} of he soluio {yi, zi, u i, vi : i =,..., } of he discree equaio (2.11) defied by Y = y, Z = z, U = u, V = v, (2.14) for, 1. Noe ha Y, Z, U ad V are measurable w.r.. he σ-algebra F whe, 1. The discree BSDEJ i (2.1), deoig c () = / ( 1), ca be wrie as Y1 = ξ (he square iegrable ad F -measurable ermial codiio), ad Y = ξ + f (c (s), Ys, Zs, Us )dc (s) Zs dw s (,1 (,1 Us dñ s Vs d M s for, 1, ad M s = W s Ñ s. Wih our oaios (2.11) becomes Y i = Y + 1 i+1 f( i, Y, Z i, U ) 1 i i Z ɛ i i+1 U η i i+1 V µ i i+1. (2.15) I Secio 2.1 ad Remar 2, we gave a umerical scheme o compue he soluio {Y i, Z i, U i : 1}. 8 (,1 (,1

10 We fially sae he covergece assumpio o he ermial codiios of he approximaig equaios: (B ) For some q > 2, sup N E ξ q +E ξ q < + ad E ξ ξ 2. Theorem 1. Uder he assumpios (A), (A ), (B) ad (B ), he se of processes (ξ, Y, Z s ds, U s ds) coverges i he J 1 -Sorohod opology, ad i probabiliy, owards he soluio (ξ, Y, Z sds, U sds) of he BS- DEJ (1.1). 3 Proofs Our heorem is ispired i he mai resul of 5. The proof follows, whe possible, he mai seps of he proof of his resul. The mai differece appears due o he fac ha he uderlyig represeaio heorem for he simple symmeric Beroulli radom wal does o ae place i our case, beig ecessary o cosider a complee sysem of orhogoal marigales i order o have his represeaio propery, as we have see i equaio (2.11). The idea is he o cosider for boh he discree ad he coiuous ime equaios he approximaios provided by he Picard s mehod. I he coiuous case deoe Y, = Z, = U, = ad defie {Y,p+1, Z,p+1, U,p+1 : 1} iducively as he soluio of he bacward differeial equaio Y,p+1 = ξ+ f(s, Y,p s, Z,p, U,p )ds s s Zs,p+1 db s (,1 Us,p+1 dñs. I he discree case, give deoe y, = z, = u, = v, = for =,..., ad defie {y,p+1, z,p+1, u,p+1, v,p+1 : =,..., } iducively as he soluio of he bacward differece equaio wih ermial codiio = ξ ad bacwards recursio defied by y,p+1 y,p+1 = y,p hf (, y,p v,p+1 µ +1 = y,p hf (, y,p, z,p, u,p, z,p, u,p ) hz,p+1 ɛ +1 u,p+1 η +1 ) m,p (3.16) 9

11 If we cosider he coiuous ime versios of he discree Picard approximaios as defied i (2.14), our mehod of proof relies o he decomposiios Y Y = (Y Y,p ) + (Y,p Y,p ) + (Y,p Y ), Z Z = (Z Z,p ) + (Z,p Z,p ) + (Z,p Z), U U = (U U,p ) + (U,p U,p ) + (U,p U), where (Y, Z, U ) (resp. (Y,p, Z,p, U,p )) are he càdlàg processes o, 1 associaed o (y, z, u ) (resp. (y,p, z,p, u,p )) as i (2.14). We prove he covergece of he discree soluio as o he soluio of (1.1), by provig he uiform covergece i he he Picard ieraio priciple, as well as he covergece of he approximaio of his soluio give by his ieraio priciple a each sep whe he ime sep is refied. 3.1 Covergece of he Picard s ieraio procedure i he discree case Wih sadard compuaios, we have ha ad E sup E p> sup Y,p 2 +,1 sup Y,p Y 2 +,1 Zs,p 2 ds + λ Z,p s Z s 2 ds + λ Us,p 2 ds U,p s < + (3.17) U s 2 ds p. We ow prese similar resuls for he discree approximaios. The ex lemma evaluaes he covergece rae of he Picard approximaio sequece (y,p, z,p, u,p ) o (y, z, u ) i he discree scheme, ad shows his rae is uiform i he ime sep 1/. I also provides he covergece of he discree auxiliary sequece (v,p ) o zero. We deoe (y, z, u) = {(y, z, u ): =,..., } (wih or wihou super- 1

12 scrip ) ad, for γ > 1 iroduce he orms (y, z, u ) 2 γ := E ( sup ) ( γ / y ) E γ / z 2 = ( ) 1 + E κ (1 κ ) γ / u 2, = 1 ) v 2 γ (κ := E (1 κ ) γ / v 2. Lemma 1. There exiss γ > 1 ad N such ha for all ad p N, ( y,p+1 y,p, z,p+1 z,p, u,p+1 u,p) 2 + γ v,p+1 vp 2 γ 1 ( y,p y,p 1, z,p z,p 1, u,p u,p 1) 2 4. γ Proof. We iroduce some oaios. As remais fixed durig he mai par of he proof, i will be omied i he oaios wheever possible. We deoe δx p+1 = := x,p+1 x,p for a quaiy x = y, z, u, v, m. Wih his oaio observe ha where m is give i (2.11). We se δf p = f (, y,p δy p+1 = δy p δf p δmp+1 +1, (3.18), z,p, u,p Now, for some β > 1 we develop he quaiy ) f (, y,p 1, z,p 1, u,p 1 ). β (δy p+1 ) 2 β (δy p+1 ) 2 = β (δy p+1 ) 2 11

13 hrough a discree (ime depede) Iô Formula (compare wih 2, VII 9): 1 ( β (δy p+1 ) 2 = β i+1 (δy p+1 i+1 )2 β i (δy p+1 i ) 2) i= 1 = (β 1) i= 1 = (β 1) i= 1 + β i= From (3.18) follows, ha 1 β i (δy p+1 i ) 2 + i= 1 β i (δy p+1 i ) 2 + 2β β i+1 ( (δy p+1 i+1 )2 (δy p+1 i ) 2) i= β i δy p+1 i (δy p+1 i+1 δyp+1 i ) β i (δy p+1 i+1 δyp+1 i ) 2. (3.19) (δy p+1 i+1 δyp+1 i ) ( ) δm p i+1 (δf p 2 i )2. Chagig sigs, usig he previous iequaliy ad (3.18) agai, from (3.19) we obai β (δy p+1 ) 2 + β 2 1 i= 1 (1 β) β ( ) i δm p+1 2 i+1 i= 1 β i (δy p+1 i ) 2 + 2β i= ( ) 1 β i δy p+1 i δf p i+1 δmp+1 i+1 + β 1 β ( i δf p 2 i+1). (3.2) 2 i= We ow use he iequaliy, for λ >, ( ) ( ) δy p+1 2β i δf p i+1 λ ( ) δy p+1 2 2β 2 ( ) i + δf p 2 λ 2 i+1 12

14 i he secod erm of (3.2) o obai β (δy p+1 ) 2 + β 2 1 i= β ( ) i δm p+1 2 i+1 1 (1 + λ β) β i (δy p+1 i ) 2 + β + 4λ 1 β i= i= 1 2β β i ( δf p i+1 i= ) 2 β i δy p+1 i δm p+1 i+1. We ow assume ha 1 + λ β < ad deoe B := β + 4λ 1 β 2 o obai β (δy p+1 ) 2 + β 2 1 i= β ( ) i δm p+1 2 i+1 B 1 β ( ) i δf p i+1 2β β i δy p+1 i δm p+1 i+1. (3.21) i= Formula (3.21) is our firs mai iequaliy. From i we obai he followig wo resuls. Firs, as he las summad is a marigale, aig expecaios wih =, we obai 1 β 2 E β ( ) i δm p+1 2 B i+1 E 1 β ( i δf p 2 i+1). (3.22) 2 i= Secod, aig supremum over =,..., we have sup β (δy p+1 ) 2 B 1 β ( i δf p 2 i+1) + 4β sup β i δy p+1 2 i= i= i= i= i δm p+1 i+1. (3.23) To obai a coveie boud i he las erm of (3.22), we use Davis (see 2, VII 3) ad aferwards Hölder iequaliies. Wih F a uiversal cosa, 13

15 we obai 4βE sup β i δy p+1 i= 4βF E sup i δm p+1 i+1 F E β (δy p+1 ) 2 i= i= β 2 E sup β (δy p+1 ) βF 2 E β 2i ( δy p+1 i β ( ) i δm p+1 2 i+1 i= ) 2 ( ) δm p+1 2 i+1 β i ( δm p+1 i+1 ) 2. Taig expecaio i (3.23), usig he previous resul, ad fially (3.22), we obai ha (1 β/2)e sup β (δy p+1 ) 2 B ( ) βF 2 β ( i δf p 2 i+1). (3.24) 2 Combiig (3.22) ad (3.24) we arrive o 1 E sup β (δy p+1 ) 2 + E β ( ) i δm p+1 2 C i+1 E 1 β ( i δf p 2 i+1), (3.25) 2 i= wih C = 2B(1 + 1/β + 64βF 2 )/(1 β/2). Usig ow he Lipschiz propery (A) we see ha here exiss a cosa K such ha δf p i 2 K ( (δy p i )2 + (δz p i )2 + κ (1 κ )(δu p i )2). (3.26) Equaios (3.25) ad (3.26) give E sup β (δy p+1 1 ) 2 + E C K i= ( E sup β ( ) i δm p+1 2 i+1 i= i= β (δy p ) E β (δz p )2 14 = 1 + κ (1 κ )E β (δu p ). )2 =

16 I remais o choose properly β ad λ as a fucio of. For some γ > 1, se β = γ 1/ ad λ = β/. The codiio 1 + λ < β is he equivale o γ > (1 1/). Thus, if γ > e, for large eough his choice is suiable wih our assumpios o β ad λ, ad C/ remais bouded as. Compuig E ( δm p+1 i+1 ) 2, we coclude he proof. Jus lie i he coiuous case, we ca use he Cauchy crierio ad he precedig lemma o ge he followig resul. Proposiio 1. Followig he oaios of (2.15) ad (3.16), ( E Y,p Y Z,p Z 2 d + λ U,p U sup 1 coverges o uiformly i as p. ) 2 d We may ow sae a global boud which will be used o prove L 2 (P) covergece i Lemma 4 below. Lemma 2. Uder Hypoheses (A), (A ), (B) ad (B ), 1 1 q E f (, y,p, z,p, u,p ) < +. I addiio, sup N sup p N 1 sup E p N = f(s, Y,p s q, Zs,p, Us,p )ds < +. Proof. Here, we deal oly wih he discree case, which is raher similar o he coiuous case whose proof is a variaio of he oe give i 3, 1. Agai, we drop he superscrip. Here, we assume i a firs ime ha he ime horizo is T ad h = T/. We se (z p, u p ) := T ( z p 2 + u p 2). =1 Sice η 2 is bouded i ad, we ge from he Burholder-Davies- Gudy iequaliy ha for some cosas C 1 ad C 2, ( ) q/2 E (z p, u p ) q/2 C 1 E m p 2 C 2 E sup m p q. =1,..., =1 15

17 O he oher had, E sup m p+1 l q E l=1,..., sup l=1,..., E ξ + T 1 q f (, y p, zp, up ) F l. Wih Hypohesis (A), for some cosa C 3 depedig oly o Λ(T ) ad K, = ( T 1 f (, y p, zp, up ) T 1 ) 1/2 f (, y p, zp, up )2 = = T 1 C 3 ( + ( y p 2 + z p 2 + u p 2 ) = C 3 T + C 3 T ) 1/2 sup y p + C 3 T (z p, u p ) 1/2. =,..., 1 Wih he Jese iequaliy for he codiioal expecaio ad he Doob iequaliy, E sup m p+1 q C 4 E ξ q + C 4 T q y p q, + C 4 T + C 4 T q E (z p, u p ) q/2 =1,..., wih y p q, := E sup y p q. =,..., 1 O he oher had, we have wih similar compuaios, for some cosa C 4 depedig oly o C 3 ad q, y p+1 E sup E ξ + T 1 q f (, y p, zp, up ) F l l=,..., 1 This proves ha for C 5 = 2C 4, y p+1 q, + E (z p+1, u p+1 ) q/2 =l C 4 E ξ q + C 4 T + C 4 T q y p q, + C 4 T q E (z p, u p ) q/2. C 5 E ξ q + C 5 T + C 5 T q ( y p q, + E (z p, u p ) q/2 ). If T is small eough so ha C 5 T < 1, he his proves ha y p q, + E (z p, u p ) q/ C 5 T (1 + C 5E ξ q ).

18 Here, we have obaied a boud whe T is small eough. Now, i order o cosider he Picard scheme o he ime ierval, 1, we may fid for each a ime T such ha C 5 T < 1 ad T = l()/ for some l for some fixed, ad solve recursively he Picard scheme o T T, T, T 2T, T T,... usig he ermial codiio ξ ad he y,p l(),... We he obai he desired boud. 3.2 Approximaio of Browia moio ad Poisso process I order o esablish covergece i probabiliy, we cosider ha all he processes are defied o he same probabiliy space. Lemma 3. (I) Le N be a Poisso process of iesiy λ ad se Ñ = N λ. The here exiss a family of idepede radom variables (η ) =1,..., whose disribuio is give by (2.6) ad he process defied by Ñ = /h 1 =1 η, is a marigale which coverges i probabiliy o Ñ i he J 1-Sorohod opology. (II) Le W be a Browia moio. The here exiss a family of realizaios idepede radom variables ɛ such ha P(ɛ = 1) = P(ɛ = 1) = 1/2 ad he process defied by W = h /h 1 i=1 ɛ i coverges uiformly i probabiliy o W. (III) The couple (W, Ñ ) coverges i he J 1 -Sorohod opology i probabiliy o (W, Ñ). Proof. (I) Le (Ω, F, P) be he probabiliy space o which N is defied. Deoe by {τ i } i=1,...,l 1 he ime jumps de N. To simplify he oaios, we se τ = ad τ l = T. Le A he he eve { } here is a mos oe jump of N A =. o ay ierval h, h( + 1)) for =,..., We deoe by A he complemeary eve of A. We se η = κ 1 wih κ = e λ/ if oe jumps occurs o h, h( + 1)) ad η = κ oherwise. The disribuio of η is give by (2.6). Le φ (, ω) be he radom piecewise liear fucio defied so ha for i =,..., l, φ (τ i ) = c (τ i ) o A ad by φ () = o A. I is easily checed ha φ () h for, 1. 17

19 The for τ i, τ i+1, Ñ φ () Ñ c (τ i ) = (e λ/ 1)(c (φ ()) c (τ i )). O he oher had, Ñ Ñτ i = λ( τ i ) ad he here exiss some cosa K such ha Ñ φ () Ñ c (τ i ) Ñ + Ñτ i Kh (3.27) for τ i, τ i+1. Besides, o A, for h small eough, Ñ c (τ i ) Ñ c (τ i ) Ñτ i Ñτ i = e λ/ 1 2λh. (3.28) Combiig (3.27) ad (3.28), oe ges ha o A, I addiio, El < + so ha O A, he φ () = ad he E sup Ñ φ () Ñ (2 + K)lh.,1 sup Ñ φ () Ñ ; A.,1 Ñ ih Ñ i 1 ih = λih h(κ 1)i + (N (j+1)h N j 1)1 {N(j+1)h N j 2}. j= From he very defiiio of A ad sice N (i+1)h N ih has he disribuio of a Poisso radom variable wih iesiy λh, P A 1 P N (i+1)h N ih 2 (1 e λh λhe λh ) λ2 T 2. i= For ay C >, P sup Ñ Ñ φ () > C,1 P A 1 + C E sup Ñ Ñ φ () ; A,1 ad his quaiy coverges o as. Thus here exiss a family (φ ) N of oe-o-oe radom ime chages from, 1 o, 1 such ha 18

20 sup,1 φ () almos surely ad sup,1 Ñ Ñ φ () i probabiliy, which meas ha Ñ coverge i he J 1 -Sorohod opology o N. Poi (II) follows from he Doser heorem, whe oe uses for example he Sorohod embeddig heorem o cosruc he ɛ s from he Browia pah 18 ad (III) holds sice W is coiuous so ha he 2-dimesioal pah (W, Ñ ) coverges i he J 1 -opology o (W, Ñ). 3.3 Covergece of marigales Le H = (W, N) be such ha W is a Browia moio ad N is a idepede Poisso process of iesiy λ. Le W ad Ñ be he oe defied i Lemma 3 ad se H = (W, Ñ ). Le (F ),1 (resp. (F ),1 ) be he filraio geeraed by H (resp. H ). Le X (resp. X ) be a of F 1 (resp. F 1 )-measurable radom variable such ha (H) EX 2 + sup N E(X ) 2 < + ad E X X. Le M (resp. M ) be he cdlg marigales M = E ( X F ) ad M = E ( X F ). (3.29) We deoe by M, M (resp. M, M) he quadraic variaio of M (resp. M) ad by M, W, M, Ñ (resp. M, W, M, Ñ) he cross variaio of M ad W (resp. Ñ ). The followig proposiio is a adapaio of Theorem 3.1 i 5, ad 7 for he covergece of filraios. Hypohesis (H) esures he uiform square iegrabiliy of M ad he he covergece of he braces. Proposiio 2. Uder he above codiios, (H, M, M, M, M, W, M, Ñ ) i probabiliy for he J 1 -Sorohod opology. (H, M, M, M, M, W, M, Ñ) Corollary 1. Se M = /h 1 =1 η ɛ, which is a marigale orhogoal o W ad Ñ. Assume i addiio o (H) ha 19

21 (H ) E X X 2. The here exis hree sequeces (Z ) 1, (V ) 1 ad (U ) 1 of F. - predicable processes, ad wo idepede (Z ) 1 ad (U ) 1 F. -predicable processes such ha M = EX +, 1, M = EX + wih E (Z Z ) 2 d + λ Z s dw s + Z s dw s + U s dñ s + U s dñs (U U ) 2 d. V s d M s Proof. The firs par is relaed o he predicable represeaio of F - marigales i erms of sochasic iegrals wih respec o hree idepede radom wals, W, Ñ ad M. The icremes of M may ae up o four differe values, which meas ha we eed hree orhogoal marigales o represe i 8. I is he easily obaied ha M is a marigale which is orhogoal o boh Ñ ad W. This is why we iroduce i. The predicable represeaio of M wih respec o W ad Ñ is classical. From he Doob iequaliy, Wih (H), sup E N 1 EM, M 1 E M 1 2 2E X 2. (Z s ) 2 dc (s) (η c (s)) 2 (U s ) 2 dc (s) (η c (s)) 2 (V s ) 2 dc (s) < +. (3.3) Sice E(η )2 λ ad U is predicable wih respec o F, oe easily ge ha lim sup E (Zs ) 2 ds + λ(us ) 2 ds < +. (3.31) 2

22 Le (F W ) be he filraio geeraed by he Browia moio. Sice W ad N are idepede, for X = X EX, EX F W 1 = Z s dw s. (3.32) Le also G be he σ-algebra geeraed by (ɛ 1,..., ɛ ). Hece, for X = X EX, I follows ha E X G = E E X G 2 = E (Zs ) 2 dc (s). Z s dw s. (3.33) Sice he ɛ s are cosruced from he rajecories of W, oe has F F1 W. Hece E E X G 2 2E E X X G 2 + 2E E X G 2. Wih he Jese iequaliy o codiioal expecaio ad (H ), oe ges ha E E X X G 2 ad E E X G 2 E E E X F1 W 2 G E E X F1 W 2. From (3.32) ad (3.33), oe ges ha lim sup E (Zs ) 2 ds Similar argumes prove ha lim sup E (Us ) 2 ds E Us 2 ds. E Zs 2 ds. (3.34) From Proposiio 2, M, W M, W i probabiliy for he J 1 -Sorohod opology, as well as M, M. The ψ () sup Z s dc (s) Z s ds (3.35) 1 21

23 i probabiliy, where ψ (). The we ge easily ha sup Zs ds Z s ds (3.36) 1 i probabiliy ad wih (3.31), i L 1 (P). O he oher had, M, Ñ M, Ñ i probabiliy for he J 1-Sorohod opology. This implies ha ψ () sup ηc 2 (s)us dc (s) λ U s ds. (3.37) 1 We ca apply he same argumes used for (3.36), Burholder-Davis-Gudy iequaliy o corol he disace bewee η 2 ad κ (1 κ ), ad he fac ha 1 κ λ/ o ge sup 1 λ U s ds U s ds (3.38) i probabiliy ad i L 1 (P). The secod par relies o he followig argume: le (g ) N { } be a sequece of fucios o, 1 Ω such ha lim sup E ad E (g (s, ω)) 2 ds sup,1 E (g (s, ω)) 2 ds (g (s, ) g (s, ))ds < +, (3.39). (3.4) For ay give fucio h i L 2 (, 1 Ω), here exiss a sequece of fucios (h ) N i L 2 (, 1 Ω) such ha h (s, ω) is of form p i=1 c i(ω)1 i, i+1 (s) ad h coverges o h i L 2 (, 1 Ω). Wih (3.39), E g (s, ω)h (s, ω)ds E g (s, ω)h m (s, ω)ds sup N g L 2 (,1 Ω h h m L 2 (,1 Ω 22

24 ad wih (3.39)-(3.4), E g (s, ω)h m (s, ω)ds E g (s, ω)h m (s, ω)ds. I follows ha g coverges wealy i L 2 (, 1 Ω) o g. I addiio, (3.39) implies ideed he srog covergece of g o g, which meas ha 1 E g (s, ω) g (s, ω) 2 ds coverges o. I is ow possible o apply his argume o boh Z ad U. 3.4 Covergece of he soluio of he BSDE The idea is ow o prove ha if (Y,p, Z,p, U,p ) coverges o (Y,p, Z,p, U,p ) i a give sese, he his is also rue for he (p + 1)-h Picard ieraio. Here, he oio of covergece is sup Y,p 1 ψ () Y,p 2 + Z,p s Z,p s 2 ds + λ U,p s U,p s 2 ds (3.41) i L 1 (P), where ψ is a radom oe-o-oe coiuous mappig from, 1 o, 1 ha coverges uiformly o almos surely. Le us se ad A,p = A,p = f (c (s), Y,p s, Z,p s, U,p s )dc (s) f(s, Y,p s, Z,p, U,p )ds. Lemma 4. If for some ieger p, (Y,p, Z,p, U,p ) coverges o (Y,p, Z,p, U,p ) i he sese of (3.41), he A,p ψ () coverges uiformly i o A,p i L 2 (P). Proof. Le us oe firs ha A,p is piecewise cosa o he iervals /, ( + 1)/). Le ξ () be he iverse of ψ (). The sup A,p,1 ψ () A,p = sup,1 A,p = sup A,p / A,p / + =,..., 1 s A,p ξ () sup =,..., 1 /,(+1)/ s sup A,p / A,p ξ (). 23

25 Because of (A), (A ) ad (3.17), we easily ge ha he las erm above coverges o i L 2 (P) uiformly i p. O he oher had, wih (A) ad (A ), A,p / A,p / K Ye le us oe ha Y,p s ( Ys,p Ys,p ds Ys,p + Zs,p Y,p s Y,p ξ (s) ds + Zs,p + Us,p Y,p Us,p )ds. ξ (s) Y s,p ds. The firs erm i he righ-had side of he previous iequaliy coverges o sice Y,p Y,p ξ coverges uiformly o i L 2 (P). Regardig he secod erm, s Ys,p is coiuous excep a he imes a which he Poisso process jumps. Hece, Y,p,p ξ (s) coverges o Ys for almos every s, 1 ad he,p Y ξ (s) Y s,p ds coverges o almos surely. Wih Lemma 2, sup,1 A,p coverges coverges o i L 2 (P). ψ () A,p Proposiio 3. Uder Hypoheses (A), (A ), (B) ad (B ), for ay p N, (Y,p, Z,p, U,p ) coverges o (Y,p, Z,p, U,p ) i he sese of (3.41). Proof. This will be doe by iducio o p. We rewrie (3.16) as Y,p+1 = ξ + A,p 1 A,p Z,p+1 s dw s U,p+1 s dñ s V s d M s. (3.42) The iducio hypohesis is ha (Y,p, Z,p, U,p ) coverges o (Y,p, Z,p, U,p ) i he sese of (3.41) so ha our aim is o prove ha he riple (Y,p+1, Z,p+1, U,p+1 ) coverges o (Y,p+1, Z,p+1, U,p+1 ) i he same sese. As (Y,, Z,, U, ) = (,, ) ad s f(s,,, ) is coiuous, he firs sep of he iducio is immediae from Corollary 1 usig (B ). Taig codiioal expecaios w.r.. F i (3.16) ad usig he fac ha Y,p+1 is F -measurable, we fid ha for c () < +1, Y,p+1 = E ξ + f (c (s), Y,p s, Z,p s, U,p s ) dc (s) F. 24

26 So ha M,p+1 :=Y,p+1 + =E ξ + =E M,p+1 1 F f (c (s), Y,p s, Z,p s, U,p s ) dc (s) = Y,p+1 + A,p s ) dc (s) F f (s, Y,p s, Z,p s, U,p is a F. marigale. Moreover, we have he represeaio = E Z,p+1 s dws + U,p+1 s dñ s + M,p+1 = = Z,p+1 s dw s + Z,p+1 s dw s + U,p+1 s dñ s + U,p+1 s dñ s + V,p+1 s d M s F V,p+1 s d M s V,p+1 s d M s. The las decomposiio correspods o he marigale represeaio heorem give i Corollary { 1. I order o apply his corollary o he sequece (M },p+1 of marigales ) 1 ; N, we have o prove he L 2 (P) covergece of M,p+1 1 (he ermial value). Usig he fac ha Y,p, Z,p ad U,p are piecewise cosa, we have ha M,p+1 1 ξ A,p 1 ξ ξ + A,p 1 A,p 1. Wih Lemma 2, Lemma 4 ad (H), his las quaiy eds o zero i L 2 (P). Applyig Corollary 1, we see ha M,p+1 coverges o M,p+1 := E i he sese ha sup M,p+1 ψ () 1 ( ξ + M,p+1 f (s, Ys p, Zs p, Us p ) ds ) F = Y,p+1 + A,p, (3.43) 2 + Z,p+1 s Zs,p+1 2 ds + λ U,p+1 s Us,p+1 2 ds i L 2 (P), where ψ is a radom oe-o-oe coiuous mappig from, 1 o, 1 ha coverges uiformly o almos surely. Wih (3.43) ad Lemma 4, he we ge he covergece of (Y,p+1, Z,p+1, U,p+1 ) o (Y,p+1, Z,p+1, U,p+1 ) i he sese of (3.41). 25

27 4 Applicaios o decoupled sysem of SDE ad BSDEJ ad o he umerical compuaios of he soluios of PIDE Le X be he soluio of he d-dimesioal SDE wih jumps X = x+ σ(s, X s )dw s + where we have assumed ha b(s, X s )ds+ c(s, X s )dñs, (4.44) b(, x) b(, y) + σ(, x) σ(, y) + c(, x) c(, y) K x y, (4.45) sup ( b(, ) + σ(, ) + c(, ) ) K (4.46),1 for all, 1 ad for alll x, y R. Of course, he BSDEJ Y = ξ + f(s, X s, Y s, Z s, U s )ds Z s σ(s, X s )dw s is lied o he o-liear PIDE (wih a = σ σ T ) by u(, x) + + d i,j=1 1 2 a i,j(, x) 2 u(, x) + x i x j d i=1 u(, x) b i (, x) x i ( u(, x + c(, x, z)) u(, x) c i (, x, z) ) Π(dz) U s dñs (4.47) u(, x) R x i = f(, x, u(, x), u(, x)σ(, x), u(, x + c(, x, )) u(, x)) (4.48) wih he ermial codiio u(1, x) = g(x). I is sadard i he heory of BSDE ha u(, X ) = Y ad hus u(, x) = Y. Oe ca compue similarly u(s, x) for ay s, ) by usig he soluio o (4.44) sarig from (s, x) isead of (, x). We use ow for Π(dz) he measure Π(dz) = x i π i δ xi, where x i belogs o he ierval I i. Followig Remar 3, we assume for he sae of simpliciy ha ideed Π(dz) = λδ ad ha he dimesio of he Browia moio W is 1. Hece, we rewrie c(, x, z) as c(, x) sice oly c(, x, ) is used. 26

28 The SDE (4.44) may be discreized he followig way for a ieger : we se χ = x ad for i =,...,, χ i+1 = χ i + hb((i + 1)h, χ i ) + hσ(((i + 1)h, χ i ))ɛ i+1 + c((i + 1)h, χ i )η i+1 (4.49) where η is a Beroulli approximaio of he compesaed Poisso process wih iesiy λ ad ɛ is a Beroulli approximaio of he Browia moio W. Of course, he χ i s are easily simulaed. This discree equaio (4.49) may be rewrie i coiuous ime as X = x + σ(s, X s )dw s + b(s, X s )dc (s) + c(s, X s )dñ s (4.5) Thas o he resuls i 22, X coverges i probabiliy i he J 1 -Sorohod opology o he soluio X o (4.44). Usig our algorihm, i is he possible o fid (y i, z i, u i ) i=1,..., adaped o (F i ) i=,..., ha solves he discree BSDE y i = y i+1 + hf((i + 1)h, χ i, y i, z i, u,i) z i ɛ i+1 u i η i+1 v i ɛ i+1η i+1 (4.51) for i =,..., 1 wih he ermial codiio y = g(χ ). We are looig for a fucio v (i, z) such ha y = v (i, χ i ) ad v solves a discree PDE. For a fucio v o {,..., } R, we defie usig (2.13) he discree operaors D v(, x) = Ev(, x) F = 1 κ ( v(, x + hb(h, x) + hσ(h, x) + κc(h, x)) 2 + v(, x + hb(h, x) ) hσ(h, x) + κc(h, x)) + κ ( v(, x + hb(h, x) + hσ(h, x) + (κ 1)c(h, x)) 2 + v(, x + hb(h, x) ) hσ(h, x) + (κ 1)c(h, x)) ad D 1 v(, x) = Ev(, x)ɛ +1 F ad D 2 v(, x) = Ev(, x)η +1 F, 27

29 for which formulae similar o he oe for D v(, x) ca be give. From hese resuls, oe ca deduce he represeaio of he soluio of a discree PDE wih he help of he χ i. This represeaio is similar o he represeaio of he soluio of he BSDEJ i erm of Y = u(, X ), where u is he soluio o he PIDE (4.48). Proposiio 4. Le v be he soluio o he discree PDE v i (i, x) = D v (i + 1, x) + hf((i + 1)h, x, v i (i, x), h 1/2 D 1 v (i + 1, χ i ), D 2 v (i + 1, x)) for i =,..., 1, x R, (4.52) wih he ermial codiio vi (1, x) = g(x). If hk < 1, he his soluio exiss ad is uique. I addiio, he soluio (y, z, u ) o he discree BSDE (2.11) wih he ermial codiio ξ = g(χ ) which we cosruc usig our algorihm saisfies yi = v (i, χ i ), zi = h 1/2 D1 v (i + 1, χ i ) ad u i = D2 v (i + 1, χ i ). Proof. As hk < 1 he exisece ad uiqueess of v (i, ) follows from he exisece of he soluio ρ(x) o ρ(x) = D v (i + 1, x) + hf((i + 1)h, x, ρ(x), h 1/2 D 1 v (i + 1, x), (1 κ) 1 D 2 v (i + 1, x)) for ay x R, oce v (i + 1, ) is ow. Thus, oe ca proceed recursively wih i = 1 dow o. Le (y, z, u ) be give by our algorihm. We assume v (i + 1, χ i+1) = y i+1, which is rue for i + 1 =. Usig (2.8), (2.9), ad he defiiios of D 1 ad D 2, z i = h 1/2 Ev (i + 1, χ i+1)ɛ i+1 F i = h 1/2 D 1 v (i + 1, χ i ), u i = Ev (i + 1, χ i+1)η i+1 F i = D 2 v (i + 1, χ i ). Taig codiioal expecaio wih respec o Fi 1, χ i ) = Ev (i + 1, χ i+1) Fi, we ge i (4.52), sice D v (i+ v (i, χ i ) = Ey i+1 F i + hf((i + 1)h, χ i, v (i, χ i ), h 1/2 D 1 v (i + 1, χ i ), D 2 v (i + 1, χ i )), (4.53) 28

30 while aig he codiioal expecaio wih respec o F i i (2.11), we ge y i = Ey i+1 F i + hf((i + 1)h, χ i, y i, h 1/2 D 1 v (i + 1, χ i ), D 2 v (i + 1, χ i )). (4.54) As hf(,,,, ) is Kh-Lipschiz i is hird argume wih Kh < 1, we obai ha yi ad v (i, χ i ) are equal. 5 A umerical example I his secio, we deal wih a umerical example. We cosider N a Poisso process wih λ = 1 ad c < 1, ad he followig BSDEJ: dy = cu d + Z dw + U (dn d), (5.55) wih ξ = N T. The explici soluio of (5.55) is give by (Y, Z, U ) = (N + (1 + c)(t ),, 1). Furhermore if ξ = he he soluio is equal o (Y, Z, U ) = (,, ). This example is borrowed from 2. We have implemeed his mehod o a sadard persoal compuig plaform (PC), ad have observed ha i performs very well usig simulaed daa, as ca be see from he simulaed daa i he Table 1 ad Figure 1. Despie he appare algebraic complexiy of he equaios (2.8), (2.9) ad (2.12) oe eeds o solve a each sep he codiioal expecaio o obai y i, he problem poses o difficuly. Usig MATLAB s simulaios ad algebra capabiliies (Versio 7. ruig o he Uiversiy of Valparaíso CIMFAV cluser) yielded bes compuig imes. I our implemeaio, ad for compuaioal coveieces we cosider he case whe T = 1. The ieraio of he algorihm begis from y = ξ = N1 a ime = T = 1 ad proceeds bacward o solve (y j, z j, u j ), where j = j/, a ime j =. Values are give wih 4 sigifica digis. I he followig able ad picure we summarize he resuls. 29

31 c =.3 c =.9 c =.1 c = Real Value Y Table 1: Numerical Scheme for dy = cu d + Z db U (dn d) wih from 1 uil = 5 seps, λ = 1, T = 1 ad differe values of c. 2.5 Moe Carlo Simulaios, =1, c=.3 2 Pahs of Y Time from > 1 Figure 1: Moe Carlo Simulaio; c =.3, λ = 1, = 1 ad T = 1. 3

32 Refereces 1 Bally, V. ad Pagès, G., A quaizaio algorihm for solvig mulidimesioal discree-ime opimal soppig problems. Beroulli 9:6 (23) Barles, G., Bucdah, R. ad Pardoux, E., BSDEs ad iegral-parial differeial equaios. Sochasics Sochasics Rep. 6:1 2 (1997) Bouchard B. ad Elie, R., Discree ime approximaio of decoupled Forward-Bacward SDE wih jumps, Sochasic Process. Appli. 118 (25) Bouchard, B. ad Touzi, N., Discree-ime approximaio ad Moe- Carlo simulaio of bacward sochasic differeial equaios. Sochasic Process. Appl. 111:2 (24) Briad, P., Delyo, B. ad Mémi, J., Doser-Type Theorem for BS- DEs. Elecro. Comm. Probab., 6 (21) Chevace, D., Numerical mehods for bacward sochasic differeial equaios. I Numerical mehods i fiace, Publ. Newo Is., Cambridge Uiv. Press, Cambridge, Coque, F., Mémi, J. ad S lomińsi, L., O wea covergece of filraios. Sémiaire de Probabiliés, XXXV, Lecure Noes i Mah Spriger, Berli, Doha, M. U., Prices i fiacial mares. The Claredo Press, Oxford Uiversiy Press, New Yor, Douglas, J., Ma, J. ad Proer, P., Numerical mehods for forwardbacward sochasic differeial equaios. A. Appl. Probab. 6:3 (1996) Elie, R., Corôle Sochasique e Méhodes Numériques, Ph.D. hesis, Uiversié Paris-Dauphie (26). 11 Gobe, E., Lemor, J-P. ad Wari, X., A regressio-based Moe Carlo mehod o solve Bacward Sochasic Differeial equaios. A. Appl. Probab. 15:3 (25)

33 12 Li, X. ad Tag, S., Necessary codiio for opimal corol of sochasic sysems wih radom jumps SIAM J. Corol Opim. 332:5 (1994), Ma J., Proer P., Sa Marí J. ad Torres S., Numerical mehod for Bacward Sochasic Differeial Equaios. A. Appl. Probab. 12:1 (22) Ma, J., Proer, P. ad Yog, J., Solvig forward-bacward sochasic differeial equaios explicily a four sep scheme. Probab. Theory Relaed Fields 98:3 (1994) Pardoux, E. ad Peg, S., Adaped soluio of bacward sochasic differeial equaio. Sysems Corol Le. 14:1 (199) Pardoux, E. ad Peg, S., Bacward sochasic differeial equaio ad quasiliear parabolic parial differeial equaios. I Sochasic parial differeial equaios ad heir applicaios (Charloe, NC, 1991), Lecure Noes i Corol ad Iform. Sci. 176 Spriger, Berli, Peg, S. ad Xu, M., Numerical algorihms for bacward sochasic differeial equaios wih 1-d Browia moio: Covergece ad simulaios. Mah. Model. Numer. Aal. 45 (211) Rogers, L.C.G. ad Williams, D., Diffusios, Marov Processes, ad Marigales: Foudaios, 2 d ed., Cambridge Uiversiy Press (2). 19 Sa Mari, J. ad Torres, S., Bacward sochasic differeial equaios: umerical mehods, Ecyclopedia of Quaiaive Fiace (21). 2 Shiryaev, A. N., Probabiliy, 2 d ed., Spriger, New Yor, Siu, R., O soluio of bacward sochasic differeial equaios wih jumps. Sochasic Process. Appl. 66:2 (1997) S lomińsi, L. Sabiliy of srog soluios of sochasic differeial equaios. Sochasic Process. Appl. 31:2 (1989) Zhag, J., A umerical scheme for BSDEs A. Appl. Probab. 14:1 (24)