A-posteriori estimates for backward SDEs

Transcription

1 A-poseror esmaes for backward SDEs Chrsan Bender 1, Jessca Sener 1 Aprl 4, 01 Suppose an approxmaon o he soluon of a backward SDE s pre-compued by some numercal algorhm. In hs paper we provde a-poseror esmaes on he L - approxmaon error beween rue soluon and approxmae soluon. These a-poseror esmaes provde upper and lower bounds for he approxmaon error. They can be expressed solely n erms of he approxmae soluon and he daa of he backward SDE, and can be esmaed conssenly by smulaon n ypcal suaons. We also llusrae by some numercal expermens n he conex of leas-squares Mone Carlo how he a-poseror esmaes can be appled n pracce. MSC 010: Prmary 65C30, secondary: 60H10, 65C05, 91G60 Keywords: BSDE, Numercal approxmaon, Mone Carlo smulaon 1 Inroducon Backward sochasc dfferenal equaons BSDEs were frs nroduced by Bsmu 1973 as adon equaons n he sochasc maxmum prncple. Apar from her applcaons n sochasc conrol for whch we refer e.g. o Yong and Zhou, 1999, BSDEs urned ou o be an exremely useful ool n mahemacal fnance see e.g. he survey paper by El Karou e al., Moreover, Feynman-Kac represenaon formulas for paral dfferenal equaons can be generalzed va BS- DEs, as shown e.g. by Pardoux and Peng 199. Movaed by hese applcaons, many numercal algorhms for backward sochasc dfferenal equaons BSDEs were developed durng he las years. They ypcally conss of wo seps. In he frs sep a me dscrezaon s performed. The mos popular choce for he me dscrezaon s an Euler-ype dscrezaon for BSDEs, whch was nroduced by Zhang 004 and Bouchard and Touz 004. The correspondng dscree BSDE sll requres o evaluae hgh order nesngs of condonal expecaons, whch n general canno be compued n closed form. Several echnques have been suggesed o approxmae hese condonal expecaons. Among hose are Mallavn Mone Carlo Bouchard and Touz, 004, leas-squares Mone Carlo Lemor e al., 006, quanzaon Bally and Pagès, 003, and cubaure on Wener 1 Saarland Unversy, Deparmen of Mahemacs, PO Box , D Saarbrücken, Germany. bender@mah.un-sb.de, sener@mah.un-sb.de. 1

2 A-poseror esmaes for dscree BSDEs space Crsan and Manolaraks, 010. For more nformaon on he rch leraure on numercal schemes for BSDEs we also refer o he survey par n Bender and Sener 01. The qualy of he dfferen numercal approxmaons menoned above s ypcally dffcul o udge and compare apar from rare suaons n some es examples where he rue soluon s known n closed form. As a way-ou, we sugges an a-poseror error creron n hs paper, whch can be expressed n erms of he approxmae soluon generaed by whaever algorhm on a gven me grd and he daa of he BSDE. The dea of he creron s o check how close he approxmae soluon s o solvng he correspondng SDE run forward n me on he grd and how well approxmaes he ermnal condon. In Secon, we show ha hs error creron s equvalen o he squared L -error beween he approxmae soluon and he unknown mplc backward Euler me dscrezaon of he rue soluon on he grd. From hs pon of vew he creron measures n he frs place how successful he numercal procedure for approxmang he condonal expecaon performs, once he me dscrezaon s already done. We hen also sudy o whch exen he me dscrezaon error can be measured by he a- poseror error creron. I s well known by he resuls of Zhang 004 ha he dscrezaon error beween rue soluon and mplc me dscrezaon heavly depends on he L -regulary of he Z-par of he soluon of he BSDE. We show ha he erm whch s relaed o hs regulary can be esmaed by our error creron under raher weak condons. In hs way we are able o derve upper and lower bounds of he L -error beween an approxmae soluon generaed by whaever algorhm and he rue connuous me soluon n Secon 3. We merely need o assume Lpschz connuy of he drver and L -negrably of he ermnal condon. No furher regulary condon needs o be mposed on he ermnal condon such as he popular assumpon ha he ermnal condon s a Lpschz funconal on he pah of a dffuson process. Several examples explan how our generc a-poseror esmaes can be appled under ypcal assumpons n he leraure on numercal approxmaons of BSDEs ncludng he suaon of rregular ermnal condons or he suaon of smooh daa. Fnally we apply he a-poseror creron o a es example n whch we compue he condonal expecaons numercally by he leas-squares Mone Carlo algorhm whch was made popular n he conex of Amercan opon prcng by Longsaff and Schwarz 001 and was suded for BSDEs n Lemor e al We explan how he a-poseror error creron can help o alor he algorhm concernng he bass choce, he number of me pons and he number of smulaon pahs. In parcular n suaons when he heorecal wors-case error esmaes by Lemor e al. 006 lead o prohbve smulaon coss n pracce, our creron can be appled o usfy he use of a more moderae sample sze or a small funcon bass. Ths paper s organzed as follows: In Secon we sudy he a-poseror error creron for dscree me BSDEs. The connuous me case s reaed n Secon 3, whle he numercal example s dscussed n Secon 4. A-poseror esmaes for dscree BSDEs In hs secon we provde a-poseror esmaes for some knd of dscree BSDEs lvng on a me grd π = { 0,..., N } whch sasfes 0 = 0 < 1 < < N = T. We assume ha Ω, F, G 0,T, P s a flered probably space whch carres a D-dmensonal Brownan mo-

3 A-poseror esmaes for dscree BSDEs on W = W 1,... W D, he sar denong ransposon. The flraon G can n general be larger han he augmened flraon generaed by he Brownan moon W whch we denoe by F. The me ncremens and he ncremens of he Brownan moon on he me grd π wll be abbrevaed by = 1 and W = W 1 W. E sands for he condonal expecaon E G wh respec o he larger flraon. The ype of dscree BSDE, whch we consder, s of he form Y π = Y π 1 f π, Y π, 1 E W M π 1 M π 1 M π, = 0,..., N 1, Y π N = ξ π. Here he daa f π, ξ π s gven and a soluon consss of a par Y π, M π of square negrable, G - adaped processes such ha M π s a G -marngale sarng n 0 and 1 s sasfed. Concernng he daa we assume: H G π ξ π s a square-negrable G T -measurable random varable. f π : π Ω R R D R s measurable and, for every π and y, z R D1, f π, y, z s G -measurable. Moreover f π s Lpschz n y, z wh consan K ndependen of π unformly n, ω and f π, 0, 0 s square-negrable for every π. I follows mmedaely from 1, ha he wo soluon processes Y π, M π are conneced va he propery M π 1 M π = Y π 1 E Y π 1. Under H G π s sraghforward o check by a conracon mappng argumen ha hs dscree BSDE adms a unque soluon f he mesh π = max{ 1 ; = 0,..., N 1} of π s suffcenly small. In fac, denong Z π = 1 E W M π 1 3 and n vew of, he dscree BSDE 1 can be rewren n erms of Y π, Z π as Y π N = ξ π, Z π N = 0, Z π = E W Y π 1, = N 1,..., 0, Y π = E Y π 1 f π, Y π, Z π, = N 1,..., 0. 4 So he dscree BSDE n 1 s us a reformulaon of he mplc me dscrezaon scheme suded e.g. by Bouchard and Touz 004. I urns ou ha hs reformulaon s more convenen for our purposes. Le us suppose now ha some approxmaon Ŷ π, ˆM π of Y π, M π s a hand. We merely assume ha Ŷ π, ˆM π s square-negrable, adaped o G and ha ˆM π s a marngale sarng n 0. No furher assumpons on he algorhm whch was used o compue Ŷ π, ˆM π are necessary a hs sage. Our purpose s o oban quanfable nformaon abou he L -error beween he approxmaon Ŷ π, ˆM π and he soluon Y π, M π, whch only requres knowledge of he approxmaon and he daa f π, ξ π. 1 3

4 A-poseror esmaes for dscree BSDEs The basc dea s o check how close Ŷ π, ˆM π s o solvng 1 nsead of checkng how close s o he soluon of 1. Ths leads o he followng error creron: E π Ŷ π, ˆM π = E 0 ξ π Ŷ π N max 1 N E 0 Ŷ π Ŷ π 1 0 f π, Ŷ π, 1 E W ˆM π 1 ˆM π The frs erm s us he squared L -error of he approxmaon a ermnal me. The second erm measures wheher he approxmaon s close o solvng he dfference equaon correspondng o 1 run forward n me. The man resul of hs secon shows ha hs error creron s equvalen o he squared L -error beween Ŷ π, ˆM π and Y π, M π. Theorem.1. Assume H G π. Then here are consans c, C > 0 such ha for suffcenly small π and for every par of square-negrable G -adaped processes Ŷ π, ˆM π, where ˆM π s a marngale sarng n 0, holds ha 1 c E πŷ π, ˆM π max E 0 Y π π Ŷ π E 0 M π N ˆM π N CE π Ŷ π, ˆM π More precsely, wh he choce c = 6 D1 K T 1 T and 6 C = D T T K D e ΓT, where he nequales hold for π Γ 1. Γ = 4K T 1D 16T K 4 1 T D, Before we presen he proof of hs heorem we explan by some generc examples how hs resul can be appled n pracce. Example.. a In hs example we assume ha here s a Markovan process X π, F akng values n R M and deermnsc funcons y π, z π such ha he soluon Y π, Z π of 4 s of he form Y π = y π, X π, Z π = z π, X π. We recall here ha F 0,T s he augmened Brownan flraon, bu could also be replaced by a larger flraon H 0,T wh respec o whch W s sll a Brownan moon. Numercal algorhms for dscree BSDEs ypcally ry o approxmae he par of funcons y π, z π. We assume ha he approxmaon of y π, z π s of he form 5 ŷ π, x; Ξ, ẑ π, x; Ξ, where Ξ s a random vecor ndependen of F T. Ths form covers Mone Carlo algorhms such as leas-squares Mone Carlo where he approxmave funcons ŷ π, ẑ π depend on ndependenly 4

5 A-poseror esmaes for dscree BSDEs smulaed sample pahs of X π whch we can collec n Ξ. We now defne he flraon G by enlargng he Brownan flraon wh he random vecor Ξ,.e. G = F σξ. Gven ŷ π, ẑ π, we consder Ŷ π = ŷ π, X π ; Ξ, Ẑ π = ẑ π, X π ; Ξ as approxmaons of Y π, Z π. An approxmaon for he marngale M π can be defned n erms of Ẑπ by ˆM π 0 = 0, ˆM π 1 ˆM π = ẑ π, X π ; Ξ W, whch clearly s a marngale wh respec o he larger flraon G, bu, n general, no wh respec o F. Then, Ẑ π = 1 E W ˆM π 1, and, consequenly, he error creron can be rewren n erms of Ŷ π, Ẑπ replacng ˆM π by Ẑπ on he lef hand sde wh a slgh abuse of noaon as E π Ŷ π, Ẑπ = E 0 ξ π Ŷ π N max E 0 1 N Ŷ π Ŷ 1 π 0 f π, Ŷ 1 π, Ẑπ Ẑ π W If an algorhm for smulang ndependen sample pahs of X π, W =0...,N s a hand, we can draw ndependen copes of Ŷ π, Ẑπ, W =0...,N gven Ξ. Now suppose ha he daa f π, ξ π s suffcenly good such ha, gven a realzaon of Ξ, we are even able o draw Λ ndependen copes λ Ŷ π, λ Ẑ π, f π, λ Ŷ π, λ Ẑ π, λw, λ ξ π ; = 0,... N λ=1,...,λ of Ŷ π, Ẑπ, f π, Ŷ π, Ẑπ, W, ξ π ; = 0,... N. Then, E π Ŷ π, Ẑπ can be esmaed conssenly by Êπ Λ Ŷ π, Ẑπ = 1 Λ Λ λ ξ π λ Ŷ π N λ=1 1 Λ max =1,...,N Λ λŷ π λ Ŷ π 1 0 f π, λ Ŷ π, λ Ẑ π 1 λẑπ λw λ=1. 6 In such suaon we can, hanks o Theorem.1, conssenly esmae upper and lower bounds for he L -approxmaon error beween he gven numercal soluon Ŷ π, ˆM π and he rue soluon Y π, M π of he dscree BSDE 1. b Suppose ha we are n he suaon of par a of hs example, bu ha, addonally, ẑ π and ŷ π are lnked va ẑ π W, x; Ξ = E ŷ π 1, X π 1 ; Ξ Ξ, Xπ = x and E ŷ π 1, X π 1 ; Ξ Ξ, X π = x 5

6 A-poseror esmaes for dscree BSDEs can be calculaed n closed form. Ths s e.g. he case n he marngale bass varan of leas-squares Mone Carlo proposed by Bender and Sener 01. Thanks o he Markov propery of X π, F we can hen defne a G -marngale ˆM π va ˆM 0 π π = 0, ˆM 1 ˆM π = ŷ π 1, X π 1 ; Ξ E ŷ π 1, X π 1 ; Ξ Ξ, X π. Then, agan, and he error creron becomes E π Ŷ π, ˆM π = E 0 ξ π Ŷ π N max Ẑ π = ẑ π, X π ; Ξ = 1 E W 1 N E 0 Ŷ π Ŷ π 1 0 ˆM π 1, f π, Ŷ π, Ẑπ ˆM π. Gven a samplng procedure for X π =0,...,N we can now, condonal on Ξ, sample ndependen copes of Ŷ π, Ẑπ, ˆM π =0...,N. Assumng agan ha he daa s good enough and so Λ ndependen copes λ Ŷ π, λ Ẑ π, f π, λ Ŷ π, λ Ẑ π, λ ˆM π, λ ξ π ; = 0,... N λ=1,...,λ of Ŷ π, Ẑπ, f π, Ŷ π, Ẑπ, ˆM π, ξ; = 0,... N are a hand, we can esmae E π Ŷ π, ˆM π analogously o par a of hs example. We now gve he proof of Theorem.1. Proof of Theorem.1. Noce frs ha he condon on he mesh sze π ensures ha a unque soluon Y π, M π o he dscree BSDE 1 exss, see e.g. Theorem 5 and Remark 6 n Bender and Denk 007. We frs prove he easer lower bound E π Ŷ π, ˆM π c max E 0 Y π π Ŷ π E 0 M π N ˆM π N. 7 In order o smplfy he noaon we se Then, Ẑ π = 1 E W E π Ŷ π, ˆM π = E 0 ξ π Ŷ π N max By 1 and 3 we ge =: A max =1,...,N B. Y π Y π N E 0 Ŷ π ˆM π 1. Ŷ π 1 0 f π, Y π, Z π M π = 0. f π, Ŷ π, Ẑπ ˆM π 6

7 A-poseror esmaes for dscree BSDEs Hence, by Young s nequaly, he Lpschz condon on f π and he marngale propery of M π ˆM π, we have for every γ > 0, B = E 0 Ŷ π Y π Ŷ 1 π 0 Y π 0 f π, Ŷ π, Ẑπ f π, Y π, Z π ˆM π M π 4 1 γ max 5 E 0 Y π π Ŷ π 61 γe 0 M π ˆM π 1 γ 1 T E 0 f π, Ŷ π, Ẑπ f π, Y π, Z π 51 γ max E 0 Y π π Ŷ π 61 γe 0 M π N ˆM π N 1 γ 1 T 1 T K max E 0 Y π π Ŷ π E 0 Z π Ẑπ. As, by he defnon of Z π and Ẑπ and he marngale propery of M π and ˆM π =1 D E 0 Z π Ẑπ = =1 we mmedaely ge =1 =1 1 E E 0 W M π 1 ˆM π 1 M π ˆM π E 0 M π 1 ˆM π 1 E 0 M π ˆM π = DE 0 M π N ˆM π N, 8 E π Ŷ π, ˆM π 61 γ DT 1 T K 1 γ 1 max π E 0 Y π Ŷ π E 0 M π N ˆM π N. Choosng γ = T 1 T K we oban he lower bound 7 wh c = 6 D1 K T 1 T. In order o derve he upper bound, we frs nroduce he process Ȳ π va Ȳ π 0 = Ŷ π 0, where agan Ẑπ ermnal condon ξ π = Ȳ π N Ȳ π 1 = Ȳ π f π, Ŷ π, Ẑπ ˆM π 1 ˆM π, = 0,..., N 1, = 1 E W ˆM π 1. Then he par Ȳ π, ˆM π solves he dscree BSDE wh and drver f π, y, z = f π, Ŷ π, z. We wll frs esmae he error beween Ȳ π, ˆM π and Y π, M π by a slgh modfcaon of he weghed a-pror esmaes n Lemma 7 n Bender and Denk 007. For some consans Γ, γ > 0, whch wll be fxed laer, we consder he weghs q = 1 1 Γ. Now recall from ha Hence, =0 M π 1 M π = Y π 1 E Y π 1, q E 0 M π 1 M π ˆM π 1 ˆM π = ˆM π 1 ˆM π = Ȳ π 1 E Ȳ π 1. =0 q E 0 Y π 1 Ȳ π 1 E Y π 1 Ȳ π 1. 7

8 A-poseror esmaes for dscree BSDEs Now, followng he argumen n sep 1 of he proof of Lemma 7 n Bender and Denk 007, we oban, =1 q E 0 M π 1 M π ˆM π 1 ˆM π q N E 0 Y π N Ȳ π N γ q E 0 Y π =0 1 T K Y π q E 0 γt =0 The argumen of sep of he same proof yelds Y max q π E 0 0 N K T 1 π Γ 1 Ȳ π Ŷ π 1 T K γ Ȳ π q N E 0 Y π N Ȳ π 1 Y π q E 0 T =0 N =0 Ŷ π =0 q E 0 Z π Ẑπ. q E 0 Z π Ẑπ. Thus, combnng hese wo nequales wh a sraghforward weghed verson of 8, we ge max q E 0 Y π 0 N Ȳ π q E 0 M π 1 M π ˆM π 1 ˆM π =1 γt q N E 0 Y π N Ȳ π N Y max q π E 0 0 N Ŷ π D γt q N E 0 Y π N Ȳ π N 1 γt K T 1 π Γ 1 =1 Y max q π E 0 Ȳ π 0 N 1 γt K T 1 π Γ 1 max q E 0 Ŷ π 0 N Ȳ π. 1 T K γ q E 0 M π 1 M π ˆM π 1 ˆM π 1 γt K T 1 π Γ 1 =1 1 T K D γ q E 0 M π 1 M π ˆM π 1 ˆM π 1 T K γ 1 1 D 1 Choosng γ = 41 T K D, Γ = 4K T 11 γt D,, hus, holds, for π Γ 1 Y max q π E 0 Ȳ π 0 N q E 0 M π 1 M π ˆM π 1 ˆM π =1 4 γt q N E 0 Y π N Ȳ π N 3 D 1 1 D 1 max 0 N q E 0 Ŷ π Ȳ π. 8

9 3 A-poseror esmaes for connuous BSDEs Now, applyng Young s nequaly wce and akng he defnon of he weghs no accoun, we have Y max E π 0 0 N Ŷ π E 0 M π N ˆM π N Y 8 γt e ΓT π E 0 N Ȳ π N ξ 16 γt e ΓT E π 0 Ŷ π N 6 D 1 1 D D 16 γt 6 D T T K D because, by he consrucon of Ȳ π, 1 e ΓT D 1 e ΓT max 0 N E 0 max E 0 Ŷ π 0 N Ȳ π e ΓT E π Ŷ π, ˆM π, Ŷ π Ȳ π Ŷ π Ȳ π = Ŷ π Ŷ π 1 0 f π, Ŷ π, 1 E W ˆM π 1 ˆM π. 9 3 A-poseror esmaes for connuous BSDEs We now urn o BSDEs n connuous me Y = ξ and assume T fs, Y s, Z s ds Z s dw s 10 H ξ s a square-negrable F T -measurable random varable. f : 0, T Ω R R D R s measurable and, for every y, z R D1, f, y, z s F -adaped. Moreover f s Lpschz n y, z wh consan K unformly n, ω and E T 0 f, 0, 0 d <. Under hs se of assumpons a classcal resul by Pardoux and Peng 1990 ensures ha here s a unque par of square negrable F -adaped processes Y, Z such ha 10 s sasfed. We now suppose ha a square-negrable approxmave soluon Ŷ π, Ẑπ π of Y, Z has been compued on some me grd π by some numercal algorhm. Agan we allow ha Ŷ π, Ẑπ s adaped o a larger flraon G and wsh o quanfy he error beween rue soluon and approxmae soluon. As, n general, may no be possble o draw sample copes of ξ and f, Ŷ π, Ẑπ, we apply he error creron 5 wh approxmave daa ξ π, f π. I now reads wh a slgh abuse of noaon, wrng agan Ẑπ nsead of he he marngale dfference of Ẑπ on he lef-hand sde, E π Ŷ π, Ẑπ = E 0 ξ π Ŷ π N max 1 N E 0 Ŷ π Ŷ π 1 0 f π, Ŷ 1 π, Ẑπ Ẑ π W, 11 9

10 3 A-poseror esmaes for connuous BSDEs where we recall ha E 0 denoes he condonal expecaon wh respec o G 0. We assume hroughou hs secon ha he larger flraon G s obaned from he augmened Brownan flraon by an enlargemen a me 0,.e. G = F σξ, where Ξ denoes a famly of random varables ndependen of F T cf. he dscusson n Example.. Concernng he approxmave daa, we suppose H F π,.e. assumpon H G π as nroduced n he prevous secon, bu wh he larger flraon G replaced by he augmened Brownan flraon F on he grd. In hs suaon we oban he followng esmaes on he squared L -error beween rue and approxmave soluon. The esmaes can be compued n erms of he approxmae soluon on he grd, he approxmae daa and he error beween rue and approxmae daa. Theorem 3.1. Suppose H and H F π and ha G = F σξ, where Ξ s ndependen of F T. Then, here are consans c, C > 0 such ha for every par of G -adaped square negrable processes Ŷ π, Ẑπ π and max E 0 Y Ŷ π π 1 E 0 =0 C E π Ŷ π, Ẑπ c E π Ŷ π, Ẑπ π E max π E 0 E ξ ξ π ξ ξ π Y Ŷ π Z Ẑπ d =0 Y Ŷ π =0 1 If, addonally, f and f π do no depend on y, hen and max E 0 Y Ŷ π π 1 E 0 =0 C E π Ŷ π, Ẑπ E E π Ŷ π, Ẑπ c max π E 0 ξ ξ π E ξ ξ π =0 =0 E 1 E 0 1 E f, Y, Z f π, Y, Z d. Y Ŷ π Z Ẑπ d f, Y, Z f π, Y, Z d Z Ẑπ d 1 Y Ŷ π =0 1 =0 E E f, Z f π, Z d E 0 1 Z Ẑπ d f, Z f π, Z d Remark 3.. If one nspecs he proof below carefully, hen he consans C and c can be made explc. They only depend on he me horzon T, he dmenson D, he Lpschz consan K of T f and on E 0 f, 0, 0 d and E ξ. 10

11 3 A-poseror esmaes for connuous BSDEs In vew of Theorem.1, he above resuls can be easly deduced from error esmaes beween Y, Z and he mplc me dscrezaon Y π, Z π correspondng o he approxmae daa ξ π, f π. I s well-known from he resuls by Zhang 004 ha such me dscrezaon crucally depends on some L -regulary of Z. For hs reason some exra condons on he daa are usually mposed. For example, ξ s ofen assumed o be a Lpschz funconal of he pah of a forward SDE X and f, y, z s supposed o depend on ω only hrough X. In our seng, he crucal observaon s ha he erm whch corresponds o hs L -regulary can be esmaed by he error creron 11 whou any such assumpons on he daa. Lemma 3.3. Suppose H and H F π and ha G = F σξ, where Ξ s ndependen of F T. Then, here s a consan C > 0 such ha for every par of G -adaped square negrable processes Ŷ π, Ẑπ π max E Y Y π 0 N C =0 E π Ŷ π, Ẑπ K y π E 1 E Z Z π d ξ ξ π =0 1 Here, K y K denoes a Lpschz consan of f π wh respec o he y-varable. E f, Y, Z f π, Y, Z d. Proof. Frs noce ha by assumpon H F π and he assumpon on he flraon G, holds Y π N = ξ π, Z π N = 0, Z π = E W Y π 1 F, = N 1,..., 0, Y π = E Y π 1 F f π, Y π, Z π, = N 1,..., 0, 1.e. he flraon G n 4 can be replaced by he augmened Brownan flraon F. The proof consss of wo seps. In he frs sep we observe ha max E Y Y π 0 N C E =0 1 ξ ξ π K y π =0 1 Z E Z π d =0 1 E Žs π dw s Z π W E f, Y, Z f π, Y, Z d, 13 where Žπ s defned on, 1, = 0,..., N 1, va he marngale represenaon heorem as 1 Ž π dw = Y π 1 E Y π 1 F. 14 Inequaly 13 can be derved by slghly modfyng he argumen n Theorem 3.1 of Bouchard and Touz 004. For he reader s convenence we provde he deals n he Appendx. 11

12 3 A-poseror esmaes for connuous BSDEs As a second sep we wll now show ha 1 E Žs π dw s Z π W CE π Ŷ π, Ẑπ, 15 =0 whch, n vew of 13, complees he proof of hs lemma. By 1 and he ndependence of G 0 and F T, Young s nequaly and Iô s somery, we oban 1 1 E Žs π dw s Z π W = E 0 Žs π dw s Z π W =0 1 E 0 = E 0 =0 1 =0 Ž π s dw s Ẑπ W Ž π s dw s Ẑπ W =0 E 0 Ẑπ Z π W =0 As n he proof of Theorem.1, we nroduce he process Ȳ π =0 E 0 Ẑπ Z π by Ȳ π 0 = Ŷ π 0, Ȳ π 1 = Ȳ π f π, Ŷ π, Ẑπ Ẑπ W, = 0,..., N 1, wh he specfc choce ˆM π 1 ˆM π = Ẑπ W for he marngale ncremen. Noce ha wh hs choce of he marngale ˆM π, E π Ŷ π, Ẑπ equals E π Ŷ π, ˆM π n he noaon of Theorem.1. Then, by 14 and 4, and applyng Young s nequaly and he Lpschz propery of f π, we ge 1 Žs π dw s Ẑπ W E 0 =0 = E 0 =0 C Y π 1 Ȳ π 1 Y π Ȳ π f π, Y π, Z π f π, Ŷ π, Ẑπ E 0 Ŷ π N Ȳ π N max π E 0 Y π Ŷ π Gaherng he above nequales and applyng 8 we have 1 E Žs π dw s Z π W =0 C E 0 Ŷ π N Ȳ π N max π E 0 =0 E 0 Z π Ẑπ. Y π Ŷ π E 0 M π N ˆM π N. Inequaly 15 now follows from Theorem.1 and 9 wh he above choce of he marngale ˆM π. 1

13 3 A-poseror esmaes for connuous BSDEs Proof of Theorem 3.1. We frs prove he frs and hrd nequaly. These wo nequales bascally follow by combnng Theorem.1 and Lemma 3.3. Denoe he marngale dfference of Ẑ π wh respec o he Brownan ncremens by ˆM π,.e. max E 0 Y Ŷ π π 1 E 0 =0 ˆM π 0 = 0, ˆM π 1 ˆM π = Ẑπ W, Then, E π Ŷ π, Ẑπ equals E π Ŷ π, ˆM π n he noaon of he prevous secon. By he ndependence of G 0 and F T and n vew of 8, we ge Z Ẑπ d max E Y Y π π =0 1 E Z Z π d max E 0 Y π π Ŷ π DE 0 M π N ˆM π N Applyng Theorem.1 and Lemma 3.3 yelds, for some consan C > 0, max E 0 Y Ŷ π π 1 E 0 Z Ẑπ d =0 C E π Ŷ π, Ẑπ K y π E ξ ξ π 1 E f, Y, Z f π, Y, Z d. If f and f π are ndependen of y, hen we can choose K y = 0 and he hrd nequaly follows. In order o complee he proof of he frs nequaly we esmae 1 E 0 Y Ŷ π d T max E 0 Y Ŷ π π 1 E Y Y d As =0 =0 =0 =0 1 E Y Y d C π e.g. by Lemma.4 n Zhang 004, we mmedaely oban he frs nequaly hanks o 16. For he second and fourh nequaly we we make use of he deny Y Y 0 = 0 16 f, Y, Z d Z dw

14 3 A-poseror esmaes for connuous BSDEs Afer nserng 17 we oban by he Iô somery, Young s nequaly and Jensen s nequaly E π Ŷ π, Ẑπ = E 0 Ŷ π N ξ π max E π 0 Ŷ 1 N Y Y 0 Ŷ π 0 c 1 1 max 0 N E 0 f, Y, Z f π, Ŷ π, Ẑπ d Ŷ π Y E ξ ξ π 1 E 0 =0 1 1 E 0 The Lpschz propery of f π and Young s nequaly yeld 1 f π, Y, Z f π, Ŷ π, Ẑπ d K y E 0 =0 E Z Ẑπ d E f, Y, Z f π, Y, Z d Z Ẑπ dw f π, Y, Z f π, Ŷ π, Ẑπ d. 18 Y Ŷ 1 π d K E 0 =0 Z Ẑπ d Combnng hs esmae wh 18 we oban he second nequaly. If f π s ndependen of y, we can agan choose K y = 0, and ge he fourh nequaly. We close hs secon wh some examples whch cover ypcal suaons n he leraure on numercal algorhms for BSDEs. Example 3.4. Suppose ha he daa ξ, f sasfes where X s he soluon of an SDE ξ = ϕx T, f, y, z = F, X, y, z dx = b, X d σ, X dw, X = x 0 wh consan nal condon x 0 and Lpschz connuous coeffcens b : 0, T R D R D and σ : 0, T R D R D D. We also assume ha ϕ s Lpschz connuous wh consan K ϕ and F s α-hölder connuous n me and Lpschz connuous n space,.e. F 1, x 1, y 1, z F, x, y, z K 1 α K x x 1 x K y 1 y K z 1 z, for some α 1/. Gven a srong order α approxmaon X π of X on a grd π,.e. max π E X X π C π α, 14

15 3 A-poseror esmaes for connuous BSDEs we defne he approxmae daa ξ π, f π by ξ π = ϕx π T, f π, y, z = F, X π, y, z. Then, E ξ ξ π =0 1 E f, Y, Z f π, Y, Z d CK ϕ π α K π α K x π Recall also from Zhang 004 ha under he above condons max =0,...N sup E Y Y C π., 1 Hence, he frs wo nequales n Theorem 3.1 can be rewren as max sup =0,...,, 1 E 0 Y Ŷ π =0 E 0 1 Z Ẑπ d C E π Ŷ π, Ẑπ π 19 and E π Ŷ π, Ẑπ c max sup =0,...,, 1 K ϕ K π α K x π E 0 Y Ŷ π =0 E 0 1 Z Ẑπ d. 0 Thus, he square roo of he error creron s, up o a erm of order 1/ n he me sep whch corresponds o he me dscrezaon error under he above assumpons, equvalen o he L - error beween rue soluon and approxmae soluon over he whole me nerval no only on he grd. If F does no depend on x, he addonal error n he me sep can be reduced o order α n he lower bound 0, f F s suffcenly regular n me and a hgher order approxmaon X π of X s appled. I vanshes compleely, when addonally F does no depend on and one can sample perfecly from X on he grd,.e. one can choose X π = X. Example 3.5. Le us now urn o he case of an rregular ermnal condon. We mpose he same assumpons as n he prevous example, bu remove he Lpschz condon on he ermnal condon ϕ. For smplcy we assume ha we can sample perfecly from X on he grd and, hence, choose he rue daa as approxmae daa ξ π = ϕx T = ξ, f π, y, z = F, X, y, z = f, y, z. In hs suaon, he frs nequaly n Theorem 3.1 becomes max E 0 Y Ŷ π π 1 E 0 =0 Y Ŷ π Z Ẑπ d C E π Ŷ π, Ẑπ π. 1 15

16 3 A-poseror esmaes for connuous BSDEs I s known ha for rregular ermnal condons and equdsan me grds, he me dscrezaon error for BSDEs can be of a smaller order han 1/ n he mesh of he me grd see e.g. he survey paper by Gess e al., 011, bu order 1/ convergence n he number of me seps may be reaned by a suable choce of a non-equdsan me grd under approprae assumpons as shown n Gobe and Makhlouf 010. Hence, under rregular ermnal condons, he error creron E π Ŷ π, Ẑπ does conan sgnfcan nformaon abou he me dscrezaon error. Example 3.6. We now consder some very specfc assumpons on he daa under whch all exra erms nvolvng he mesh sze π of π vansh, namely ξ = ϕx T, f, x, y, z = F z for some Lpschz connuous funcon F and a funcon ϕ whch s no necessarly Lpschz. Ths ype of drver f may occur e.g. n he conex of g-expecaons as nroduced n Peng We mpose he same assumpons on X as n he prevous example and addonally suppose ha we can sample perfecly from X on he grd. We can hen choose ξ π, f π = ξ, f. As f and f π do no depend on y we can rewre he hrd and fourh nequaly n Theorem 3.1 as 1 c E πŷ π, Ẑπ max E 0 Y Ŷ π π 1 E 0 =0 Z Ẑπ d CE π Ŷ π, Ẑπ Here he squared L -error beween rue soluon and approxmae soluon s equvalen o he error creron even n connuous me. Ths s somehow surprsng, because he error creron can be compued solely n erms of he approxmae soluon on he grd. The L -error of he Y -par s, however, only consdered on he grd π n, whereas was esmaed on he whole nerval n he prevous examples. Example 3.7. Our las example reas he case of smooh daa. Precsely, n he seng of Example 3.4 we suppose ha all coeffcen funcons ϕ, F, b, σ are suffcenly smooh and bounded wh bounded dervaves. We also assume for he momen ha we can sample perfecly from X on he grd π, whch s here aken as equdsan. We can hence choose ξ π = ϕx T = ξ, f π, y, z = F, X, y, z = f, y, z. By he resuls of Gobe and Labar 007 follows ha max E Y Y π π E Z Z π C π. =0 A sraghforward combnaon wh Theorem.1, akng 8 no accoun, leads o he upper bound max E 0 Y Ŷ π π E 0 Z Ẑπ C =0 E π Ŷ π, Ẑπ π. 3 Here, we assume, of course, ha he assumpons of Theorem 3.1 are n force and ha Ŷ π, Ẑπ s G -adaped. Compared o 19, he exra erm s now of order 1 n he mesh sze nsead of 16

17 4 Numercal examples order 1/. The prce o pay for hs, s ha he L -error beween rue soluon and approxmae soluon s esmaed on he grd only. I s sraghforward o check ha 3 sll holds rue when he approxmae daa s of he form ξ π = ϕx π T, f π, y, z = F, X π, y, z, where X π s a srong order 1 approxmaon of X such as a Mlsen scheme. Indeed, one mus us compare he error crera based on he rue daa and he approxmae daa. 4 Numercal examples In hs secon we apply he a-poseror esmaes o a numercal example. The es BSDE n hs example s a slgh modfcaon of he one suggesed by Bender and Zhang 008 n he conex of coupled FBSDEs. We here consder he followng Markovan BSDE where X d, = x d,0 Y = 0 D snx d,t d=1 x R = R x R σ D snx d,u dw d,u, d = 1,..., D, d =1 T 1 σ Y u 3 D 3du D d=1 T Z d,u dw d,u, 4 s he runcaon funcon a level ±R, W = W 1,..., W D s a D-dmensonal Brownan moon and σ > 0 and x d,0, d = 1,..., D, are consans. The runcaon funcon was merely mplemened n order o ensure he Lpschz connuy of he drver. In hs example, he rue soluon for Y, Z s relaed o X by D Y = y, X = snx d,, d=1 Z d, = z d, X = σ cosx d, D snx d,, 5 d =1 whch can be verfed by Iô s formula. Ths relaon wll be used laer o compare he error creron proposed n hs paper and he squared L -error beween rue soluon and approxmaon quanavely. For he evaluaon of he error creron we apply as approxmae daa f π, y, z = f, y, z = 1 σ y 3 D 3, ξ π = D snxd,t π where X π = X π,e or X π = X π,ms s eher he Euler scheme or he Mlsen scheme of X wh respec o he paron π. We denoe he equdsan paron of 0, T no N subnervals by π N. d=1 17

18 4 Numercal examples For he numercal approxmaon of y, z we wll use leas-squares Mone Carlo as suggesed by Lemor e al. 006 based on he Euler scheme. To hs end we frs sample a famly Ξ = λ X π N 1, λ W λ = 1,..., Λ, = 0,..., N 1 6 of Λ ndependen copes of X π N,E 1, W =0,...,. Furhermore, we choose funcon bases η, x = {η 1, x,..., η K, x}, = 0,..., N 1, where K s he number of bass funcons a each me sep. In prncple, dfferen bass funcons can be used for he Y -par and he Z-par of he soluon. Bu below we apply, for smplcy, he same bass funcons for boh pars of he soluon. Inalzng he algorhm a ermnal me N = T by ŷ π N N, x = φx π N N one defnes eravely, for = N 1 o 0, ẑ π N ˆα d, = arg mn α R K Λ λ=1 λw d, ŷ π N 1 η, λ X π N α, d = 1,..., D, d, x = η, xˆα d,, d = 1,..., D, Λ ˆα 0, = arg mn ŷ π N 1 λ X π N 1 f π, ŷ π N 1 λ X π N α R K ŷ π N, x = η, xˆα 0,. We fnally oban λ=1 Ŷ π N = ŷ π N, X π N,E, Ẑ π N = ẑ π N, X π N,E 1, ẑ π N λ X π N η, λ X π N α, as approxmaons for Y, Z on he grd π N. They are exended by pecewse consan nerpolaon on he whole me nerval 0, T. In he muldmensonal case we also apply a slgh modfcaon of hs procedure, whch has a flavour of he classcal conrol varae echnque n he presen non-lnear BSDE seng and can easly be appled o varous oher BSDEs n a smlar way. Insead of approxmang Y drecly, we approxmae Y u, X for a funcon u, for whch we hope ha u, X explans a sgnfcan par of Y. In he presen example we consruc u as follows: We freeze he dffuson coeffcen of X and consder he smple BSDE wh he same ermnal funcon as n he orgnal BSDE,.e. X d, = x d,0 σ D snx d,0 W d,, d = 1,..., D, Y = d =1 D sn X d,t d=1 D d=1 T Z d,u dw d,u. 18

19 4 Numercal examples A drec calculaon shows ha Y = u, X for D u, x 1,..., x D = exp 1 D σ snx d,0 T snx d d =1 Then, applyng Iô s formula o u, X, we observe ha Y = Y V u, X D Z d, = Zd, V σ D snx d, exp 1 σ snx d,0 T cosx d,, d =1 where Y V, Z V solves he followng BSDE wh zero ermnal condon: { T Y V 1 D = σ us, X s Ys V 3 D 3 us, X s snx d,0 D snx d,s } ds D d=1 T d =1 d=1 Z V d,s dw d,s. 7 When we say below, ha he non-lnear conrol varae echnque s appled, hs means ha he par Y V, Z V s approxmaed by leas-squares Mone-Carlo nsead of Y, Z. Then he correspondng approxmaons o Y, Z are defned va Ŷ V,π N u, X π N,E D σ Ẑ V,π N d, d =1 snx π N,E d, d=1 d=1 D exp 1 σ snx d,0 T cosxπ N,E 4.1 Case 1: One-dmensonal Brownan moon and local bass funcons In he frs case we fx he parameers as follows: D = 1, T = 1, x 0,1 = π/, σ = 0.4. d =1 d,. Le K 3 be he number of bass funcons. We consder a local bass of ndcaor funcons whch paron he nerval 0, 3 no equdsan subnervals. Precsely, we se η 1, x = 1 {x<0} x, η d,k = 1 {x 3} x, η k, x = 1 {x 3k 1/K, 3k/K } x, k = 1,..., K, for = 0,..., N 1. The numercal procedure now depends of he number of me seps N, he dmenson of he funcon bass K and he sample sze Λ. For = 1,..., 11 and l = 3,..., 5 hey are fxed as N = 1, K = max { 1, 3 }, Λ = l 1, 19

20 4 Numercal examples where a s he closes neger o a and a s he smalles neger larger or equal o a. To be precse, we wll dscuss hree dfferen choces of l, n whch we smulaneously ncrease he parameers N, K and Λ hrough her dependence on. The resuls n Lemor e al. 006 sugges ha n hs onedmensonal seng he L -approxmaon error converges o zero a a rae of N β/ for 0 < β 1 up o a logarhmc facor, f he number of sample pahs Λ s proporonal o N β K. Wh our choce of he parameers, we hence expec convergence of order 1/ for l = 5. The choce l = 4 s us on he hreshold of he heorecal convergence resuls, and we hence canno expec convergence for l = 3. In order o llusrae he compuaonal effor, he able below dsplays he number of smulaed pahs for he dfferen choces of l n dependence of he number of me seps N. l Table 1: Sample sze Λ n dependence of N and l N Noe ha he choce l = 5, whch corresponds o convergence of order 1/, yelds remendous smulaon coss. Gven hese parameers, we compue he coeffcens ˆα π N 0, and ˆα π N 1, for he lnear combnaon of he bass funcon by leas-squares Mone Carlo and and receve he approxmae soluon by seng Ŷ π N = η, X π N,E ˆα 0,, π Ẑ π N = η, X π N,E ˆα π 1,. We here recall ha X π N,E denoes he Euler scheme. Thanks o 5 we observe ha he squared approxmaon error on he grd s gven by max E 0 Y Ŷ π N 0 N T N E 0 Z Ẑπ N =1 = max E 0 snx π N,MS 0 N Ŷ π N T =1 N E 0 σ cosx π N,MS snx π N,MS Ẑπ N ON, where X π N,MS ndcaes he approxmaon of X by he Mlsen scheme. Here agan E 0 denoes he condonal expecaon gven G 0, where G = F σξ, and Ξ s he collecon of random varables generaed o deermne he coeffcens for he leas-squares Mone Carlo esmaor. Fgure 1 dsplays a log-log-plo of he rgh-hand sde of 8 wh he expecaons replaced by he emprcal mean usng 1000N ndependen pahs. We observe ha he convergence behavour n hs example s much beer han he heorecal bounds n Lemor e al. 006 sugges. For he cases l = 4 and l = 5, Fgure 1 ndcaes ha he L -approxmaon error converges a a rae of 8 0

21 4 Numercal examples l = 3 l = 4 l = Number of meseps, N =,..., 64 Fgure 1: Squared approxmaon error for dfferen choces of l. 1/. For l = 3, he fgure s less conclusve. The scheme seems o converge n hs case as well conrarly o wha we expeced n vew of he heorecal resuls, bu possbly a a lower rae. We now show how o recover hese resuls by applyng he error creron, whch we nroduced n hs paper, whou makng use of he explc form of he soluon n 5. To hs end we approxmae he ermnal condon based on he Mlsen scheme and hence he error creron becomes E πn Ŷ π N, Ẑπ N = E 0 snx π N,MS N =0,...,, 1 Ŷ π N N max 1 N E 0 =0 Ŷ π N Ŷ π N σ Ŷ π N 3 D 3 Ẑ π N W. By Example 3.4 here are consans c and C such ha squared approxmaon error on he whole nerval sasfes max sup E 0 Y Ŷ 1 π N E 0 Z Ẑπ N d C E πn Ŷ π N, Ẑπ N 1 N and max sup =0,...,, 1 E 0 Y Ŷ π N =0 E 0 1 Z Ẑπ N d c E πn Ŷ π N, Ẑπ N 1 N for suffcenly large N. Consequenly, he squared approxmaon error on he whole nerval s equvalen o he error creron for suffcenly large N, f E πn Ŷ π N, Ẑπ N cons. 1 N. 1

22 4 Numercal examples Fgure dsplays he a-poseror error creron for he hree cases l = 3, 4, 5. As before, he l = 3 l = 4 l = Number of meseps, N =,..., 64 Fgure : A-poseror error creron for dfferen choces of l. expecaon s replaced by a sample mean over 1000N ndependen pahs. Comparng Fgures 1 and we observe ha he squared approxmaon error on he grd Fgure 1 and he error creron Fgure look almos dencally, no only qualavely, bu also quanavely. We can derve from Fgure ha he a-poseror error creron converges o zero a a rae of N 1 for he cases l = 4, 5 and, consequenly, he approxmaon error beween approxmae soluon and rue soluon on he whole nerval converges o zero a rae of N 1/ for hese wo cases. In parcular, we can conclude ha s unnecessary o run he exensve scheme wh l = 5 n hs example, as an approxmaon of almos he same qualy can be acheved wh moderae smulaon coss n he case l = 4. The cheap scheme wh l = 3 leads, however, o a sgnfcanly larger error. 4. Case : Three-dmensonal Brownan moon and global bass funcons We now consder he case of a hree dmensonal drvng Brownan moon and apply a small global bass conssng of us a few monomals. The number of smulaed pahs wll be adused n a way ha he heorecal resuls n Lemor e al. 006 suppor convergence of he smulaon error of order 1/ n he number of me seps. Apparenly wh he bass funcons fxed, he scheme canno converge o he rue soluon, bu evenually, he proecon error due o he choce of he bass wll domnae he me dscrezaon error and he smulaon error. We now demonsrae how he error creron can be appled o check wheher a small global bass s suffcenly good compared o he choce of he me grd. As parameers of he BSDE we choose D = 3, T = 1, s 1,0 = s 3,0 = π/, s,0 = π/, σ = 0.4.

23 4 Numercal examples The bass consss of K = 7 funcons, he consan funcon wh value 1, he hree monomals of order 1 and he hree mxed monomal of order. The smulaon parameers are gven by N = 1, Λ = 3 1, whch corresponds o a me dscrezaon error and a smulaon error ha decrease wh rae N 1/. We apply he a-poseror error creron o he orgnal leas-squares Mone Carlo scheme and o he modfed one, whch makes use of he non-lnear conrol varae echnque as descrbed and desgned a he begnnng of hs chaper. In conras o he resuls n he prevous secon, he approxmae ermnal condon n he error creron s based on he Euler scheme,.e. ξ π N = 3 d=1 snxπ N,E d, N orgnal leas squares Mone Carlo leas squares Mone Carlo wh non lnear conrol varaes Number of me seps, N =,..., 56 Fgure 3: A-poseror error creron - orgnal leas squares Mone Carlo vs. leas squares Mone Carlo wh non-lnear conrol varaes Fgure 3 shows a log-log-plo for he error creron of boh mplemenaons wh and whou non-lnear conrol varaes where, as before, he expecaon s replaced by a sample mean wh 1000N ndependen copes. In he case whou conrol varaes, he error creron decreases roughly wh a rae of N 1 for small values of N roughly up o N = 45 me seps. Ths corresponds o an approxmaon error of order 1/ whch sems from he me dscrezaon and he smulaon. Sarng from N = 64 he error creron does no decrease sgnfcanly anymore, whch suggess ha he proecon error domnaes he wo oher error sources. A sgnfcan mprovemen of he approxmave soluon can, hence, no be acheved by ncreasng he number of me seps or he number of smulaed pahs, bu a beer choce of he bass would be called for. The suaon urns ou o be que dfferen for he mplemenaon wh non-lnear conrol varaes. Here, he error creron decreases wh a rae of N 1 for he whole range of me seps whch we consder n hs example up o N = 56. Ths suggess ha he small global bass conssng of 3

24 A Proof of nequaly 13 seven monomals s suffcenly good n hs suaon and he domnang error sources are he me dscrezaon and he smulaon error even for a raher fne me grd conssng of 56 pons. Ths example demonsraes on he one hand how he error creron can be appled n order o udge he success of a small global bass. On he oher hand shows ha he non-lnear conrol varae echnque can mpressvely mprove he qualy of he leas-squares Mone Carlo scheme. A Proof of nequaly 13 The proof follows he lnes of Theorem 3.1 n Bouchard and Touz 004. process ˇY π on, 1, = 0,..., N 1, by ˇY π = E Y π 1 F f π, Y π, Z π 1 We frs defne he wh ˇY T π = ξπ. Then, ˇY π = Y π for π by 1. Moreover, hanks o 14, he pars Y, Z and ˇY π, Žπ solve on, 1 he followng dfferenal equaons Y = Y 1 ˇY π = Y π fs, Y s, Z s ds f π, Y π, Z π d By Iô s formula we hen oban Y E ˇY π 1 E Y1 Y π 1 = I II. E Zs Žπ s ds Z s dw s, Ž π s dw s, E Y s ˇY π s fs, Y s, Z s f π, Y π, Z π ds Young s nequaly and he Lpschz condon on f π yeld for some γ > 0 o be fxed laer, 1 Ys II γ E ˇY π s ds 1 f E π, Y s, Z s f π, Y π γ, Z π ds 1 E fs, Y s, Z s f π, Y s, Z s ds γ 1 γ C E 1 Y s ˇY s π ds 4 1 γ Ky E Y s Y π E fs, Y s, Z s f π, Y s, Z s ds. Due o Zhang 004, Lemma.4 we have E Y s Y π E Y s Y E Y Y π 1 C π C E Z Y d E Y π. K E Z s Z π ds 4

25 A Proof of nequaly 13 Applyng Iô s somery and Young s nequaly once more, we ge 1 II γ E Ys ˇY s π ds 8K γ K y C π C 1 C 1 E Y E Y π Z d CE E fs, Y s, Z s f π, Y s, Z s ds 1 Ys =: γ E ˇY π s ds 8K γ A B. Summarzng, we have E Y ˇY π E Y ˇY π 1 E Y1 Y π 1 γ E Z s Žπ s 1 E 1 Zs E Žπ s ds 1 Žs π dw s Z π W ds Ys ˇY s π ds 8K γ A B. By Gronwall s lemma follows ha E Y ˇY π e γ E Y 1 Y π 1 8K A /γ B. Inserng hs resul no he second nequaly of 9 yelds Y E Y π 1 Z E Žπ d 1 γ e γ E Y1 Y π 1 8K γ A B 1 Cγ E Y1 Y π 1 8K γ A B for π small enough. Choosng γ = 3K and π 1/Cγ leads o E Y Y π 1 1 Y E Y π 1 E Z Žπ d 1 Cγ E Y1 Y π 1 Z E Žπ d. 1 B 9 5

26 A Proof of nequaly 13 Hence, for π small enough Y E Y π 1 1 { 1 C E Z E Žπ d Y1 Y π 1 K y C π C 1 1 CE Žs π dw s Z π W C 1 E f, Y, Z f π, Y, Z } d. E Z d 30 Thanks o he dscree Gronwall lemma we ge for some larger consan C Y E Y π { C E ξ ξ π 1 K y π E Žs π dw s Z π W = 1 = E f, Y, Z f π, Y, Z } d, 31 because T 0 E Z d <. Nex we sum 30 up from = 0 o N 1 and oban =0 1 Z E Žπ d { C E ξ ξ π 1 K y π E Žs π dw s Z π W = 1 = E f, Y, Z f π, Y, Z d max E Y Y π } π 3 Nong ha =0 1 Z E Z π d =0 1 nequaly 13 s a drec consequence of 31 and 3. Acknowledgemen Z E Žπ 1 d E Žs π dw s Z π W, Fnancal suppor by he Deusche Forschungsgemenschaf under gran BE3933/3-1 s graefully acknowledged. 6

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