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L -variation of Le´vy driven BSDEs with 2 3 non-smooth terminal conditions 1 0 2 n Christel Geiss and Alexander Steinicke a J 1 Department of Mathematics 3 University of Innsbruck ] R Technikerstraße 13/7 P . A-6020 Innsbruck h t Austria a m [email protected] [email protected] [ 2 v 0 February 1, 2013 2 4 3 . 2 Abstract 1 2 We consider the L -regularity of solutions to backward stochastic dif- 2 1 ferential equations (BSDEs) with Lipschitz generators driven by a Brown- : v ian motion and a Poisson random measure associated with a Le´vy process Xi (Xt)t [0,T].TheterminalconditionmaybeaBorelfunctionoffinitelymany ∈ r increments of the Le´vy process which is not necessarily Lipschitz but only a satisfies a fractional smoothness condition. The results are obtained by in- vestigatinghowthespecialstructureappearinginthechaosexpansionofthe terminalcondition isinherited bythesolution totheBSDE. 1 Introduction The main objective of this paper consists in studying the relation between frac- tionalsmoothnessoftheterminalconditionofaBSDE and theL -variationofits 2 according solution. 1 A motivationto investigate this relation arises from the fact that the convergence rate of the discretization error of BSDEs with Lipschitz generator is determined by the convergence of the discretized terminal condition to its limit and by the L -variationpropertiesoftheexactsolution (Y,Z): 2 In the Brownian scenario the discretization of BSDEs has been studied by many authors, see for example, [3], [21], [29], [10], [18], [9] and [13]. If the BSDE is givenby T T Y = ξ + F(s,Y ,Z )ds Z dW , 0 t T, t s s s s − ≤ ≤ Zt Zt wedefinetheL -variation p 1 n ti 2 var (ξ,f,τ) := sup sup Y Y + Z Z 2dt p k s − ti−1kp k t − ti−1kp 1≤i≤n ti−1<s≤ti Xi=1 Zti−1 ! where τ = (t )n is adeterministictime-net 0 = t < < t = T and i i=0 0 ··· n 1 ti Z := E Z ds . ti−1 t t s |Fti−1 i − i−1 (cid:20)Zti−1 (cid:21) GobetandMakhlouf[19]consideredL -regularityof (Y,Z)foraterminalcondi- 2 tiongivenbyξ = g(X )whereg doesnotneedtobeLipschitzandX denotesthe T forward process. In [13] the L -regularity of (Y,Z) for p 2 was shown if the p ≥ terminal condition depends on the forward process at finitely many time points, ξ = g(X ,...,X ), and satisfies a path-dependent Malliavin fractional smooth- r1 rm ness condition which is weaker than the Lipschitz condition on g. Using these results and followingtheideas of [10], onecan showthat the convergence rate of the error between the discretizations (Yτ,Zτ) and the solution (Y,Z) is of order 1,i.e. 2 T 1 Errτ,2(Y,Z) := sup E Yt Y¯tτ 2 + E Zt Z¯tτ 2dt 2 c τ 21 | − | | − | ≤ | | n0<t≤T Z0 o provided that the time grid for the discretization is chosen in an appropriate way (likein[13]),andthediscretizedterminalconditionconvergesinthisorder. With- out any assumptions on the dependence of the terminal condition ξ on a forward processX,Hu,NualartandSong[20]haveshowntheconvergencerate 1 suppos- 2 ingMalliavindifferentiabilitypropertiesof ξ (and ofthegenerator). 2 For a BSDE driven by a Poisson random measure, Bouchard and Elie [8] proved thattheconvergencerateisoforder 1 (inthenon-degeneratecase)iftheterminal 2 conditionisgivenbyξ = g(X ) where g is Lipschitz. T Here we studywhetherit is possibleto achievetheconvergence rate 1 witha ter- 2 minalconditionξ = g(X ,...,X )andwhethertheLipschitzconditionongcan r1 rm beweakened to a Malliavinfractional smoothnesscondition. The methodwe use allowsto answerthisquestionin thecase where X istheLe´vyprocess itself. Thepaperis organized as follows. In Section 2 we describe thesetting and recall someneededfacts. InSection3aresultaboutMalliavindifferentiabilityoftheso- lution(Y,Z)oftheBSDE isstated. ItsproofwhichfollowsideasofPardouxand Peng [23] can be found in the appendix. Our method to show the L -regularity 2 of solutions to BSDEs exploits the fact that their Malliavin derivative is piece- wiseconstantintime. Thisisensuredbyselectingaterminalconditionwhichhas this property. For this purpose we introduce a space of suitable terminal conditions and investigate the chaos expansion of the according solution in Section 4. Section 5 contains ourmain result, equivalences and implicationsconcerning the L -regularity of (Y,Z). An important fact, which will be considered in Section 2 6, is a sufficient condition for the L -regularity of the solution: a certain Malli- 2 avin fractional smoothness of the terminal condition (in addition to our standing assumptionthatthegeneratorisLipschitz). 2 Setting Let X = (X ) be an L -Le´vy-process on a complete probability space t t [0,T] 2 (Ω, ,P) with Le∈´vy-measure ν. We will denote the augmented natural filtration F ofX by F := ( ) and letL := L (Ω, ,P). Ft t [0,T] 2 2 FT ∈ TheLe´vy-Itoˆ decompositonofanL -Le´vy-process X can bewrittenas 2 ˜ X = γt+σW + xN(ds,dx), t t Z]0,t] (R 0 ) × \{ } whereσ 0,W isaBrownianmotionandN˜ isthecompensatedPoissonrandom ≥ measurecorrespondingto X. Wedefinetherandommeasure M by M(dt,dx) := σdW δ (dx)+xN˜(dt,dx). (1) t 0 3 ThenEM(B)2 = (dt,dx)forB ([0,T] R) where Bm ∈ B × R (dt,dx) = (λ µ)(dt,dx) m ⊗ with µ(dx) = σ2δ (dx)+x2ν(dx) 0 and λdenotes theLebesguemeasure. For 0 t T weconsidertheBSDE ≤ ≤ T Y = ξ + F s,Y ,Z¯ ds Z M(ds,dx), (2) t s s s,x − Zt Z]t,T] R × (cid:0) (cid:1) where Z¯ = Z κ(dx) s s,x R Z and κ(dx) := κ(x)µ(dx) such that κ L (R, (R),µ).We will use the follow- ′ ′ 2 ∈ B ingassumptions: (A ) ξ L , ξ 2 ∈ (A ) F: Ω [0,T] R2 R is jointly measurable, adapted to ( ) , F t t [0,T] × × → F ∈ Lipschitz-continuousinthelasttwovariables,uniformly inω andt, i.e. F(t,y ,z ) F(t,y ,z ) L ( y y + z z ), (3) 1 1 2 2 f 1 2 1 2 | − | ≤ | − | | − | andsatisfies T E F(t,0,0) 2dt < . | | ∞ Z0 Forlateruseweintroducespaces ofstochasticprocesses. Definition2.1. 1. Let S denotethespace ofall ( )-adapted and ca`dla`g pro- t F cesses (y ) such that t 0 t T ≤ ≤ y 2 := E sup y 2 < . k kS | t| ∞ 0 t T ≤ ≤ 2. We define H as the space of all random fields z: Ω [0,T] R R × × → which are measurablewith respect to (R) (where denotes thepre- P ⊗B P dictableσ-algebraonΩ [0,T]generatedbytheleft-continuousF-adapted × processes)suchthat z 2 := E z 2 (ds,dx)< . k kH Z[0,T] R| s,x| m ∞ × ThespaceS H isequippedwiththenorm (y,z) := y 2 + z 2 21. × k kS×H k kS k kH (cid:0) (cid:1) 4 A pair(Y,Z) S H whichsatisfies (2)iscalled asolutiontotheBSDE (2). ∈ × Theorem 2.2. Assume (ξ,F) satisfies (A ) and (A ). Then the BSDE (2) has a ξ F uniquesolution(Y,Z) S H. ∈ × ThisresultcanbefoundinTangandLi[28,Lemma2.4]andinBarles,Buckdahn and Pardoux [5, Theorem 2.1]. We next cite the stability result of [5] comparing thedistancebetweensolutionstotheBSDE (2)withdifferentterminalconditions and generators. Theorem 2.3([5],Proposition2.2). Let(ξ,F)and(ξ ,F )satisfy(A )and(A ). ′ ′ ξ F Forthecorrespondingsolutions(Y,Z) and(Y ,Z )to (2)itholds ′ ′ Y Y 2 + Z Z 2 k − ′kS k − ′kH T C ξ ξ 2 + F s,Y ,Z¯ F s,Y ,Z¯ 2 ds , ≤ k − ′kL2 k s s − ′ s s kL2 (cid:18) Z0 (cid:19) (cid:0) (cid:1) (cid:0) (cid:1) whereC = C(L ) = C(L ,µ). F′ F′ Remark2.4. Accordingto[26,Proposition3](seealso[22,Proposition 3]or[1, Lemma2.2])forany z L (P )thereexistsaprocess 2 ∈ ⊗m pz L (Ω [0,T] R, (R),P ) 2 ∈ × × P ⊗B ⊗m such that for any fixed x R the function (pz) is a version of the predictable ,x ∈ · projection (in the classical sense) of z . In the followingwe will always use this ,x · resulttoget predictableprojectionswhich aremeasurable w.r.t. aparameter. 3 Malliavin differentiability of (Y,Z) We will use the Malliavin derivative which is defined via chaos expansions (see, forexample,[27]). Any G L hasauniquechaosexpansion 2 ∈ ∞ G = I (f ) n n n=0 X and itholds ∞ G 2 := G 2 = n! f 2 k k k kL2 k nkLn2 n=0 X 5 where the f Ln = L (([0,T] R)n, n) are symmetric functions and I n ∈ 2 2 × m⊗ n denotes the n-th multiple integral with respect to M defined in (1). The space D consistsofall randomvariables G L such that 1,2 2 ∈ ∞ G 2 := (n+1)! f 2 < . k kD1,2 k nkLn2 ∞ n=0 X ForG D theMalliavinderivative 1,2 ∈ G L (P ):= L Ω [0,T] R, ([0,T] R),P 2 2 T D ∈ ⊗m × × F ⊗B × ⊗m isgivenby (cid:0) (cid:1) ∞ G(ω) := nI (f ((t,x), ))(ω), t,x n 1 n D − · n=1 X forP -a.a. (ω,t,x) Ω [0,T] R. ⊗m ∈ × × The following theorem is concerned with Malliavin differentiability of the solu- tiontoaBSDE oftheform T Y = ξ + f s,X ,Y ,Z¯ ds Z M(ds,dx), 0 t T (4) t s s s s,x − ≤ ≤ Zt Z]t,T] R × (cid:0) (cid:1) wherewewillassume (A ) f ([0,T] R3) isLipschitz-continuousin (x,y,z),uniformlyin t, i.e. f ∈ C × f(t,x ,y ,z ) f(t,x ,y ,z ) L ( x x + y y + z z ). (5) 1 1 1 2 2 2 f 1 2 1 2 1 2 | − | ≤ | − | | − | | − | (A 1) f satisfies (A )and f 0,1,1,1([0,T] R3). f f ∈ C × Notethat(4)isaspecialformof(2),andF(ω,t,y,z) := f(t,X (ω),y,z)satisfies t (A )iff does satisfy(A ). F f Theorem 3.1. Let ξ D and assume (A 1). Then the following assertions 1,2 f ∈ hold. (i) For -a.e. (r,v) [0,T] R there exists a unique solution (Ur,v,Vr,v) m ∈ × ∈ S H totheBSDE × T Ur,v = ξ + F s,Ur,v,V¯r,v ds t Dr,v r,v s s Zt (cid:0) (cid:1) 6 Vr,vM(ds,dx), 0 r t T (6) − s,x ≤ ≤ ≤ Z]t,T] R × whereV¯r,v := Vr,vκ(dx), s R s,x F (s,Ur,0,RV¯r,0) := f s,X ,Y ,Z¯ , 1I (s),Ur,0,V¯r,0 , r,0 s s ∇ s s s [r,T] s s with = (∂ ,∂ ,∂ ), (cid:10) (cid:0) (cid:1) (cid:0) (cid:1)(cid:11) x y z ∇ 1 F (s,Ur,v,V¯r,v) := f s,X +v1I (s),Y +vUr,v,Z¯ +vV¯r,v r,v s s v s [r,T] s s s s (cid:20) (cid:0) (cid:1) f s,X ,Y ,Z¯ forv = 0, s s s − 6 (cid:21) (cid:0) (cid:1) andUr,v = Vr,v = 0 for r > s. s s,x (ii) Forthesolution(Y,Z)of (4) itholds Y L ([0,T];D ), Z L ([0,T] R;D ), (7) 2 1,2 2 1,2 ∈ ∈ × and Y admitsa ca`dla`gversionfor -a.e. (r,v) [0,T] R. r,v D m ∈ × (iii) ( Y, Z)is aversionof (U,V),i.e. for -a.e. (r,v) itsolves D D m T Y = ξ + F s, Y , Z κ(dx) ds r,v t r,v r,v r,v s r,v s,x D D D D Zt (cid:18) ZR (cid:19) Z M(ds,dx), 0 r t T. (8) r,v s,x − D ≤ ≤ ≤ Z]t,T] R × p (iv) Theprocess ( Y ) is aversion ofZ wherewe set Dr,v r r [0,T],v R ∈ ∈ (cid:16) (cid:17) Y (ω) := lim Y (ω) r,v r r,v t D t rD ց forall(r,v,ω)forwhich Y is ca`dla`gand Y (ω) := 0 otherwise. r,v r,v r D D In the setting of time delayed BSDEs a similar result was proved by Delong and Imkeller[12]assumingthatthetimehorizonT ortheLipschitzconstantL ofthe f generator are sufficiently small. For the convenience of the reader, we present a proofofTheorem 3.1in theappendix. 7 4 Chaotic representation of (Y,Z) Thegoal of thissection is to investigateproperties ofthechaos expansionsof the solution(Y,Z)to (4)withterminalvalues ξ ofthefollowingform: Let r = 0 < r < < r = T be a partitionof [0,T].Define Λ := ]r ,r ] 0 1 m k k 1 k for k = 1,...,m an·d··Vn := 1,...,m n. The cuboids in ]0,T]n induced−by the m { } partitionwewilldenoteby Λ := Λ Λ , α = (α ,...,α ) Vn. α α1 ×···× αn 1 n ∈ m Furthermorewelet ∞ H := ξ = I (f ) L : gα L (Rn,µ n) suchthat  n n ∈ 2 ∃ n ∈ 2 ⊗  Xn=0  f ((t ,x ),...,(t ,x )) = gα(x ,...,x )1I (t ,...,t ) . n 1 1 n n n 1 n Λα 1 n  αX∈Vmn  Hence on each cuboid Λ the function f is constant in (t ,...,t ). In particular, α n 1 n thisspacecontainsrandomvariablesoftheform g(X X ,...,X X ) L rm − rm−1 r1 − 0 ∈ 2 where g is aBorel function(see[4]) . Remark 4.1. By convolution with mollifiers we construct for any function f ∈ ([0,T] R3) satisfying (A ) a sequence (f ) converging uniformly to f in f n n 0 C([0,T]×R3),suchthatforalln Nandt [0,≥T]wehavef (t, ) (R3), n ∞ C × ∈ ∈ · ∈ C and f satisfiestheLipschitz-condition(5)withthesameconstantL forall n. n f Let (ξ ) D be a sequence converging to ξ in L . By (Yn,Zn) we will n n 0 1,2 2 ≥ ⊆ denotethesolutionto (4)withterminalcondition ξ and generator f .Then The- n n orem2.3 impliesthat (Yn,Zn) (Y,Z) if n in S H. (9) → → ∞ × Forξ Hthesolution(Y,Z)hasachaosexpansionwhichresemblesthoseofthe ∈ elementsofH : 8 Theorem 4.2. Let (A ) hold. For ξ H thechaos expansion of (Y,Z) S H f ∈ ∈ × hastherepresentation ∞ Y = I ϕα(t)1I , P λ-a.e., (10) t n n Λα∩]0,t]n ⊗ n=0 α Vn X X∈ m   whereϕα isjointlymeasurable,ϕα(t) L (Rn,µ n)and n n ∈ 2 ⊗ ∞ Z = I ψα(t,x)1I , P -a.e., (11) t,x n n Λα∩]0,t]n ⊗m n=0 α Vn X X∈ m   whereψα isjointlymeasurableandψα(t) L (Rn+1,µ n+1). n n ∈ 2 ⊗ Proof. We may use Remark 4.1 and approximate ξ H by a sequence (ξ ) n n ∈ ⊆ H D andf by(f ) satisfying(A 1).SincetheconvergenceinS H implies 1,2 n n f ∩ × convergencew.r.t. thenorm 1 (y,z) := y 2 + z 2 2 , (12) |k |k k kL2(P λ) k kH ⊗ andthespaceofprocesses(y,z)wi(cid:0)threpresentations(1(cid:1)0)and(11)isclosedwith respect to the norm (12) we only need to show that the assertion holds for any solution (Yn,Zn) w.r.t. (ξ ,f ). Hence we may assume that ξ H D and n n 1,2 ∈ ∩ f 0,1,1,1([0,T] R3)suchthat∂ f,∂ f and∂ f areboundedbyL . According x y z f ∈ C × to Theorem 3.1 we can differentiate (4) and obtain for -a.e. (t,x) and all s m ∈ [t,T]that T Y = ξ + f(r,X ,Y ,Z¯ )dr Z M(dr,dy). t,x s t,x t,x r r r t,x r,y D D D − D Zs Z]s,T] R × Theorem3.1 yieldsthat Z isaversionof p( Y ),hence t,x t D Z = Y P -a.e. t,x t,x t D ⊗m Wedefinetherecursion Y0 := 0, Z0 := 0, s s,y T k+1 := ξ + f r,X ,Yk,Z¯k dr, Yk+1 :=o( k+1), (13) Ys r r r Y Zs (cid:0) (cid:1) 9 where o denotes the optional projection, which is according to [11, Theorem 47 and Remark 50]ca`dla`g. Since Yk+1 = E k+1 P-a.s. wedefinefor u uYu T Yk+1 = E ξ +E f(r,X ,Yk,Z¯k)dr, s < u (14) Ds,y u uDs,y u Ds,y r r r Zu theset A := (s,y) [0,T] R : theRHS of(14) isca`dla`g on[s,T] P-a.s. k { ∈ × } and assume a pathwise ca`dla`g version of Yk+1 for any (s,y) A while we s,y k D ∈ let Yk+1 bezero otherwise. In thissensewecan set s,y D k+1 := lim Yk+1, Zk+1 :=p( k+1) (15) Zs,y t sDs,y t Z ց fork = 0,1,2,... TheprocessYk+1hasaca`dla`gversion,therefore,(Yk,Zk) S H forallk N. ∈ × ∈ In the proof of [28, Theorem 2.2] it is shown that (Yk,Zk) converges to (Y,Z) withrespect tothenorm(12). Consequently,weonlyneed to showthat (10) and(11)holdfor(Yk,Zk). Forfixed t ]0,T[wedescribe(10)byintroducingthespace ∈ ∞ H := I gα1I L : gα L (Rn,µ n) . t n n Λα∩]0,t]n ∈ 2 n ∈ 2 ⊗ (cid:26)n=0 (cid:18)α Vn (cid:19) (cid:27) X X∈ m From [4]onecan concludethefollowingfact: Lemma 4.3. For any Borel function h : Rd R and ξ ,...,ξ H such that 1 d t → ∈ h(ξ ,...,ξ ) L it holdsh(ξ ,...,ξ ) H . 1 d 2 1 d t ∈ ∈ Assumenowthat(10)and (11) holdforYk and Zk,respectively. Wehave ∞ Z¯k = I ψα(t,x)1I κ(dx) t R n n Λα∩[0,t]⊗n Z n=0 α Vn X X∈ m   ∞ = I ψα(t,x)κ(dx)1I (16) n R n Λα∩[0,t]⊗n n=0 α VnZ X X∈ m   ∞ = I ψ¯α(t)1I H . n n Λα∩[0,t]⊗n ∈ t n=0 α Vn X X∈ m   10