Discrete time approximation of decoupled Forward-Backward SDE with jumps

Transcription

1 Dscree me approxmaon of decoupled Forward-Backward SD wh jumps Bruno Bouchard, Romuald le To ce hs verson: Bruno Bouchard, Romuald le Dscree me approxmaon of decoupled Forward-Backward SD wh jumps Sochasc Processes and her Applcaons, lsever, 28, 118 1, pp53-75 <hal > HAL Id: hal hps://halarchves-ouveresfr/hal Submed on 8 Dec 25 HAL s a mul-dscplnary open access archve for he depos and dssemnaon of scenfc research documens, wheher hey are publshed or no The documens may come from eachng and research nsuons n France or abroad, or from publc or prvae research ceners L archve ouvere plurdscplnare HAL, es desnée au dépô e à la dffuson de documens scenfques de nveau recherche, publés ou non, émanan des éablssemens d ensegnemen e de recherche franças ou érangers, des laboraores publcs ou prvés

2 Dscree me approxmaon of decoupled Forward-Backward SD wh jumps Bruno BOUCHARD LPMA CNRS, UMR 7599 Unversé Pars 6 e-mal: bouchard@ccrjusseufr and CRST Romuald LI CRMAD CNRS, UMR 7534 Unversé Pars 9 e-mal: ele@ensaefr and CRST-NSA Ths verson : November 25 Absrac We sudy a dscree-me approxmaon for soluons of sysems of decoupled forward-backward sochasc dfferenal equaons wh jumps Assumng ha he coeffcens are Lpschz-connuous, we prove he convergence of he scheme when he number of me seps n goes o nfny When he jump coeffcen of he frs varaon process of he forward componen sasfes a non-degeneracy condon whch ensures s nversbly, we oban he opmal convergence rae n 1/2 The proof s based on a generalzaon of a remarkable resul on he pah-regulary of he soluon of he backward equaon derved by Zhang 28, 29 n he nojump case A smlar resuls obaned whou he non-degeneracy assumpon whenever he coeffcens are Cb 1 wh Lpschz dervaves Several exensons of hese resuls are dscussed In parcular, we propose a convergen scheme for he resoluon of sysems of coupled semlnear parabolc PD s Key words : Dscree-me approxmaon, forward-backward SD s wh jumps, Mallavn calculus MSC Classfcaon 2: 65C99, 6H7, 6J75 1

3 1 Inroducon In hs paper, we sudy a dscree me approxmaon scheme for he soluon of a sysem of decoupled Forward-Backward Sochasc Dfferenal quaons FBSD n shor wh jumps of he form { X = X + bx rdr + σx rdw r + Y = gx T + T h Θ r dr T Z r dw r T where Θ := X, Y, Z, Γ wh Γ := ρeueλde βx r, e µde, dr, U re µde, dr 11 Here, W s a d-dmensonal Brownan moon and µ an ndependen compensaed Posson measure µde, dr = µde, dr λdedr Such equaons naurally appear n hedgng problems, see eg yraud-losel 13, or n sochasc conrol, see eg Tang and L 26 and he recen paper Becherer 4 for an applcaon o exponenal uly maxmsaon n fnance Under sandard Lpschz assumpons on he coeffcens b, σ, β, g and h, exsence and unqueness of he soluon have been proved by Tang and L 26, hus generalzng he semnal paper of Pardoux and Peng 21 The man movaon for sudyng dscree me approxmaons of sysems of he above form s ha hey provde an alernave o classcal numercal schemes for a large class of deermnsc PD s of he form where Lu, x + h, x, u, x, σ, x x u, x, Iu, x =, ut, x = gx, 12 Lu, x := u, x + xu, xbx + 1 d σσ x j 2 u, x 2 x xj,j=1 + {u, x + βx, e u, x x u, xβx, e} λde, Iu, x := {u, x + βx, e u, x} ρe λde Indeed, s well known ha, under mld assumpons on he coeffcens, he componen Y of he soluon can be relaed o he vscosy soluon u of 12 n he sense ha Y = u, X, see eg 2 Thus solvng 11 or 12 s essenally he same In he socalled four-seps scheme, hs relaon allows o approxmae he soluon of 11 by frs esmang numercally u, see 11 and 18 Here, we follow he converse approach Snce classcal numercal schemes for PD s generally do no perform well n hgh dmenson, we wan o esmae drecly he soluon of 11 so as o provde an approxmaon of u 2

4 In he no-jump case, e β =, he numercal approxmaon of 11 has already been suded n he leraure, see eg Zhang 29, Bally and Pages 3, Bouchard and Touz 6 or Gobe e al 15 In 6, he auhors sugges he followng mplc scheme Gven a regular grd π = { = T/n, =,, n}, hey approxmae X by s uler scheme X π and Y, Z by he dscree-me process Ȳ π, Z π n defned backward by Z π π = n Ȳ +1 W +1 F Ȳ π = Ȳ π +1 F + 1 n h X π, Ȳ π, Z π where Ȳ π n := gx π n and W +1 := W +1 W In he no-jump case, urns ou ha he dscrezaon error rr n Y, Z := { max <n sup,+1 s nmaely relaed o he quany n 1 +1 = Y Ȳ π 2 n 1 + = +1 Z Z 2 d where Z := n Z Z π 2 d +1 Z d F Under Lpschz connuy condons on he coeffcens, Zhang 27 was able o prove ha he laer s of order of n 1 } 1 2 Ths remarkable resul allows o derve he bound rr n Y, Z Cn 1/2, e he above approxmaon acheves he opmal convergence rae n 1/2 In hs paper, we exend he approach of Bouchard and Touz 6 and approxmae he soluon of 11 by he backward scheme Z π π = n Ȳ +1 W +1 F Γ π π = n Ȳ ρe µde, +1, +1 F Ȳ π π = Ȳ +1 F + 1 h X π n, Ȳ π, Z π, Γ π where Ȳ π n := gx π n By adapng he argumens of Gobe e al 15, we frs prove ha our dscrezaon error rr n Y, Z, U := { max <n sup,+1 Y Ȳ π 2 n 1 + = +1 Z Z π 2 + Γ Γ π 2 d converges o as he dscresaon sep T/n ends o We hen provde upper bounds on max <n sup,+1 Y Y 2 n 1 + = +1 Z Z 2 + Γ Γ 2 d, 3 } 1 2

5 where Γ +1 := n Γ d F We frs show ha max <n sup,+1 Y Y 2 n 1 + = +1 Γ Γ 2 d C n 1 whenever he coeffcens are Lpschz connuous Under some addonal condons on he nversbly of β + I d, see H 1, or on he regulary of he coeffcen, see H 2, we hen prove ha n 1 +1 = Z Z 2 d C n 1 Ths exends o our framework he remarkable resul derved by Zhang 28, 29 n he no-jump case and allows us o show ha our dscree-me scheme acheves he opmal convergence rae n 1/2 Observe ha, n opposon o algorhms based on he approxmaon of he Brownan moon by dscree processes akng a fne number of possble values see 1, 8, 9, 1 and 17, our scheme does no provde a fully mplemenable numercal procedure snce nvolves he compuaon of a large number of condonal expecaons However, he mplemenaon of he above menoned schemes n hgh dmenson s quesonable and, n our seng, hs ssue can be solved by approxmang he condonal expecaon operaors numercally n an effcen way In he no-jump case, Bouchard and Touz 6 use he Mallavn calculus o rewre condonal expecaons as he rao of wo uncondonal expecaons whch can be esmaed by sandard Mone-Carlo mehods In he refleced case where h does no depend on Z, Bally and Pages 3 use a quanzaon approach Fnally, Gobe e al 15 have suggesed an adapaon of he so-called Longsaff and Schwarz algorhm based on non-paramerc regressons, see 16, whch also works n he case where β bu he drver does no depend on U Snce hs s no he man ssue of hs paper, we leave he heorecal sudy and numercal mplemenaon of such mehods n our seng for furher research The res of he paper s organzed as follows In Secon 2, we descrbe he approxmaon scheme and sae our man convergence resul We also dscuss several possble exensons In parcular, we propose a convergen scheme for he resoluon of sysems of coupled semlnear parabolc PD s Secon 3 conans some resuls on he Mallavn dervaves of Forward and Backward SD s Applyng hese resuls n Secon 4, we derve some regulary properes for he soluon of he backward equaon under addonal smoohness assumpons on he coeffcens We fnally use an approxmaon argumen o conclude he proof of our man heorem 4

6 Noaons : Any elemen x R d wll be denfed o a column vecor wh -h componen x and ucldan norm x For x R d, n and d N, we defne x 1,, x n as he column vecor assocaed o x 1 1,, x d 1 1,, x 1 n,, x dn n The scalar produc on R d s denoed by x y For a m d-dmensonal marx M, we noe M := sup{ Mx ; x R d, x = 1}, M s ranspose and we wre M M d f m = d Gven p N and a measured space A, A, µ A, we denoe by L p A, A, µ A ; R d, or smply L p A, A or L p A f no confuson s possble, he se of p-negrable R d -valued measurable maps on A, A, µ A For p =, L A, A, µ A ; R d s he se of essenally bounded R d -valued measurable maps The se of k-mes dfferenable maps wh bounded dervaves up o order k s denoed by C k b and C b := k 1 C k b For a map b : Rd R k, we denoe by b s Jacoban marx whenever exss In he followng, we shall use hese noaons whou specfyng he dmenson when s clearly gven by he conex 2 Dscree me approxmaon of decoupled FBSD wh jumps 21 Decoupled forward backward SD s Le Ω, F, F = F T, P be a sochasc bass such ha F conans he P-null ses, F T = F and F sasfes he usual assumpons We assume ha F s generaed by a d-dmensonal sandard Brownan moon W and an ndependen Posson measure µ on, T where = R m for some m 1 We denoe by F W = F W T resp F µ = F µ T he P-augmenaon of he naural flraon of W resp µ We assume ha he compensaor ν of µ has he form νd, de = λded for some fne measure λ on, endowed wh s Borel rbe, and denoe by µ := µ ν he compensaed measure Gven K >, wo K-Lpschz connuous funcons b : R d R d and σ : R d M d, and a measurable map β : R d R d such ha sup e β, e K and sup βx, e βx, e K x x x, x R d, 21 e we defne X as he soluon on, T of X = X + bx r dr + σx r dw r + βx r, e µde, dr, 22 for some nal condon X R d The exsence and unqueness of such a soluon s 5

7 well known under he above assumpons, see eg 14 and he Appendx for sandard esmaes for soluons of such SD Before nroducng he backward SD, we need o defne some addonal noaons Gven s and some real number p 2, we denoe by S p s, he se of real valued adaped càdlàg processes Y such ha Y S p s, 1 := sup Y r p p s r <, H p s,s he se of progressvely measurable Rd -valued processes Z such ha Z H p s, := s p 1 p Z r 2 2 dr <, L p λ,s, s he se of P measurable maps U : Ω, T R such ha U L p λ,s, := s 1 U s e p p λdeds < wh P defned as he σ-algebra of F-predcable subses of Ω, T The space B p s, := S p s, Hp s, Lp λ,s, s endowed wh he norm Y, Z, U B p s, := 1 Y p + Z p + U p p S p H p L p s, s, λ,s, In he sequel, we shall om he subscrp s, n hese noaons when s, =, T For ease of noaons, we shall somemes wre ha an R n -valued process s n S p s, or L p λ,s, meanng ha each componens n he correspondng space Smlarly an elemen of M m s sad o belong o H p s, f each column belongs o H p s, The norms are hen naurally exended o such processes The am of hs paper s o sudy a dscree me approxmaon of he rple Y, Z, U soluon on, T of he backward sochasc dfferenal equaon Y = gx T + T h Θ r dr T Z r dw r T where Θ := X, Y, Z, Γ and Γ s defned by Γ := ρeueλde, 6 U r e µde, dr, 23

8 for some measurable map ρ : R m sasfyng sup ρe K 24 e By a soluon, we mean an F-adaped rple Y, Z, U B 2 sasfyng 23 In order o ensure he exsence and unqueness of a soluon o 23, we assume ha he map g : R d R and h : R d R R d R m R are K-Lpschz connuous see Lemma 52 n he Appendx For ease of noaons, we shall denoe by C p a generc consan dependng only on p and he consans K, λ, b, σ, h, g and T We wre Cp f also depends on X In hs paper, p wll always denoe a real number greaer han 2 Remark 21 For he convenence of he reader, we have colleced n he Appendx sandard esmaes for he soluons of Forward and Backward SD s In parcular, hey mply X, Y, Z, U p S p B p C p 1 + X p, p 2 25 The esmae on X s sandard, see 54 of Lemma 51 n he Appendx Pluggng hs n 58 of Lemma 52 leads o he bound on Y, Z, U B p Usng 55 of Lemma 51, we also deduce ha sup X u X s p s u C p 1 + X p s, 26 whle he prevous esmaes on X combned wh 59 of Lemma 52 mples { } sup Y u Y s p C p 1 + X p s p + Z p + U p 27 H p L p s u s, λ,s, 22 Dscree me approxmaon We frs fx a regular grd π := { := T/n, =,, n} on, T and approxmae X by s uler scheme X π defned by { X π := X X π +1 := X π + 1 n bxπ + σx π W +1 + βxπ, e µde,, where W +1 := W +1 W Is well known ha max sup X X π <n 2,+1 C 2 n

9 We hen approxmae Y, Z, Γ by Ȳ π, Z π, Γ π defned by he backward mplc scheme Z π π := n Ȳ +1 W +1 F Γ π π := n Ȳ ρe µde, +1, +1 F Ȳ π π := Ȳ +1 F + 1 h 21 X π n, Ȳ π, Z π, Γ π on each nerval, +1, where Ȳ π n := gx π n Observe ha he resoluon of he las equaon n 21 may nvolve he use of a fxed pon procedure However, h beng Lpschz and mulpled by 1/n, he approxmaon error can be negleced for large values of n Remark 22 The above backward scheme, whch s a naural exenson of he one consdered n 6 n he case β =, can be undersood as follows On each nerval, +1, we wan o replace he argumens X, Y, Z, Γ of h n 23 by F -measurable random varables X, Ỹ, Z, Γ Is naural o ake X = X π Takng condonal expecaon, we oban he approxmaon Y 1 = Y+1 F + n h X π, Ỹ, Z, Γ Ths leads o a backward mplc scheme for Y of he form Ȳ π π = Ȳ +1 F + 1 n h X π, Ȳ π, Z, Γ 211 I remans o choose Z and Γ n erms of Ȳ π +1 By he represenaon heorem, here exs wo processes Z π H 2 and U π L 2 λ sasfyng Ȳ π π +1 Ȳ +1 F = +1 Z π s dw s + +1 U π s e µds, de Observe ha hey do no depend on he way Ȳ π s defned and ha Z π and Γ π defned n 21 sasfy Z π = n Zs π ds F and Γπ = n Γ π s ds F 212 and herefore concde wh he bes H 2,+1 -approxmaons of Zπ <+1 and Γ π <+1 := ρeu π eλde <+1 by F -measurable random varables vewed as consan processes on, +1, e +1 Z π Z π 2 d +1 Γ π Γ π 2 d +1 = nf Z Z L 2 π Z 2 d Ω,F +1 = nf Γ π Γ 2 d Γ L 2 Ω,F 8

10 Thus, s naural o ake Z, Γ = Z π, Γ π n 211, so ha Ȳ π = Ȳ π n h X π, Ȳ π, Z π, Γ π +1 Z π s dw s Fnally, observe ha, f we defne Y π on, +1 by seng Y π := Ȳ π hx π, Ȳ π, Z π, Γ π + we oban n +1 Y π d F Z π s dw s + +1 U π s e µds, de U π s e µds, de, π = Ȳ +1 F + 1 n h X π, Ȳ π, Z π, Γ π = Y π = Ȳ π Thus, n hs scheme, Ȳ π s he bes H 2,+1 -approxmaon of Y π on, +1 by F - measurable random varables vewed as consan processes on, +1 Ths explans he noaon Ȳ π whch s conssen wh he defnon of Z π and Γ π Remark 23 One could also use an explc scheme as n eg 3 or 15 In hs case, 21 has o be replaced Z π π := n Ỹ +1 W +1 F Γ π π := n Ỹ ρe µde, +1, +1 F Ỹ π π := Ỹ +1 F + h 1 X π n, Ỹ π +1, Z π, Γ π F 213 wh he ermnal condon Ỹ π n = gx π n The advanage of hs scheme s ha does no requre a fxed pon procedure However, from a numercal pon of vew, addng a erm n he condonal expecaon defnng Ỹ π makes more dffcul o esmae We herefore hnk ha he mplc scheme may be more racable n pracce The convergence of he explc scheme wll be dscussed n Remarks 26 and 28 below 23 Convergence of he approxmaon scheme In hs subsecon, we show ha he approxmaon error rr n Y, Z, U := { sup Y Ȳ π 2 } 1 + Z Z π 2 H + Γ Γ π H 2 T converges o Before o sae hs resul, le us nroduce he processes Z, Γ defned on each nerval, +1 by Z := n +1 Z s ds F and 9 Γ +1 := n Γ s ds F

11 Remark 24 Observe ha Z and Γ are he counerpars of Z π and Γ π for he orgnal backward SD They can also be nerpreed as he bes H 2,+1 -approxmaons of Z <+1 and Γ <+1 by F -measurable random varables vewed as consan processes on, +1, e +1 Z Z 2 d +1 Proposon 21 We have n 1 = +1 Γ Γ 2 d +1 = nf Z L 2 Ω,F = nf Γ L 2 Ω,F +1 Z Z 2 d Γ Γ 2 d Y Y 2 d C 2 n 1 and Z Z H 2 + Γ Γ H 2 ɛn 214 where ɛn as n Moreover, so ha rr n Y, Z, U C 2 n 1/2 + Z Z H 2 + Γ Γ H 2, 215 rr n Y, Z, U n Proof We adap he argumens of 6 Recall from Remark 22 ha Y π = Ȳ π hx π, Ȳ π, Z π, Γ π + Zs π dw s + Us π e µds, de on, +1 and ha Ȳ π = Y π For L = Y, Z or U, we se δl := L L π I follows from he defnon of Z π and Ū π n 212, Jensen s nequaly and he bound on ρ ha Z Z π 2 + Γ Γ π 2 C 2 n δz 2 H + δu 2 2 L 216,+1 2 λ,,+1 For, +1, we deduce from Iô s Lemma, he Lpschz propery of h, 29 and 216 ha δy 2 + δz 2 H + δu 2 2 L,+1 2 λ,,+1 +1 δy α δy s 2 ds + C 2 n 2 + α B + B π 217 where α s some posve consan o be chosen laer, and B, B π s defned as B := +1 Ys Y 2 + Z s Z s 2 + Γ s Γ s 2 ds B π := n 1 δy 2 + δz 2 H + δu 2 2 L,+1 2 λ,,+1 1

12 Usng Gronwall s Lemma, follows ha δy 2 δy C 2 n 2 + α B + B π e α/n 218 Pluggng hs nequaly n 217 and akng α and n large enough leads o δy 2 + η δz 2 H + δu 2 2 L 1 + C 2,+1 2 λ,,+1 n δy C2 n 2 + B + n 1 δy 2, wh η > For n large enough, combnng he lasnequaly wh he deny δy n = gx T gxt π and he esmae 29 leads o δy 2 C 2 n 1 + B n 1 where B := j= B j, 22 whch plugged no 219 mples δy 2 + η δz 2 H + δu 2 2 L,+1 2 λ,,+1 δy C 2 n 2 + B n + B Summng up over and usng 218 and 22, we fnally oban rr n Y, Z, U 2 C 2 n 1 + B 221 Snce Y solves 23, Y Y 2 C 2 hx r, Y r, Z r, Γ r 2 + Z r 2 + U r e 2 λde dr Combnng he Lpschz propery of h wh 25, follows ha n 1 +1 = Y Y 2 d C 2 n Ths s exacly he frs par of 214 whch combned wh 221 leads o 215 I remans o prove he second par of 214 Snce Z s F-adaped, here s a sequence of adaped processes Z n n such ha Z n = Z n on each, +1 and Z n converges o Z n H 2 By Remark 24, we observe ha Z Z 2 H 2 Z Zn 2 H 2, and applyng he same reasonng o Γ concludes he proof 11

13 Remark 25 If σ =, whch mples Z =, or h does no dependen on Z, he erm B n he above proof reduces o B = +1 Ys Y 2 + Γ s Γ s 2 ds In hs case, he asseron 215 of Proposon 21 can be replaced by rr n Y, Z, U C 2 n 1/2 + Γ Γ H Remark 26 In hs Remark, we explan how o adap he proof of Proposon 21 o he explc scheme defned n 213 Frs, we can fnd some Ẑπ H 2 and Û π L 2 λ such ha Ỹ π π +1 = Ỹ +1 F + Ẑs π dw s + Ûs π e µde, ds We hen defne Ŷ π on, +1 by Ŷ π = Ỹ π h X π, Ỹ π +1, Z π, Γ π Observe ha Ŷ π +1 = Ỹ π +1 Z π = n for all < n Moreover and +1 Ẑ π s ds F F + hx s, Y s, Z s, Γ s = hx, Y +1, Z, Γ F Ẑ π s dw s + +1, Γπ = n ˆΓπ s ds F + hx, Y, Z, Γ hx, Y +1, Z, Γ F + hx s, Y s, Z s, Γ s hx, Y, Z, Γ where by he Lpschz connuy of h and of Theorem 21 below hx, Y, Z, Γ hx, Y, Z, Γ F 2 +1 C2/n, and +1 hxs, Y s, Z s, Γ s hx, Y, Z, Γ 2 ds +1 C2 n 2 + Z s Z 2 + Γ s Γ 2 ds, Û π s e µde, ds by of Theorem 21 and 26 Usng hese remarks, he proof of Proposon 21 can be adaped n a sraghforward way Ths mples ha he approxmaon error due o he explc scheme s also upper-bounded by C2 n 1/2 + Z Z H 2 + Γ Γ H 2 12

14 24 Pah-regulary and convergence rae under addonal assumpons In vew of Proposon 21, he dscrezaon error converges o zero In order o conrol s speed of convergence, remans o sudy Z Z 2 H + Γ Γ 2 2 H In hs secon, 2 we shall appeal o one of he addonal assumpons : H 1 : For each e, he map x R d βx, e adms a Jacoban marx βx, e such ha he funcon x, ξ R d R d ax, ξ; e := ξ βx, e + I d ξ sasfes one of he followng condon unformly n x, ξ R d R d ax, ξ; e ξ 2 K 1 or ax, ξ; e ξ 2 K 1 H 2 : σ, b, β, e, h and g are Cb 1 unformly n e funcons wh K-Lpschz connuous dervaves, Remark 27 Observe for laer use ha he condon H 1 mples ha, for each x, e R d, he marx βx, e + I d s nverble wh nverse bounded by K Ths ensure he nversbly of he frs varaon process X of X, see Remark 32 Moreover, f q s a smooh densy on R d wh compac suppor, hen he approxmang funcons β k, k N, defned by β k x, e := k d β x, eqkx xd x R d are smooh and also sasfy H 1 We can now sae he man resul of hs paper Theorem 21 The followng holds For all < n sup Y Y 2 C2 n 1 and sup Γ Γ 2,+1,+1 C 2 n so ha Γ Γ 2 S C 2 2 n 1 and Γ Γ 2 H C 2 2 n 1 Assume ha H 1 holds Then n 1 +1 = Z Z 2 d C2 n

15 so ha Z Z 2 H C 2 2 n 1 Assume ha H 2 holds Then, for all < n and, +1, Z Z 2 C2 n 1, 225 so ha Z Z 2 H C 2 2 n 1 Ths regulary propery wll be proved n he subsequen secons Combned wh Proposon 21 and Remark 25, provdes an upper bound for he convergence rae of our backward mplc scheme Corollary 21 Assume ha eher H 1 holds, or H 2 holds, or σ =, or h s ndependen of Z Then, rr n Y, Z, U C2 n 1/2 Remark 28 In vew of Remark 26, he resul of Corollary 21 can be exended o he explc scheme defned n Possble xensons I wll be clear from he proofs ha all he resuls of hs paper hold f we le he maps b, σ, β, and h depend on whenever hese funcons are 1/2-Hölder n and he oher assumpons are sasfed unformly n In hs case, he backward scheme 21 s modfed by seng Ȳ π π = Ȳ +1 F + 1 n h, X π, Ȳ π, Z π, Γ π The uler approxmaon X π of X could be replaced by any oher adaped approxmaon sasfyng 29 Le M be he soluon of he SD M = M + b M M r dr + β M M r, e µde, dr where b M : R k R k and β M, e : R k R k, k 1, are Lpschz connuous unformly n e wh β M, bounded, and consder he sysem { X = X + bm r, X r dr + σm r, X r dw r + Y = gm T, X T + T h M r, Θ r dr T Z r dw r T βm r, X r, e µde, dr U 226 re µde, dr 14

16 where b, σ, β, e and h are K-Lspchz, unformly n e and β, s bounded Here, he dscree-me approxmaon of Y s gven by Ȳ π n = gm π n, X π n, Ȳ π π = Ȳ +1 F + 1 n h M π, X π, Ȳ π, Z π, Γ π, where M π, X π s he uler scheme of M, X Consderng M, X as an R k+d dmensonal forward process, we can clearly apply he resuls of Proposon 21 Moreover, we clam ha Theorem 21 holds as well as resp f H 1 resp H 2 holds for bm,, σm,, βm,, gm, and hm, as funcons of x, y, z, γ unformly n m R k Ths comes from he fac ha he dynamcs of M are ndependen of X and ha he Mallavn dervave of M wh respec o he Brownan moon equals zero Ths parcular feaure mples ha he proofs of Secon 33 and Secon 4 work whou any modfcaon n hs conex v In 22, see also 25, he auhors consder a sysem of he form { X = X + bm r, X r dr + σm r, X r dw r Y = gm T, X T + T h M r, Θ r dr T Z r dw r T U re µde, dr 227 where M s an F µ -adaped purely dsconnuous jump process In 22, s shown ha a large class of sysems of coupled semlnear parabolc paral dfferenal equaons can be rewren n erms of sysems of BSD of he form 227, where he backward componens are decoupled However, her parcular consrucon mples ha b, σ, h and g are no Lpschz n her frs varable m In hs remark, we explan how o consder hs parcular framework Hereafer, we assume ha he pah of M can be smulaed exacly, whch s he case n 22 Then, recallng ha λ < so ha µ has as only a fne number of jumps on, T, we can nclude he jump mes of M n he uler scheme X π of X Thus, even f b and σ are no Lpschz n her frs varable m, we can sll defne an approxmang scheme X π of X such ha sup X X π 2,+1 C 2 +1 whenever bm, and σm, are Lpschz n x and bm, + σm, s bounded, unformly n m We now explan how o consruc a convergen scheme for he backward componen even when g and h are no Lpschz n m We assume ha hm, s Lpschz and hm, s bounded, unformly n m We make he same assumpon on gm, The 15

17 approxmaon s defned as follows: Z π π := n Ȳ +1 W +1 F Γ π π := n Ȳ ρe µde, +1, +1 F Ȳ π π +1 := Ȳ +1 F + h M s, X π, Ȳ π +1, Z π, Γ π ds F 228 for, +1, wh he ermnal condon Ȳ π n = gm n, X π n Wh hs scheme he proof of Proposon 21 can be modfed as follows We keep he same defnon for Z π and U π bu we now defne Y π as Y π = Ȳ π n + Zs π dw s + +1 h M s, X π, Ȳ π +1, Z π, Γ π ds F U π s e µds, de Le us nroduce he processes H T and H T defned, for, +1, by H := hm, X, Y, Z, Γ +1, H := n h M s, X, Y, Z, Γ ds F Observe ha hm, Θ n +1 h M s, X, Y +1, Z, Γ ds F can be wren as hm, Θ H + H H + H +1 n h M s, X, Y +1, Z, Γ ds F Recall from of hs secon ha of Theorem 21 holds for 227 Followng he argumens of Remark 26, we ge H +1 n By of Theorem 21 and 26, +1 hm, Θ H 2 d C 2 h M s, X, Y +1, Z, Γ ds F 2 C 2 n +1 n 2 + Z Z 2 + Γ Γ 2 d We hen deduce from he same argumens as n he proof of Proposon 21 ha rr n Y, Z, U C2 n 1/2 + Z Z H 2 + Γ Γ H 2 + H H H 2, where Z Z H 2 + Γ Γ H 2 + H H H 2 ɛn for some map ɛ such ha ɛn when n Ths shows ha he approxmaon scheme s convergen Recall from of hs secon ha he resuls of Theorem 21 for hs sysem Snce here β =, follows ha Z Z H 2 + Γ Γ H 2 of hs secon We leave he sudy of H H H 2 16 o furher research C 2n 1 2, see

18 3 Mallavn calculus for FBSD In hs secon, we prove ha he soluon Y, Z, U of 23 s smooh n he Mallavn sense under he addonal assumpons C X 1 : b, σ and β, e are C 1 b C Y 1 : g and h are C 1 b unformly n e We shall also show ha her dervaves are smooh under he sronger assumpons C X 2 : b, σ and β, e are Cb 2 wh second dervaves bounded by K, unformly n e C Y 2 : g and h are Cb 2 wh second dervaves bounded by K Ths wll allow us o provde represenaon and regulary resuls for Y, Z and U n Secon 4 Under C X 1 -C Y 1, hese resuls wll mmedaely mply of Theorem 21, whle of Theorem 21 wll be obaned by adapng he argumens of 29 under he addonal assumpon H 1 Under C X 2 -C Y 2, hese resuls wll also drecly mply of Theorem 21 The proof of Theorem 21 wll hen be compleed by appealng o an approxmaon argumen Ths secon s organzed as follows Frs we derve some properes for he Mallavn dervaves of sochasc negrals wh respec o µ Nex, we recall some well known resuls on he Mallavn dervaves of he forward process X Fnally, we dscuss he Mallavn dfferenably of he soluon of Generales We sar by nroducng some addonal noaons We denoe by D he Mallavn dervave operaor wh respec o he Brownan moon and by ID 1,2 he space of random varables H L 2 Ω, F T, P; R such ha D H exss for all T and sasfy H 2 ID := H 2 T + D 1,2 s H 2 ds < As usual we exend hese noaons o vecor or marx valued processes by akng he Mallavn dervave componenwse and by consderng he suable norm We hen defne H 2 ID 1,2 as he se of elemens ξ H 2 such ha ξ ID 1,2 for almos all T and such ha, afer possbly passng o a measurable verson, T ξ 2 H 2 ID 1,2 := ξ 2 H + D 2 s ξ 2 H 2ds < We also defne L 2 λ ID1,2 as he compleon of he se { } L 2 λ ID 1,2 := Vec ψ = ξϑ : ξ H 2 ID 1,2, F W, ϑ L 2 λf µ, ψ L 2 λ ID 1,2 < 17

19 for he norm ψ 2 L 2 λ ID1,2 := ψ 2 L 2 λ + T D s ψ 2 L 2 λds Here, H 2 ID 1,2, F W resp L 2 λ Fµ denoes he se of F W -adaped resp F µ -adaped elemens of H 2 ID 1,2 resp L 2 λ Moreover, we exend he defnon of H 2 and L 2 λ o processes wh values n M d and R d n a naural way The wo followng Lemmas are generalzaons of Lemma 33 and Lemma 34 n 22 whch correspond o he case where s fne, see also Lemma 23 n 21 for he case of Iô negrals Lemma 31 Assume ha ψ L 2 λ ID1,2 Then, and D s H := H := T T ψ e µde, d ID 1,2 D s ψ e µde, d for all s T Proof Assume ha ψ = ξϑ where ξ H 2 ID 1,2, F W, ϑ L 2 λ Fµ and ψ L 2 λ ID 1,2 < Then, T ψ e µde, d = T ξ ϑ eµde, d Snce λ <, we oban by condonng by µ ha T D s ξ ϑ eµde, d = T T ξ D s ξ ϑ eµde, d, ϑ eλded whle, see 2, T D s ξ ϑ eλded = T D s ξ ϑ eλded = T D s ξ ϑ eλded Ths proves he requred resul when ψ L 2 λ ID 1,2 For he general case, we consder a sequence ψ n n n L 2 λ ID 1,2 whch converges n L 2 λ ID1,2 o ψ Then H n := T ψn e µde, d s a Cauchy sequence n ID 1,2 whch converges o H Thus, H ID 1,2 Snce D s H n s T converges n H 2 o T D sψ e µde, d s T, hs proves he requred resul 18

20 Lemma 32 Fx ξ, ψ H 2 L 2 λ and assume ha T T H := ξ dw + ψ e µde, d ID 1,2 Then, ξ, ψ H 2 ID 1,2 L 2 λ ID1,2 and D s H := ξ s + T d =1 where ξ denoes he ranspose of ξ T D s ξ dw + D s ψ e µde, d, Proof Le SW denoe he se of random varables of he form T T H W = φ f 1 dw,, f κ dw wh κ 1, φ C b and f :, T R d s a bounded measurable map for each κ Then, he se H := Vec { H W H µ : H W SW, H µ L Ω, F µ T, H W H µ = } s dense n ID 1,2 {H L 2 Ω, F, P : H = } for ID 1,2 Thus, suffces o prove he resul for H of he form H W H µ where H W SW, H µ L Ω, F µ T and H W H µ = By he represenaon heorem, here exss ψ L 2 λ such ha H µ = H µ + T ψ e µde, d and by Ocone s formula, see eg Proposon 135 n 19, H W = H W + Thus follows from Iô s Lemma ha H = T H µ D H W F W T D H W F W T dw + H W dw ψ e µde, d where H µ = H µ F and H W = H W F Furhermore he wo negrands belong respecvely o H 2 ID 1,2 and L 2 λ ID1,2 Thus, Lemma 31 above and 146 n 2 conclude he proof 19

21 32 Mallavn calculus on he Forward SD In hs secon, we recall well-known properes concernng he dfferenably n he Mallavn sense of he soluon of a Forward SD In he case where β = he followng resuls saed n eg 19 The exenson o he case β s easly obaned by condonng by µ, see eg 24 for explanaons n he case where s fne, or by combnng Lemma 31 wh a fxed pon procedure as n he proof of Theorem 221 n 19, see also Proposon 32 below From now on, gven a marx A, we shall denoe by A s -h column For k d, we denoe by D k he Mallavn dervave wh respec o W k Proposon 31 Assume ha C X 1 holds, hen X ID 1,2 for all T For all s T and k d, Ds k X adms a verson χ s,k whch solves on s, T χ s,k = σ k X s + s bx r χ s,k r dr+ s d j=1 σ j X r χ s,k r dw j r + s βx r, eχ s,k r µdr, de If moreover C X 2 holds, hen Ds k X ID 1,2 for all s, T and k d For all u T and l d, DuD l s k X adms a verson χ u,l,s,k whch solves on u s, T χ u,l,s,k = σ k X s χ u,l s + σ l X u χ s,k u d + bx r χ u,l,s,k r + bx r χ u,l r χ s,k r dr + + s s s =1 d σ j X r χ u,l,s,k r + j=1 d =1 βx r, eχ u,l,s,k r + σ j X r χ u,l r χ s,k r dwr j 31 d =1 βx r, e χ u,l r χ s,k r µdr, de Remark 31 Fx p 2 and r s u T Under C X 1, follows from Lemma 51 appled o X and χ s ha χ s p S p C p 1 + X p 32 χ s u χ s p C p u 1 + X p 33 χ s χ r p S p C p s r 1 + X p 34 If moreover C X 2 holds hen smlar argumens show ha χ r,s p S p C p 1 + X 2p, 35 where χ r,s = χ r,l,s,k l,k d 2

22 Remark 32 Under C X 1, we can defne he frs varaon process X of X whch solves on, T X = I d + + bx r X r dr + d σ j X r X r dwr j j=1 βx r, e X r µdr, de 36 Moreover, under H 1, see Remark 27, X 1 s well defned and solves on, T d X 1 = I d X 1 r bx r σ j X r σ j X r dr + X 1 r j=1 βx r, eλdedr d j=1 X 1 r σ j X r dw j r X 1 r βx r, e + I d 1 βx r, eµde, dr 37 Ths can be checked by smply applyng Iô s Lemma o he produc X X 1, see 19 p 19 for he case where β = Remark 33 Fx p 2 Under H 1 -C X 1, follows from Remark 27 and Lemma 51 appled o X and X 1 ha X S p + X 1 S p C p 38 Remark 34 Assume ha H 1 -C X 1 holds and observe ha χ s = χ s,k k d and X solve he same equaon up o he condon a me s By unqueness of he soluon on, T, follows ha χ s r = X r X s 1 σx s 1 s r for all s, r T Mallavn calculus on he Backward SD In hs secon, we generalze he resul of Proposon 31 n 22 Le us denoe by B 2 ID 1,2 he se of rples Y, Z, U B 2 such ha Y ID 1,2 for all T and Z, U H 2 ID 1,2 L 2 λ ID1,2 Proposon 32 Assume ha C X 1 -C Y 1 holds The rples Y, Z, U belongs o B 2 ID 1,2 For each s T and k d, he equaon Υ s,k T = gx T χ s,k T + T hθ r Φ s,k r dr 21 T ζr s,k dw r Vr s,k e µde, dr 31

23 wh Φ s,k := χ s,k, Υ s,k, ζ s,k, Γ s,k and Γ s,k := ρev s,k eλde, adms a unque soluon Moreover, Υ s,k, ζ s,k, V s,k s, T s a verson of Ds k Y, Ds k Z, Ds k U s, T Assume furher ha C X 2 -C Y 2 holds Then, for each s T and k d, D k s Y, D k s Z, D k s U belongs o B 2 ID 1,2 For each u T and l d, he equaon Υ u,l,s,k = + χ u,l T T T HgXT χ s,k T + gx T χ u,l,s,k T hθ r Φ u,l,s,k + D luθ r HhΘr D ks Θ r dr ζ u,l,s,k dw r T V u,l,s,k r e µde, dr 311 where Φ u,l,s,k := χ u,l,s,k, Υ u,l,s,k, ζ u,l,s,k, Γ u,l,s,k wh Γ u,l,s,k := ρev u,l,s,k eλde, and Hg resp Hh denoes he Hessan marx of g resp h, adms a unque soluon Moreover, Υ u,l,s,k, ζ u,l,s,k, V u,l,s,k u,s, T s a verson of DuD l s k Y, Z, U u,s, T Proof For ease of noaons, we only consder he case d = 1 and om he ndexes k and l n he above noaons We proceed as n Proposon 53 n 12 Combned wh C 1 X -C1 Y and 32, Lemma 52 mples ha Υ s, ζ s, V s s well defned for each s T and ha we have sup Υ s, ζ s, V s p B C p p 1 + X p for all p s T We now defne he sequence Θ n := X, Y n, Z n, Γ n as follows Frs, we se Y, Z, U :=,, Then, gven Θ n 1, we defne Y n, Z n, U n as he unque soluon n B 2 of T Y n = gx T + T T hθ n 1 r dr Zr n dw r Ur n e µde, dr and se Γ n = ρeu n eλde From he proof of Lemma 24 n 26, Y n, Z n, U n n s a Cauchy sequence n B 2 whch converges o Y, Z, U Moreover, usng Lemma 32 and an nducve argumen, one obans ha Y n, Z n, U n B 2 ID 1,2 For s T, se Υ s,n, ζ s,n, V s,n := D s Y n, D s Z n, D s U n, Φ s,n := χ s, Υ s,n, ζ s,n, Γ s,n, Ξ s,n := χ s, Υ s,n, ζ s,n, U s,n and Ξ s := χ s, Υ s, ζ s, U s, where Γ s,n := ρev s,n eλde By Lemma 32 agan, we have T Υ s,n = gx T χ s T + hθ n 1 r Φ s,n 1 r dr T T ζr s,n dw r Vr s,n e µde, dr

24 Fx I N o be chosen laer, se δ := T/I and τ := δ for I By 51 of Lemma 52, we have G s,n := Ξ s Ξ s,n 4 S 4 B 4 τ,τ +1 where C 4 Υ s τ +1 Υ s,n τ A s,n 1 + B s,n A s,n 1 := { hθ n 1 hθ}φ s 4 H 4 τ,τ +1 τ+1 B s,n 1 := τ hθ n 1 r 4 {Φ s r Φ s,n 1 }dr r Recallng ha ρ and he dervaves of h are bounded, we deduce from Cauchy-Schwarz and Jensen s nequaly ha B s,n 1 C 4 δ 2 G s,n 1, 315 whch combned wh an nducve argumen and leads o sup G s,n < for all n 316 s T Snce he dervaves of h are also connuous and Θ n 1 converges o Θ n S 2 B 2, we deduce from ha, afer possbly passng o a subsequence, lm sup A s,n 1 = 317 n s T I follows from ha for I large enough here s some α < 1 such ha for any ε > we can fnd N, ndependen of s, such ha G s,n C 4 Υ s τ +1 Υ s,n 1 τ ε + αg s,n 1 for n N 318 Snce Υ s T = Υs,n 1 T, we deduce ha for = I 1 and n N sup G s,n s T I 1 ε + α n N sup s T G s,n I 1 By 316, follows ha sup s T G s,n I 1 as n In vew of 318, a sraghforward nducon argumen shows ha, for all I 1, sup s T G s,n as n so ha, summng up over, we ge sup Ξ s Ξ s,n S 4 B 4 s T n 319 Snce Y n, Z n, U n converges o Y, Z, U n B 2, hs shows ha Y, Z, U B 2 ID 1,2 and ha here s a verson of DY, DZ, DU gven by Υ, ζ, V 23

25 In vew of and C X 2 -C Y 2, follows from Lemma 52 ha Υ u,s, ζ u,s, V u,s s well defned for u, s T and ha we have sup Υ u,s, ζ u,s, V u,s p B C p p 1 + X 2p for all p 2 32 u,s T Usng Lemma 32, 313 and an nducve argumen, we hen deduce ha DY n, DZ n, DU n B 2 ID 1,2 and T Υ u,s,n = χ u T HgX T χ s T + gx T χ u,s T + + T Φ u,n 1 r HhΘ n 1 r Φ s,n 1 r dr T hθ n 1 r Φr u,s,n 1 dr T ζr u,s,n dw r Vr u,s,n e µde, dr, where Υ u,s,n, ζ u,s,n, V u,s,n, Φ u,s,n := D u Υ s,n, ζ s,n, V s,n, Φ s,n By, Y n, Z n, U n goes o Y, Z, U n B 2 and Υ s,n, ζ s,n, V s,n converges o Υ s, ζ s, V s n B 4 Moreover, 319 mples sup n 1 so ha, by domnaed convergence, C Y 2 and 32, sup Υ s,n, ζ s,n, V s,n 4 B <, s T Φ u,n HhΘ n Φ s,n Φ u HhΘΦ s H 2 + hθ n hθ Φ u,s H 2, n afer possbly passng o a subsequence The res of he proof follows sep by sep he argumens of excep ha we now work on S 2 B 2 nsead of S 4 B 4 Proposon 33 Assume ha C X 1 -C Y 1 holds For each k d, he equaon T T T Y k = gx T XT k + hθ r Φ k rdr Zr k dw r Ur k e µde, dr 322 wh Φ k = X k, Y k, Z k, Γ k and Γ k := ρe U k eλde, adms a unque soluon Y k, Z k, U k Moreover, here s a verson of ζ s,k, Υ s,k, V s,k s, T gven by { Y, Z, U X s 1 σ k X s 1 s } s, T where Y s he marx whose k-column s gven by Y k and Z, U are defned smlarly Proof In vew of Proposon 32 and 39, hs follows mmedaely from he unqueness of he soluon of 31 Remark 35 I follows from Lemma 52 and 38 ha Y, Z, U B p C p for all p

26 4 Represenaon resuls and pah regulary for he BSD In hs secon, we use he above resuls o oban some regulary for he soluon of he BSD 23 under C X 1 -C Y 1 -H 1 or C X 2 -C Y 2 Smlar resuls under H 1 or H 2 wll hen be obaned by usng an approxmaon argumen Fx u, s,, x, T 3 R d and k, l d In he sequel, we shall denoe by X, x he soluon of 22 on, T wh nal condon X, x = x, and by Y, x, Z, x, U, x he soluon of 23 wh X, x n place of X We defne smlarly Υ s,k, x, ζ s,k, x, V s,k, x, Y, x, Z, x, U, x and Υ u,l,s,k, x, ζ u,l,s,k, x, V u,l,s,k, x Observe ha, wh hese noaons, we have X, X, Y, X, Z, X, U, X = X, Y, Z, U 41 Represenaon We sar hs secon by provng useful bounds for he deermnsc maps defned on, T R d by u, x := Y, x, u, x := Y, x, v s,k, x := Υ s,k, x and w u,l,s,k, x := Υ u,l,s,k, x, where u, s, T 2 and k, l d Proposon 41 Assume ha C X 1 and C Y 1 hold, hen, u, x + v s,k, x C x and u, x C 2 41 for all s, T, k d and x R d Assume ha C X 2 and C Y 2 hold, hen, w u,l,s,k, x C x 2, 42 for all u, s, T, l, k d and x R d Proof When, x =, X, he resul follows from 25 n Remark 21, 312, 32 and 323 The general case s obaned smlarly by changng he nal condon on X Proposon 42 Assume ha C X 1 and C Y 1 hold There s a verson of Z gven by Υ T whch sasfes Z p S p C p 1 + X p 43 25

27 Assume furher ha C X 2 and C Y 2 hold, hen, for each k d, here s a verson of ζ s,k s, T gven by Υ,l,s,k l d s, T whch sasfes sup ζ s,k p S C p p 1 + X 2p 44 s T Proof Here agan we only consder he case d = 1 and om he ndexes k, l By Proposon 32, Y, Z, U belongs o B 2 ID 1,2 and follows from Lemma 32 ha D s Y = Z s s hθ r D s Θ r dr + s D s Z r dw r + s D s U r e µde, dr, 45 for < s T Takng s = leads o he represenaon of Z Thus, afer possbly passng o a suable verson, we have Z = D Y = Υ By unqueness of he soluon of for any nal condon n L 2 Ω, F a, we have Υ = v, X The bound on Z hen follows from Proposon 41 combned wh 25 of Remark 21 Under C X 2 and C Y 2, he same argumens appled o Υ s, ζ s, V s nsead of Y, Z, U leads o he second clam, see of Proposon 32, of Proposon 41 and recall 25 Proposon 43 Defne Ũ by Ũ e := u, X + βx, e lm r u r, X r Then Ũ s a verson of U and sasfes Assume ha C X 1 and C Y 1 holds Defne Ũ by sup Ue p S C p p 1 + X p 46 e Ũe := u, X + βx, e lm r u r, X r Then Ũ s a verson of U and sasfes sup Ue p S C p p 47 e Assume ha C X 1 and C Y 1 holds, hen, for each k d, here s a verson of V s,k s, T gven by Ṽ s,k s, T defned as I sasfes Ṽ s,k e := v s,k, X + βx, e lm r v s,k r, X r sup sup V s,k e p S C p p 1 + X p 48 e s T 26

28 Remark 41 We wll see n Proposon 44 below ha u s connuous under C X 1 so ha C Y 1 and U e := u, X + βx, e u, X One could smlarly show ha v s,k and u are connuous under C X 2 and C Y 2 so ha V s,k e := v s,k, X + βx, e v s,k, X Ũe := u, X + βx, e u, X However, snce hs resuls no requred for our man heorem, we do no provde s proof Proof of Proposon 43 By unqueness of he soluon of for any nal condon n L 2 Ω, F, P; R d a me, U eµde, {} = Y Y = Hence, T Ũe U e 2 µde, d =, whch, by akng expecaon, mples T Ũe U e 2 λded Ũ eµde, {} = The bound on U follows from 41 and 25 The wo oher clams are proved smlarly by usng Pah regulary Proposon 44 Assume ha C X 1 and C Y 1 hold Then, { u 1, x 1 u 2, x 2 2 C x x 1 x 2 2} for all 1 2 T and x 1, x 2 R 2d Proof For A denong X,Y,Z or U we se A := A, x for = 1, 2 and δa := A 1 A 2 By 56 of Lemma 51, we derve δx 2 S 2 2,T C 2 { x1 x x } 49 27

29 Pluggng hs esmae n 51 of Lemma 52 leads o δy, δz, δu 2 B 2 2,T C 2 { x1 x x } 41 Now, observe ha u 1, x 1 u 2, x 2 2 = Y 1 1 Y Y 1 C 2 2 Y Y 1 2 Y Pluggng 43 and 46 n 27, we ge Y 1 2 Y C x , whch, combned wh 41, leads o he frs clam Corollary 41 Assume ha C X 1 and C Y 1 hold There s a verson of Y, U such ha sup Y r Y s 2 + sup sup U r e U s e 2 r s, e r s, C X 2 s, for all s T If moreover C X 2 and C Y 2 hold, hen here s a verson of Z such ha Z Z s 2 C X 4 s, for all s T Proof Observe ha Y = u, X by unqueness of he soluon of Thus, pluggng 25 and 26 n Proposon 44 gves he upper-bound on sup r s, Y r Y s 2 The upper-bound on sup e sup r s, U r e U s e 2 s obaned smlarly by usng he represenaon of U gven n Remark 41 By Proposon 42, a verson of Z s gven by Υ so ha Z Z s 2 C 2 Υ Υ s 2 + Υ s Υ s s 2 By 59 of Lemma 52, 32, 44 and 48, we have Υ s Υ s s 2 C X 4 s By pluggng 34 n 51 of Lemma 52, we hen deduce ha Υ Υ s 2 C X 2 s 28

30 Proposon 45 Assume ha H 1 -C X 1 -C Y 1 hold Then for all n 1 n 1 +1 = Z Z 2 C 2 n 1 Proof 1 We denoe by x h resp y h, z h, γ h he graden of h wh respec o s x varable resp y, z, γ We frsnroduce he processes Λ and M defned by Λ := exp y hθ r dr, M := 1 + M r z hθ r dw r Snce h has bounded dervaves, follows from Iô s Lemma and Proposon 42 ha T Λ M Z = M T Λ T gx T χ T + x hθ r χ r + γ hθ r Γ r Λr dr F By Remark 34 and Proposon 33, we deduce ha T Λ M Z = M T Λ T gx T X T + F r Λ r dr F X 1 σx where he process F s defned by F r = x hθ r X r + γ hθ r Γ r for r T I follows ha Λ M Z = { G F } F r Λ r dr X 1 σx 411 where G := M T Λ T gx T X T + T F r Λ r dr By 38 and 47, we deduce ha G p Cp for all p Se m s := G F s and le ζ, Ṽ H2 L 2 λ wh values n Md R d be defned such ha m s = G T ζ r dw r T s s Ṽ r e µde, dr Applyng 412 and Lemma 52 o m, ζ, Ṽ mples ha m, ζ, Ṽ B p C p for all p

31 Usng C X 1, 38, 47, 413, applyng Lemma 51 o M 1 and usng Iô s Lemma, we deduce from he las asseron ha can be wren as where Z := ΛM 1 m Z = Z + µ r dr + F r Λ r dr X 1 σ r dw r + and µ, σ and β are adaped processes sasfyng where β r e µde, dr, Z p S p C p for all p 2, 414 A p,t C p for all p A p s, := µ p + σ p + β p H p H p L p s, s, λ,s,, s T 2 Observe ha Z = Z σx P as snce he probably of havng a jump a me s equal o zero I follows ha, for all n and, +1, Z Z 2 C 2 I 1, + I 2, 416 where I 1, := Z Z 2 σx 2 and I 2, := σx σx 2 Z 2 Observng ha I 1, = Z Z 2 F σx 2 +1 C 2 µ r 2 + σ r 2 + β r e 2 λde dr σx 2 we deduce from Hölder nequaly, 25 and he lnear growh assumpon on σ ha n 1 +1 = T I 1,d C 2 n 1 µ r 2 + σ r 2 + β r e 2 λde dr sup σx 2 T C 2A 4,T 1 2 n

32 Usng he Lpschz connuy of σ, we oban I 2, C 2 X X 2 Z Now observe ha for each k, l d X k X k 2 Z l 2 C 2 Z l Z l 2 X k 2 + X k Z l X k Zl Argung as above, we oban n 1 +1 = Z l Z l 2 X k 2 C2 1 + A 4,T 1 2 n 1 42 Moreover, we deduce from he lnear growh condon on b, σ, β and 25, 414 and 415 ha X k Zl can be wren as X k Z l = X k Z l + ˆµ kl r dr + ˆσ kl r dw r + ˆβ kl r e µde, dr where ˆµ kl, ˆσ kl and ˆβ kl are adaped processes sasfyng ˆµ kl H 2 + ˆσ kl H 2 + ˆβ kl L 2 λ C 2 I follows ha n 1 +1 = X k Z l X k Zl 2 whch combned wh 418, 419 and 42 leads o n 1 +1 = C 2 n 1 ˆµ kl 2 H + 2 ˆσkl 2 H + ˆβ kl 2 2 L 2 λ I 2,d C 21 + A 4,T 1 2 n The proof s concluded by pluggng n 416 and recallng 415 We now complee he proof of Theorem 21 Proof of Theorem 21 1 We frs prove Observe ha he second asseron s a drec consequence of 224 and Remark 24 We frs show ha 224 holds under H 1 and C Y 1 We consder a Cb densy q on R d wh compac suppor and se b k, σ k, β k, ex = k R d b, σ, β, e x q kx x d x d For large k N, hese funcons are bounded by 2K a Moreover, hey are K-Lpschz and Cb 1 Usng he connuy of σ, one also easly checks ha σk s sll nverble By H 1 and Remark 27, for each e and x R d, I d + β k x, e s nverble wh 31

33 unformly bounded nverse We denoe by X k, Y k, Z k, U k he soluon of wh b, σ, β replaced by b k, σ k, β k Snce b k, σ k, β k converges ponwse o b, σ, β, one easly deduces from Lemma 51 and Lemma 52 ha X k, Y k, Z k, U k converges o X, Y, Z, U n S 2 B 2 Snce he resul of Proposon 45 holds for X k, Y k, Z k, U k unformly n k, hs shows ha holds under H 1 and C Y 1 We now prove ha 224 holds under H 1 Le X, Y k, Z k, U k be he soluon of wh h k nsead of h, where h k s consruced by consderng a sequence of molfers as above For large k, h k s bounded by 2K By Lemma 52, Y k, Z k, U k converges o Y, Z, U n S 2 B 2 whch mples by argung as above 2 The same approxmaon argumen shows ha of Corollary 41 holds rue whou C X 1 -C Y 1 Snce ρ s bounded and λ <, hs leads o 223 Now observe ha sup Γ Γ 2 2 sup Γ Γ Γ Γ 2,+1,+1 where, by Jensen s nequaly and he fac ha Γ Γ Γ 2 +1 n 2 Γ Γ s ds n s F -measurable, +1 Thus, 223 mples Γ Γ 2 S 2 C 2 n 1 and Γ Γ 2 H 2 C 2 n 1 Γ Γ s 2 ds 3 Iem s proved smlarly by usng of Corollary 41 5 Appendx: A pror esmaes For sake of compleeness, we provde n hs secon some a pror esmaes on soluons of forward and backward SD s wh jumps provde all he deals The proofs beng sandard, we do no Proposon 51 Gven ψ L 2 λ, le M be defned on, T by M = ψ se µds, de Then, for all p 2, k p ψ p L p λ,,t M p S p,t K p ψ p L p λ,,t 51 where k p, K p are posve numbers ha depend only on p, λ and T Proof 1 We frs prove he lef hand-sde Observe ha for a sequence a I of non-negave numbers we have α 1 α a α max a a a for all α 1 52 I I I 32 I

34 I follows ha ψ p L p λ,,t T = ψ s e p µde, ds T ψ s e 2 µde, ds p 2, snce p/2 1, and he resul follows from Burkholder-Davs-Gundy nequaly see eg 23 p We now prove he rgh hand-sde nequaly for p 1 We follow he nducve argumen of 5 For p 1, 2, we deduce from Burkholder-Davs-Gundy nequaly and 52 ha sup M s p s T T K p p ψ s e 2 2 µde, ds snce 2/p 1 Ths mples he requred resul T K p ψ s e p µde, ds We now assume ha he nequaly s vald from some p > 1 and prove has also rue for 2p Se M = ψ se 2 µde, ds for, T Then, M, M T = M T + T ψ se 2 λdeds Applyng Burkholder-Davs-Gundy nequaly, we oban sup s T M s 2p M, M p T where T p M, M p T K p M T p + ψ s e 2 λdeds and K p denoes a generc posve number ha depends only on p Applyng 51 o M, we oban M T p T K p ψ s e 2p λdeds On he oher hand, follows from Hölder nequaly ha T ψ s e 2 λdeds T 1 ψ s e 2p p λdeds T λ 1 q where q = p/p 1, recall ha p > 1 Combnng he wo lasnequales leads o he requred resul We now consder some measurable maps b : Ω, T R d R d σ : Ω, T R d M d β : Ω, T R d R d f : Ω, T R R d L 2,, λ; R, = 1, 2 33

35 Here L 2,, λ; R s endowed wh he naural norm ae 2 λde 1 2 Omng he dependence of hese maps wh respec o ω Ω, we assume ha for each T b,, σ,, β,, e and f, are as K-Lpschz connuous unformly n e for β We also assume ha f,, b, s F-progressvely measurable, and σ,, β, s F-predcable, = 1, 2 Gven some real number p 2, we assume ha b,, σ, and f, are n H p, and ha β,, s n L p λ For 1 2 T, X L 2 Ω, F, P; R d for = 1, 2, we now denoe by X he soluon on, T of X = X + b s, X sds + σ s, X sdw s + β s, e, X s µde, ds 53 Lemma 51 { } X 1 p C S p p X 1 p + b 1, p + σ 1, p + β 1,, p H p H p L p 1,T 1,T 1,T λ, 1,T 54 Moreover, for all 1 s T, sup Xu 1 Xs 1 p s u where A 1 p s defned as X 1 p + sup b 1 s, p + 1 s T sup σ 1 s, p + 1 s T C p A 1 p s, 55 { } sup β 1 s,, e p λde 1 s T, and, for 2 T, δx p S p 2,T C p X 1 X 2 p + A 1 p 2 1 T p + C p δ b d + δ σ p + δ β p H p L p 2,T λ, 2,T 2 56 where δx := X 1 X 2, δ b = b 1 b 2, X 1 and δ σ, δ β are defned smlarly Lemma 52 Le f be equal o f 1 or f 2 Gven Ỹ Lp Ω, F T, P; R, he backward SD T Y = Ỹ + fs, Y s, Z s, U s ds + T Z s dw s + T 34 U s e µde, ds 57

36 has a unque soluon Y, Z, U n B 2 I sasfes T Y, Z, U p B C p p Ỹ p + Moreover, f A p := Ỹ p + sup T f, p <, hen p f, d { } sup Y u Y s p C p A p s p + Z p + U p H p L p s u s, λ,s, Fx Ỹ 1 and Ỹ 2 n L p Ω, F T, P; R and le Y, Z, U be he soluon of 58 wh Ỹ, f n place of Ỹ, f, = 1, 2 Then, for all T, T δy, δz, δu p C B p p δỹ p +,T p δ f r dr where δỹ := Ỹ 1 Ỹ 2, δy := Y 1 Y 2, δz := Z 1 Z 2, δu := U 1 U 2 and δ f := f 1 f 2, Y 1, Z 1, U 1 51 Proof of Lemma 51 Applyng Burkholder-Davs-Gundy nequaly see eg 23 p 175 and usng Proposon 51, we ge T sup Xs 1 p C p X 1 p + s 1,T 1 p b 1 s, Xs 1 ds + C p σ 1, X 1 p + β 1, X 1, p H p L p 1,T λ, 1,T The esmae 54 s hen deduced by usng he Lpschz properes of b 1, σ 1, β 1 and Gronwall s Lemma The esmae 55 s obaned by applyng he same argumens o he process X 1 Xs 1 p on s, To oban he las asseron 56, we frs apply he above argumen o δx = X 1 X 2 on 2, T Then, decomposng b 1, X 1 b 2, X 2 as δ b + b 2, X 1 b 2, X 2 and dong he same for σ and β, he Lpschz properes of b2, σ 2, β 2 combned wh Gronwall s lemma leads o T p sup δx s p C p X 1 2 X 2 p + δ b d + δ σ p + δ β p H p L p s 2,T 2 2,T λ, 2,T We hen conclude by usng he 55 Proof of Lemma 52 See 26 and 2 for exsence and unqueness We dvde, T n N nervals τ, τ +1 of equal lengh δ := T/N For τ s τ +1 τ+1 Y s Y τ+1 + fr, Y r, Z r, U r dr F s, 35

37 whch, by Doob and Jensen s nequales, mples sup Y s p C p Y τ+1 p + s τ +1 τ+1 p fr, Y r, Z r, U r dr Moreover, follows from Burkholder-Davs-Gundy nequaly see eg 23 p 175 and Proposon 51 ha Z p + U p H p L p,τ +1 λ,,τ +1 τ+1 p C p Y τ+1 p + fr, Y r, Z r, U r dr + sup Y s p s τ +1 Thus, usng Hölder and Jensen s nequales, we oban τ+1 p Y, Z, U p C B p p Y τ+1 p + fr, Y r, Z r, U r dr,τ +1 T p C p { Y τ+1 p + f, τ+1 d + Y p S p u,τ +1 du } + δ p/2 Z p + U p H p L p,τ +1 λ,,τ +1 by he Lpschz connuy assumpon on f For δ smaller han 2C p 2/p, we hen ge Y, Z, U p B p,τ +1 C p { Y τ+1 p T p + d f, + Usng Gronwall s Lemma, we deduce ha Y p S p τ,τ +1 C p { Y τ+1 p T + τ+1 p } f, d Pluggng hs esmae no he prevous upper bound, we fnally ge Ths leads o 58 Y, Z, U p B p τ,τ +1 T p C p Y τ+1 p + f, d By Burkholder-Davs-Gundy nequaly and Proposon 51, we have p sup Y u Y s p C p fr, Y r, Z r, U r dr s u s { } + C p Z p + U p H p L p s, λ,s, } Y p S p u,τ +1 du Usng he Lpschz connuy assumpon on f ogeher wh 58 leads o 59 The esmae 51 s obaned by applyng smlar argumens o δy, δz, δu 36

38 References 1 Anonell F and A Kohasu-Hga 2 Flraon sably of backward SD s Sochasc Analyss and Is Applcaons, 18, Barles G, R Buckdahn and Pardoux 1997 Backward sochasc dfferenal equaons and negral-paral dfferenal equaons Sochascs Sochascs Repors, 6, Bally V, and G Pages 22 A quanzaon algorhm for solvng dscree me muldmensonal opmal soppng problems Bernoull, 9 6, Becherer D 25 Bounded soluons o Backward SD s wh Jumps for Uly Opmzaon and Indfference Hedgng Preprn, Imperal College London 5 Bcheler K and J Jacod 1983 Calcul de Mallavn pour des dffusons avec sau: exsence d une densé dans le cas undmensonel Sémnare de Probablé, 17, Bouchard B and N Touz 24 Dscree-Tme Approxmaon and Mone- Carlo Smulaon of Backward Sochasc Dfferenal quaons Sochasc Processes and her Applcaons, 111 2, Brémaud P 1981 Pon Processes and Queues - Marngale Dynamcs, Sprnger- Verlag, New-York 8 Brand P, B Delyon, and J Mémn 21 Donsker-ype heorem for BSD s lecronc Communcaons n Probably, 6, Chevance D 1997 Numercal Mehods for Backward Sochasc Dfferenal quaons In Numercal mehods n fnance, d LCG Rogers and D Talay, Cambrdge Unversy Press, Coque F, V Mackevčus, and J Mémn 1998 Sably n D of marngales and backward equaons under dscrezaon of flraon Sochasc Processes and her Applcaons, 75, Douglas J Jr, J Ma, and P Proer 1996 Numercal Mehods for Forward- Backward Sochasc Dfferenal quaons Annals of Appled Probably, 6, l Karou N, S Peng and M-C Quenez 1997 Backward sochasc dfferenal equaons n fnance Mahemacal fnance, 7 1,

39 13 yraud-losel A 25 Backward Sochasc Dfferenal quaons wh nlarged Flraon Opon Hedgng of an nsder rader n a fnancal marke wh Jumps To appear n Sochasc processes and her Applcaons 14 Fujwara T and H Kuna 1989 Sochasc dfferenal equaons of Jump ype and Lévy processes n dffeomorphsm group J Mah Kyoo Unv, 25 1, Gobe, JP Lemor and X Warn 25 Rae of convergence of emprcal regresson mehod for solvng generalzed BSD preprn cole Polyechnque, RI Longsaff F A and R S Schwarz 21 Valung Amercan Opons By Smulaon : A smple Leas-Square Approach Revew of Fnancal Sudes, 14, Ma J, P Proer, J San Marn, and S Torres 22 Numercal Mehod for Backward Sochasc Dfferenal quaons Annals of Appled Probably, 12 1, Ma J, P Proer, and J Yong 1994 Solvng forward-backward sochasc dfferenal equaons explcly - a four sep scheme Probably Theory and Relaed Felds, 98, Nualar D 1995 The Mallavn Calculus and Relaed Topcs Sprnger Verlag, Berln 2 Nualar D and Pardoux 1988 Sochasc calculus wh ancpang negrands Prob Theory and Rel Felds, 78, Pardoux and S Peng 1992 Backward sochasc dfferenal equaons and quaslnear parabolc paral dfferenal equaons Lecure Noes n Conrol and Inform Sc, 176, Pardoux, F Pradelles and Z Rao 1997 Probablsc nerpreaon for a sysem of semlnear parabolc paral dfferenal equaons Ann Ins H Poncare, 33 4, Proer P 199 Sochasc negraon and dfferenal equaons Sprnger Verlag, Berln 24 Forser B, Lükebohmer and J Techmann 25 Calculaon of he greeks for jump-dffusons Preprn 38