Malliavin Calculus for Backward Stochastic Differential Equations And

Home , Malliavin calculus

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Since then there have been extensive studies of this equation. We refer to the review paper by El Karoui, Peng and Quenez [7], to the books of El Karoui and Mazliak [6] and of Ma and Yong [12] and the references therein for more comprehensive presentation of the theory. A current important topic in the applications of BSDEs is the numerical approximation schemes. In most work on numerical simulations, a certain forward stochastic differential equation of the following form: t t (1.2) Xt = X0 + b(r, Xr,Yr) dr + σ(r, Xr) dWr Z0 Z0 is needed. Usually it is assumed that the generator f in (1.1) depends on Xr at the time r: f(r, Yr,Zr)= f(r, Xr,Yr,Zr), where f(r,x,y,z) is a deterministic function of (r,x,y,z), and f is global Lipschitz in (x,y,z). If in addition the terminal value ξ is of the form ξ = h(XT ), where h is a deterministic function, a so-called four-step numerical scheme has been developed by Ma, Protter and Yong in [11]. A basic ingredient in this paper is that the solution {Yt}0≤t≤T to the BSDE is of the form Yt = u(t, Xt), where u(t,x) is determined by a quasi-linear partial differential equation of parabolic type. Recently, Bouchard and Touzi [4] propose a Monte-Carlo approach which may be more suitable for high-dimensional problems. Again in this forward– backward setting, if the generator f has a quadratic growth in Z, a numerical approximation is developed by Imkeller and Dos Reis [9] in which a trunca- tion procedure is applied. In the case where the terminal value ξ is a functional of the path of the forward diffusion X, namely, ξ = g(X·), different approaches to construct numerical methods have been proposed. We refer to Bally [1] for a scheme with a random time partition. In the work by Zhang [16], the L2-regularity of Z is obtained, which allows one to use deterministic time partitions as well as to obtain the rate estimate (see Bender and Denk [2], Gobet, Lemor and Warin [8] and Zhang [16] for different algorithms). We should also mention the works by Briand, Delyon and Mémin [5] and Ma et al. [10], where the Brownian motion is replaced by a scaled random walk. The purpose of the present paper is to construct numerical schemes for the general BSDE (1.1), without assuming any particular form for the terminal value ξ and generator f. This means that ξ can be an arbitrary random variable, and f(r, y, z) can be an arbitrary Fr-measurable random variable (see Assumption 2.2 in Section 2 for precise conditions on ξ and f). The natural tool that we shall use is the Malliavin calculus. We emphasize that the main difficulty in constructing a numerical scheme for BSDEs is usually the approximation of the process Z. It is necessary to obtain some regularity properties for the trajectories of this process Z. The Malliavin calculus turns out to be a suitable tool to handle these problems because the random MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 3 variable Zt can be expressed in terms of the trace of the Malliavin derivative of Yt, namely, Zt = DtYt. This relationship was proved in the paper by El Karoui, Peng and Quenez [7] and was used by these authors to obtain estimates for the moments of Zt. We shall further exploit this identity to obtain the Lp-Hölder continuity of the process Z, which is the critical ingredient for the rate estimate of our numerical schemes. Our first numerical scheme was inspired by the paper of Zhang [16], where the author considers a class of BSDEs whose terminal value ξ takes the form g(X·), where X is a forward diffusion of the form (1.2), and g satisfies a Lipschitz condition with respect to the L∞ or L1 norms (similar assumptions for f). The discretization scheme is based on the regularity of the process Z in the mean square sense; that is, for any partition π = {0= t0

n−1 ti+1 E 2 2 (1.3) [|Zt − Zti | + |Zt − Zti+1 | ] dt ≤ K|π|, ti Xi=0 Z where |π| = max0≤i≤n−1(ti+1 − ti), and K is a constant independent of the partition π. We consider the case of a general terminal value ξ which is twice differentiable in the sense of Malliavin calculus, and the first and second derivatives satisfy some integrability conditions; we also made similar assumptions for the generator f (see Assumption 2.2 in Section 2 for details). In this sense our framework extends that of [13] and is also natural. In this framework, we are able to obtain an estimate of the form p p/2 (1.4) E|Zt − Zs| ≤ K|t − s| , where K is a constant independent of s and t. Clearly, (1.4) with p = 2 implies (1.3). Moreover, (1.4) implies the existence of a γ-Hölder continuous 1 1 version of the process Z for any γ < 2 − p . Notice that, up to now the path regularity of Z has been studied only when the terminal value and the generator are functional of a forward diffusion. After establishing the regularity of Z, we consider different types of numerical schemes. First we analyze a scheme similar to the one proposed in [16] [see (3.2)]. In this case we obtain a rate of convergence of the following type: T E π 2 E π 2 E π 2 sup |Yt − Yt | + |Zt − Zt | dt ≤ K(|π| + |ξ − ξ | ). 0≤t≤T Z0 Notice that this result is stronger than that in [16] which can be stated as (when ξπ = ξ) T E π 2 E π 2 sup |Yt − Yt | + |Zt − Zt | dt ≤ K|π|. 0≤t≤T Z0 4 Y. HU, D. NUALART AND X. SONG

We also propose and study an “implicit” numerical scheme [see (4.1) in Section 4 for the details]. For this scheme we obtain a much better result on the rate of convergence, T p/2 E π p E π 2 p/2 E π p sup |Yt − Yt | + |Zt − Zt | dt ≤ K(|π| + |ξ − ξ | ), 0≤t≤T Z0 where p> 1 depends on the assumptions imposed on the terminal value and the coefficients. In both schemes, the integral of the process Z is used in each iteration, and for this reason they are not completely discrete schemes. In order to implement the scheme on computers, one must replace an integral of the form ti+1 Zπ ds by discrete sums, and then the convergence of the obtained ti s scheme is hardly guaranteed. To avoid this discretization we propose a truly R discrete numerical scheme using our representation of Zt as the trace of the Malliavin derivative of Yt (see Section 5 for details). For this new scheme, we obtain a rate of convergence result of the form E π p π p p/2−ε max {|Yti − Yt | + |Zti − Zt | }≤ K|π| 0≤i≤n i i for any ε> 0. In fact, we have a slightly better rate of convergence (see Theorem 5.2), p/2 E π p π p p/2−p/(2 log(1/|π|)) 1 max {|Yti − Yt | + |Zti − Zt | }≤ K|π| log . 0≤i≤n i i |π| However, this type of result on the rate of convergence applies only to some classes of BSDEs, and thus this scheme remains to be further investigated. In the computer realization of our schemes or any other schemes, an ex- tremely important procedure is to compute the conditional expectation of E form (Y |Fti ). In this paper we shall not discuss this issue but only mention the papers [2, 4] and [8]. The paper is organized as follows. In Section 2 we obtain a representation of the martingale integrand Z in terms of the trace of the Malliavin derivative of Y , and then we get the Lp-Hölder continuity of Z by using this representation. The conditions that we assume on the terminal value ξ and the generator f are also specified in this section. Some examples of application are presented to explain the validity of the conditions. Section 3 is devoted to the analysis of the approximation scheme similar to the one introduced in [16]. Under some differentiability and integrability conditions in the sense of Malliavin calculus on ξ and the nonlinear coefficient f, we establish a better rate of convergence for this scheme. In Section 4, we introduce an “implicit” scheme and obtain the rate of convergence in the Lp norm. A completely discrete scheme is proposed and analyzed in Section 5. Throughout the paper for simplicity we consider only scalar BSDEs. The results obtained in this paper can be easily extended to multi-dimensional BSDEs. MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 5

2. The Malliavin calculus for BSDEs.

2.1. Notations and preliminaries. Let W = {Wt}0≤t≤T be a one-dimensional standard Brownian motion defined on some complete filtered probability space (Ω, F, P, {Ft}0≤t≤T ). We assume that {Ft}0≤t≤T is the filtration generated by the Brownian motion and the P -null sets, and F = FT . We denote by P the progressive σ-field on the product space [0, T ] × Ω. For any p ≥ 1 we consider the following classes of processes: • M 2,p, for any p ≥ 2, denotes the class of square integrable random variables F with a stochastic integral representation of the form T F = EF + ut dWt, Z0 E p where u is a progressively measurable process satisfying sup0≤t≤T |ut| < ∞. p • HF ([0, T ]) denotes the Banach space of all progressively measurable processes ϕ : ([0, T ] × Ω, P) → (R, B) with norm T p/2 1/p 2 kϕkHp = E |ϕt| dt < ∞. 0 p Z • SF ([0, T ]) denotes the Banach space of all the RCLL (right continuous with left limits) adapted processes ϕ : ([0, T ] × Ω, P) → (R, B) with norm 1/p p kϕkSp = E sup |ϕt| < ∞. 0≤t≤T Next, we present some preliminaries on Malliavin calculus, and we refer the reader to the book by Nualart [14] for more details. Let H = L2([0, T ]) be the separable Hilbert space of all square integrable real-valued functions on the interval [0, T ] with scalar product denoted by h·, ·iH. The norm of an element h ∈ H will be denoted by khkH. For any H T h ∈ we put W (h)= 0 h(t) dWt. ∞ Rn We denote by Cp ( ) the set of all infinitely continuously differentiable functions g : Rn → R suchR that g and all of its partial derivatives have poly- nomial growth. We make use of the notation ∂ g = ∂g whenever g ∈ C1(Rn). i ∂xi Let S denote the class of smooth random variables such that a random variable F ∈ S has the form

(2.1) F = g(W (h1),...,W (hn)), ∞ Rn H where g belongs to Cp ( ), h1,...,hn are in and n ≥ 1. The Malliavin derivative of a smooth random variable F of the form (2.1) is the H-valued random variable given by n DF = ∂ig(W (h1),...,W (hn))hi. Xi=1 6 Y. HU, D. NUALART AND X. SONG

For any p ≥ 1 we will denote the domain of D in Lp(Ω) by D1,p, meaning that D1,p is the closure of the class of smooth random variables S with respect to the norm p p 1/p kF k1,p = (E|F | + EkDF kH) . We can deﬁne the iteration of the operator D in such a way that for a smooth random variable F , the iterated derivative DkF is a random variable with values in H⊗k. Then for every p ≥ 1 and any natural number k ≥ 1 we introduce the seminorm on S deﬁned by

k 1/p E p E j p kF kk,p = |F | + kD F kH⊗j . ! Xj=1 We will denote by Dk,p the completion of the family of smooth random variables S with respect to the norm k · kk,p. Let µ be the Lebesgue measure on [0, T ]. For any k ≥ 1 and F ∈ Dk,p, the derivative k k D F = {Dt1,...,tk F,ti ∈ [0, T ], i = 1,...,k} is a measurable function on the product space [0, T ]k × Ω, which is deﬁned a.e. with respect to the measure µk × P . 1,p We use La to denote the set of real-valued progressively measurable processes u = {ut}0≤t≤T such that: 1,p (i) For almost all t ∈ [0, T ], ut ∈ D . E T 2 p/2 T T 2 p/2 (ii) (( 0 |ut| dt) + ( 0 0 |Dθut| dθ dt) ) < ∞. Notice thatR we can choose a progressivelyR R measurable version of the H-valued process {Dut}0≤t≤T .

2.2. Estimates on the solutions of BSDEs. The generator f in the BSDE (1.1) is a measurable function f : ([0, T ] × Ω × R × R, P⊗B⊗B) → (R, B), and the terminal value ξ is an FT -measurable random variable.

Definition 2.1. A solution to the BSDE (1.1) is a pair of progressively T 2 T measurable processes (Y,Z) such that 0 |Zt| dt < ∞, 0 |f(t,Yt,Zt)| dt < ∞, a.s. and R R T T Yt = ξ + f(r, Yr,Zr) dr − Zr dWr, 0 ≤ t ≤ T. Zt Zt The next lemma provides a useful estimate on the solution to the BSDE (1.1). MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 7

q q Lemma 2.2. Fix q ≥ 2. Suppose that ξ ∈ L (Ω), f(t, 0, 0) ∈ HF ([0, T ]) and f is uniformly Lipschitz in (y, z); namely, there exists a positive number L such that µ × P a.e.

|f(t,y1, z1) − f(t,y2, z2)|≤ L(|y1 − y2| + |z1 − z2|) for all y1,y2 ∈ R and z1, z2 ∈ R. Then there exists a unique solution pair q q (Y,Z) ∈ SF ([0, T ]) × HF ([0, T ]) to (1.1). Moreover, we have the following estimate for the solution: T q/2 q 2 E sup |Yt| + E |Zt| dt 0≤t≤T 0 (2.2) Z T q/2 ≤ K E|ξ|q + E |f(t, 0, 0)|2 dt , Z0 where K is a constant depending only on L, q and T .

Proof. The proof of the existence and uniqueness of the solution (Y,Z) can be found in [7], Theorem 5.1, with the local martingale M ≡ 0, since the ﬁltration here is the ﬁltration generated by the Brownian motion W . Esti- mate (2.2) can be easily obtained from Proposition 5.1 in [7] with (f 1,ξ1)= (f,ξ) and (f 2,ξ2) = (0, 0).

As we will see later, for a given BSDE the process Z will be expressed in terms of the Malliavin derivative of the solution Y , which will satisfy a linear BSDE with random coeﬃcients. To study the properties of Z we need to analyze a class of linear BSDEs. Let {αt}0≤t≤T and {βt}0≤t≤T be two progressively measurable processes. We will make use of the following integrability conditions:

Assumption 2.1. (H1) For any λ> 0, T E 2 Cλ := exp λ (|αt| + βt ) dt < ∞. Z0 (H2) For any p ≥ 1, p p Kp := sup E(|αt| + |βt| ) < ∞. 0≤t≤T

Under condition (H1), we denote by {ρt}0≤t≤T the solution of the linear stochastic diﬀerential equation dρ = α ρ dt + β ρ dW , 0 ≤ t ≤ T , (2.3) t t t t t t ρ = 1. 0 The following theorem is a critical tool for the proof of the main theorem in this section, and it has also its own interest. 8 Y. HU, D. NUALART AND X. SONG

q q Theorem 2.3. Let q>p ≥ 2 and let ξ ∈ L (Ω) and f ∈ HF ([0, T ]). As- sume that {αt}0≤t≤T and {βt}0≤t≤T are two progressively measurable processes satisfying conditions (H1) and (H2) in Assumption 2.1. Suppose that T 2,q the random variables ξρT and 0 ρtft dt belong to M , where {ρt}0≤t≤T is the solution to (2.3). Then the following linear BSDE, R T T (2.4) Yt = ξ + [αrYr + βrZr + fr] dr − Zr dWr, 0 ≤ t ≤ T, Zt Zt has a unique solution pair (Y,Z), and there is a constant K> 0 such that p p/2 (2.5) E|Yt − Ys| ≤ K|t − s| for all s,t ∈ [0, T ].

We need the following lemma to prove the above result.

Lemma 2.4. Let {αt}0≤t≤T and {βt}0≤t≤T be two progressively measurable processes satisfying condition (H1) in Assumption 2.1, and {ρt}0≤t≤T be the solution of (2.3). Then, for any r ∈ R we have E r (2.6) sup ρt < ∞. 0≤t≤T

Proof. Let t ∈ [0, T ]. The solution to (2.3) can be written as t β2 t ρ = exp α − s ds + β dW . t s 2 s s Z0 Z0 For any real number r, we have t β2 t E sup ρr = E sup exp r α − s ds + r β dW t s 2 s s 0≤t≤T 0≤t≤T Z0 Z0 T 1 T ≤ E exp |r| |α | ds + (|r| + r2) β2 ds s 2 s Z0 Z0 t r2 t × sup exp r β dW − β2 ds . s s 2 s 0≤t≤T Z0 Z0 Then, ﬁxing any p> 1 and using H¨older’s inequality, we obtain t pr2 t 1/p (2.7) E sup ρr ≤ C E sup exp rp β dW − β2 ds , t s s 2 s 0≤t≤T 0≤t≤T Z0 Z0 where T q T 1/q C = E exp q|r| |α | ds + (|r| + r2) β2 ds s 2 s Z0 Z0 1 1 and p + q = 1. MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 9

t r2 t 2 Set Mt = exp{r 0 βs dWs − 2 0 βs ds}. Then {Mt}0≤t≤T is a martingale due to (H1). We can rewrite (2.7) into R R 1/p E r E p (2.8) sup ρt ≤ C sup Mt . 0≤t≤T 0≤t≤T By Doob’s maximal inequality, we have E p E p (2.9) sup Mt ≤ cp MT 0≤t≤T for some constant cp > 0 depending only on p. Finally, choosing any γ> 1, 1 1 λ> 1 such that γ + λ = 1 and applying again the H¨older inequality yield T γ T EM p = E exp rp β dW − p2r2 β2 ds T s s 2 s Z0 Z0 γp − 1 T × exp pr2 β2 ds 2 s Z0 T 1 T 1/γ ≤ E exp rpγ β dW − γ2p2r2 β2 ds s s 2 s Z0 Z0 λ(γp − 1) T 1/λ × E exp pr2 β2 ds 2 s Z0 λ(γp − 1) T 1/λ = E exp pr2 β2 ds < ∞. 2 s Z0 Combining this inequality with (2.8) and (2.9) we complete the proof.

Proof of Theorem 2.3. The existence and uniqueness is well known. −1 We are going to prove (2.5). Let t ∈ [0, T ]. Denote γt = ρt , where {ρt}0≤t≤T is the solution to (2.3). Then {γt}0≤t≤T satisfies the following linear stochastic differential equation: 2 dγt = (−αt + βt )γt dt − βtγt dWt, 0 ≤ t ≤ T , γ = 1. 0 For any 0 ≤ s ≤ t ≤ T and any positive number r ≥ 1, we have, using (H2), the Hölder inequality, the Burkholder–Davis–Gundy inequality and Lem- ma 2.4 applied to the process {γt}0≤t≤T , t t r E r E 2 |γt − γs| = (−αu + βu)γu du − βuγu dWu Zs Zs

t r t r/2 r −1 E 2 E 2 2 (2.10) ≤ 2 (−αu+βu)γu du +Cr βuγu du Zs Zs

≤ C(t − s )r/2,

10 Y. HU, D. NUALART AND X. SONG where Cr is a constant depending only on r, and C is a constant depending on T , r and the constants appearing in conditions (H1) and (H2). From (2.3), (2.4) and by Itˆo’s formula, we obtain

d(Ytρt)= −ρtft dt + (βtρtYt + ρtZt) dWt. As a consequence, T T −1E E (2.11) Yt = ρt ξρT + ρrfr dr Ft = ξρt,T + ρt,rfr dr Ft , t t Z Z −1 where we write ρt,r = ρt ρr = γtρr for any 0 ≤ t ≤ r ≤ T . Now, ﬁx 0 ≤ s ≤ t ≤ T . We have T T p p E|Yt − Ys| = E E ξρt,T + ρt,rfr dr Ft − E ξρs,T + ρs,rfr dr Fs t s Z Z

p −1 p ≤ 2 E|E(ξρt,T |Ft) − E(ξρs,T |Fs)| T T p + E E ρt,rfr dr Ft − E ρs,rfr dr Fs Zt Zs

= 2p−1(I + I ). 1 2 First we estimate I1. We have p I1 = E|E(ξρt,T |Ft) − E(ξρs,T |Fs)| p = E|E(ξρt,T |Ft) − E(ξρs,T |Ft)+ E(ξρs,T |Ft) − E(ξρs,T |Fs)| p−1 p p ≤ 2 [E|E(ξρt,T |Ft) − E(ξρs,T |Ft)| + E|E(ξρs,T |Ft) − E(ξρs,T |Fs)| ] p−1 p p ≤ 2 [E|ξ(ρt,T − ρs,T )| + E|E(ξρs,T |Ft) − E(ξρs,T |Fs)| ] p−1 = 2 (I3 + I4). Using the H¨older inequality, Lemma 2.4 and the estimate (2.10) with r = 2pq q−p , the term I3 can be estimated as follows: q p/q pq/(q−p) (q−p)/q I3 ≤ (E|ξ| ) (E|ρt,T − ρs,T | ) E q p/q E 2pq/(q−p) (q−p)/(2q) E 2pq/(q−p) (q−p)/(2q) ≤ ( |ξ| ) ( |γt − γs| ) ( ρT ) ≤ C|t − s|p/2, where C is a constant depending only on p,q,T , E|ξ|q and the constants appearing in conditions (H1) and (H2). In order to estimate the term I4 we will make use of the condition ξρT ∈ M 2,q. This condition implies that T ξρT = E(ξρT )+ ur dWr, Z0 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 11

E q where u is a progressively measurable process satisfying sup0≤t≤T |ut| < ∞. Therefore, by the Burkholder–Davis–Gundy inequality, we have q E|E(ξρT |Ft) − E(ξρT |Fs)| t q t q/2 E E 2 = ur dWr ≤ Cq ur dr Zs Zs t (q−2) /2 q ≤ C q(t − s) E |ur| dr Zs q/2 q ≤ Cq(t − s) sup E|ut| . 0≤t≤T

As a consequence, from the deﬁnition of I4 we have p I4 = E|γs[E(ξρT |Ft) − E(ξρT |Fs)]| E pq/(q−p) (q−p)/q E E E q p/q ≤ ( γs ) ( | (ξρT |Ft) − (ξρT |Fs)| ) ≤ C|t − s|p/2, E q where C is a constant depending on p,q,T, sup0≤t≤T |ut| < ∞ and the constants appearing in conditions (H1) and (H2). The term I2 can be decomposed as follows: T T p I2 = E E ρt,rfr dr Ft − E ρs,rfr dr Fs t s Z Z T T p p −1 ≤ 3 E E ρt,rfr dr Ft − E ρs,rfr dr Ft t t Z Z T T p

+ E E ρs,rfr dr Ft − E ρs,rfr dr F t t s Z Z T T p

+ E E ρs,rfr dr Ft − E ρs,rfr dr Fs Zs Zs

= 3p−1(I + I + I ). 5 6 7 ′ Let us ﬁrst estimate the term I5. Suppose that p

12 Y. HU, D. NUALART AND X. SONG

T p′q/(2(q−p′)) p(q−p′)/(p′q) p/2 E 2 ≤ C|t − s| ρr dr Zt T q/2 p/q E 2 × fr dr Zt p/2 p ≤ C|t − s| kfkHq , where C is a constant depending on p,p′, q, T and the constants appearing in conditions (H1)b and (H2). ′ Nowb we estimate I6. Suppose that p

= E E ρ rfr dr Ft − E ρ rfr dr Fs Z0 Z0 t q q/2 q = E vr dWr ≤ Cq(t − s) sup E|v t| . Zs 0≤t≤T

MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 13

Finally, we estimate I7 as follows: T T p I7 = E E ρs,rfr dr Ft − E ρs,rfr dr Fs s s Z Z T T p E −1 E E = ρs ρrfr dr Ft − ρrfr dr Fs Zs Zs

E −pq/(q−p) (q−p)/p ≤ { ρs } (2.12) T T q p/q × E E ρrfr dr Ft − E ρrfr dr Fs s s Z Z q p/q T T ≤ C E E ρrfr dr Ft − E ρrfr dr Fs Zs Zs

p/2 ≤ C|t − s| , where C is a constant depending on p, q, T , sup E|v |q and the con- b 0≤t≤T t stants appearing in conditions (H1) and (H2). As ab consequence, we obtain for all s,t ∈ [0, T ] p p/2 E|Yt − Ys| ≤ K|t − s| , where K is a constant independent of s and t.

2.3. The Malliavin calculus for BSDEs. We return to the study of (1.1). The main assumptions we make on the terminal value ξ and generator f are the following:

q Assumption 2.2. Fix 2 ≤ p< 2 . (i) ξ ∈ D2,q, and there exists L> 0, such that for all θ,θ′ ∈ [0, T ], p ′ p/2 (2.13) E|Dθξ − Dθ′ ξ| ≤ L|θ − θ | , q (2.14) sup E|Dθξ| < ∞ 0≤θ≤T and q (2.15) sup sup E|DuDθξ| < ∞. 0≤θ≤T 0≤u≤T (ii) The generator f(t,y,z) has continuous and uniformly bounded ﬁrst- and second-order partial derivatives with respect to y and z, and f(·, 0, 0) ∈ q HF ([0, T ]). (iii) Assume that ξ and f satisfy the above conditions (i) and (ii). Let (Y,Z) be the unique solution to (1.1) with terminal value ξ and generator f. 1,q For each (y, z) ∈ R × R, f(·,y,z), ∂yf(·,y,z) and ∂zf(·,y,z) belong to La , 14 Y. HU, D. NUALART AND X. SONG and the Malliavin derivatives Df(·,y,z), D∂yf(·,y,z) and D∂zf(·,y,z) satisfy

T p/2 2 ′ p/2 (2.19) E |Dθf(r, Yr,Zr) − Dθ′ f(r, Yr,Zr)| dr ≤ L|θ − θ | . Zt 1,q For each θ ∈ [0, T ], and each pair of (y, z), Dθf(·,y,z) ∈ La and it has continuous partial derivatives with respect to y, z, which are denoted by ∂yDθf(t,y,z) and ∂zDθf(t,y,z), and the Malliavin derivative DuDθf(t,y,z) satisﬁes T q/2 2 (2.20) sup sup E |DuDθf(t,Yt,Zt)| dt < ∞. 0≤θ≤T 0≤u≤T Zθ∨u The following property is easy to check and we omit the proof.

Remark 2.5. Conditions (2.17) and (2.18) imply

T q/2 2 sup E |∂yDθf(t,Yt,Zt)| dt < ∞ 0≤θ≤T Zθ and T q/2 2 sup E |∂zDθf(t,Yt,Zt)| dt < ∞, 0≤θ≤T Zθ respectively.

The following is the main result of this section.

Theorem 2.6. Let Assumption 2.2 be satisﬁed.

(a) There exists a unique solution pair {(Yt,Zt)}0≤t≤T to the BSDE (1.1), 1,q and Y,Z are in La . A version of the Malliavin derivatives {(DθYt, MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 15

DθZt)}0≤θ,t≤T of the solution pair satisﬁes the following linear BSDE: T DθYt = Dθξ + [∂yf(r, Yr,Zr)DθYr Zt (2.21) + ∂zf(r, Yr,Zr)DθZr + Dθf(r, Yr,Zr)] dr T − DθZr dWr, 0 ≤ θ ≤ t ≤ T ; Zt (2.22) DθYt = 0, DθZt = 0, 0 ≤ t<θ ≤ T.

Moreover, {DtYt}0≤t≤T deﬁned by (2.21) gives a version of {Zt}0≤t≤T , namely, µ × P a.e.

(2.23) Zt = DtYt. (b) There exists a constant K> 0, such that, for all s,t ∈ [0, T ], p p/2 (2.24) E|Zt − Zs| ≤ K|t − s| . Proof. Part (a): The proof of the existence and uniqueness of the solu- 1,2 tion (Y,Z), and Y,Z ∈ La is similar to that of Proposition 5.3 in [7], and also the fact that (DθYt, DθZt) is given by (2.21) and (2.22). In Proposition 5.3 T in [7] the exponent q is equal to 4, and one assumes that 0 kDθf(·, Y, Z)k2 dθ < ∞, which is a consequence of (2.16) and the fact that Y,Z ∈ L1,2. H2 R a Furthermore, from conditions (2.14) and (2.16) and the estimate in Lem- ma 2.2, we obtain T q/2 q 2 (2.25) sup E sup |DθYt| + E |DθZt| dt < ∞. 0≤θ≤T θ≤t≤T Zθ 1,q Hence, by Proposition 1.5.5 in [14], Y and Z belong to La . Part (b): Let 0 ≤ s ≤ t ≤ T . In this proof, C> 0 will be a constant independent of s and t, and may vary from line to line. By representation (2.23) we have

Denote αu = ∂yf(u, Yu,Zu) and βu = ∂zf(u, Yu,Zu) for all u ∈ [0, T ]. Then, by Assumption 2.2(ii), the processes α and β satisfy conditions (H1) and (H2) in Assumption 2.1, and from (2.21) we have for r ∈ [s, T ] T T DsYr = Dsξ + [αuDsYu + βuDsZu + Dsf(u, Yu,Zu)] du − DsZu dWu. Zr Zr p ′ Next, we are going to use Theorem 2.3 to estimate E|DsYt − DsYs| . Fix p ′ q ′ q p′ with p

+ ∂zf(θ,Yθ,Zθ) r + (∂yyf(u, Yu,Zu) − ∂yzf(u, Yu,Zu)βu)DθYu du θ Z r + (∂yzf(u, Yu,Zu) − ∂zzf(u, Yu,Zu)βu)DθZu du θ Z r + (Dθ∂yf(u, Yu,Zu) − βuDθ∂zf(u, Yu,Zu)) du . Zθ By the boundedness of the ﬁrst- and second-order partial derivatives of f with respect to y and z, (2.17), (2.18), (2.25), Lemma 2.4, the H¨older inequality and the Burkholder–Davis–Gundy inequality, it is easy to show that for any p′′

= E ρrDsf(r, Yr,Zr) dr Zs T T + E [DθρrDsf(r, Yr,Zr) Z0 Zs + ρr∂yDsf(r, Yr,Zr)DθYr

+ ρr∂zDsf(r, Yr,Zr)DθZr

+ ρrDθDsf(r, Yr,Zr)] dr Fθ dWθ T T E s = ρrDsf(r, Yr,Zr) dr + vθ dWθ. Zs Z0 E s p′ E s p′ We claim that sup0≤θ≤T |uθ| < ∞ and sup0≤θ≤T |vθ| < ∞. In fact, E s p′ E E p′ |uθ| = | (DθρT Dsξ + ρT DθDsξ|Fθ)| p′−1 p′ p′ ≤ 2 (E|DθρT Dsξ| + E|ρT DθDsξ| ) p′−1 p′q/(q−p′) (q−p′)/q q p′/q ≤ 2 ((E|DθρT | ) (E|Dsξ| ) E p′q/(q−p′) (q−p′)/q E q p′/q + ( ρT ) ( |DθDsξ| ) ). E s p′ By (2.14), (2.15), (2.28) and Lemma 2.4, we have sup0≤s≤T sup0≤θ≤T |uθ| < ∞. On the other hand, T E s p′ E E |vθ| = [DθρrDsf(r, Yr,Zr) Zs

+ ρr∂yDsf(r, Yr,Zr)DθYr

+ ρr∂zDsf(r, Yr,Zr)DθZr p′ + ρrDθDsf(r, Yr,Zr)] dr Fθ ′ p −1 ≤ 4 [J1 + J2 + J3 + J4], 18 Y. HU, D. NUALART AND X. SONG where T p′ J1 = E DθρrDsf(r, Yr,Zr) dr , Zs ′ T p J2 = E ρr∂yDsf(r, Yr,Zr)DθY r dr , Zs ′ T p J3 = E ρr∂zDsf(r, Yr,Zr)DθZr dr Zs and T p′ J4 = E ρrDθDsf(r, Yr,Zr) dr . Zs

For J1, we have

Corollary 2.7. Under the assumptions in Theorem 2.2, let (Y,Z) ∈ q q SF ([0, T ]) × HF ([0, T ]) be the unique solution pair to (1.1). If E q sup0≤t≤T |Zt| < ∞, then there exists a constant C, such that, for any s,t ∈ [0, T ], q q/2 (2.30) E|Yt − Ys| ≤ C|t − s| . 20 Y. HU, D. NUALART AND X. SONG

Proof. Without loss of generality we assume 0 ≤ s ≤ t ≤ T . C > 0 is a constant independent of s and t, which may vary from line to line. Since t t Ys = Yt + f(r, Yr,Zr) dr − Zr dWr, Zs Zs we have, by the Lipschitz condition on f, t t q q E|Yt − Ys| = E f(r, Yr,Zr) dr − Zr dWr Zs Zs t q t q q −1 ≤ 2 E f(r, Yr,Zr) dr + E Zr dWr Zs Zs t q/2 t q/2 q/2 2 2 ≤ Cq |t − s| E |f(r, Y r,Zr)| dr + E |Zr| dr Zs Zs t q/2 t q/2 q/2 2 2 ≤ C |t − s| E |Yr| dr + E |Zr| dr Zs Zs t q/2 + E |f(r, 0, 0)|2 dr Zs q/2 q + |t − s| sup E|Zr| 0≤r≤T ≤ C|t − s|q/2. The proof is complete.

Remark 2.8. From Theorem 2.6 we know that {(DθYt, DθZt)}0≤θ≤t≤T satisfies equation (2.21) and Zt = DtYt, µ × P a.e. Moreover, since (2.14) and (2.16) hold, we can apply the estimate (2.2) in Lemma 2.2 to the linear E q BSDE (2.21) and deduce sup0≤t≤T |Zt| < ∞. Therefore, by Lemma 2.7, the process Y satisfies the inequality (2.30). By Kolmogorov’s continuity criterion this implies that Y has Hölder continuous trajectories of order γ 1 1 for any γ< 2 − q .

2.4. Examples. In this section we discuss three particular examples where Assumption 2.2 is satisﬁed.

Example 2.9. Consider equation (1.1). Assume that: (a) f(t,y,z) : [0, T ] × R × R → R is a deterministic function that has uniformly bounded ﬁrst- and second-order partial derivatives with respect to y T 2 and z, and 0 f(t, 0, 0) dt < ∞. R MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 21

(b) The terminal value ξ is a multiple stochastic integral of the form

(2.31) ξ = g(t1,...,tn) dWt1 · · · dWtn , n Z[0,T ] where n ≥ 2 is an integer and g(t1,...,tn) is a symmetric function in L2([0, T ]n), such that

2 sup g(t1,...,tn−1, u) dt1 · · · dtn−1 < ∞, n−1 0≤u≤T Z[0,T ] 2 sup g(t1,...,tn−2, u, v) dt1 · · · dtn−2 < ∞, n−2 0≤u,v≤T Z[0,T ] and there exists a constant L> 0 such that for any u, v ∈ [0, T ]

2 |g(t1,...,tn−1, u) − g(t1,...,tn−1, v)| dt1 · · · dtn−1

Duξ = n g(t1,...,tn−1, u) dWt1 · · · dWtn−1 . n−1 Z[0,T ] The above assumption implies Assumption 2.2, and therefore, Z satisﬁes the H¨older continuity property (2.24).

Example 2.10. Let Ω= C0([0, 1]) equipped with the Borel σ-field and Wiener measure. Then, Ω is a Banach space with supremum norm k · k∞, and Wt = ω(t) is the canonical Wiener process. Consider equation (1.1) on the interval [0, 1]. Assume that: (g1) f(t,y,z) : [0, 1] × R × R → R is a deterministic function that has uniformly bounded first- and second-order partial derivatives with respect to y 1 2 and z, and 0 f(t, 0, 0) dt < ∞. (g2) ξ = ϕ(W ), where ϕ :Ω → R is twice Fréchet differentiable, and the R first- and second-order Fréchet derivatives δϕ and δ2ϕ satisfy 2 r |ϕ(ω)| + kδϕ(ω)k + kδ ϕ(ω)k≤ C1 exp {C2kωk∞} for all ω ∈ Ω and some constants C1 > 0, C2 > 0 and 0 0 such that for all 0 ≤ θ ≤ θ′ ≤ 1, E|λ((θ,θ′])|p ≤ L|θ − θ′|p/2 for some p ≥ 2. 22 Y. HU, D. NUALART AND X. SONG

It is easy to show that Dθξ = λ((θ, 1]) and DuDθξ = ν((θ, 1]×(u, 1]), where ν denotes the signed measure on [0, 1] × [0, 1] associated with δ2ϕ. From the above assumptions and Fernique’s theorem, we can get Assumption 2.2, and therefore, the H¨older continuity property (2.24) of Z.

Example 2.11. Consider the following forward–backward stochastic diﬀerential equation (FBSDE for short): t t Xt = X0 + b(r, Xr) dr + σ(r, Xr) dWr, (2.32)  Z0 Z0  T T T  2  Yt = ϕ Xr dr + f(r, Xr,Yr,Zr) dr − Zr dWr, Z0 Zt Zt where b, σ, ϕ and f are deterministic functions, and X ∈ R.  0 We make the following assumptions: (h1) b and σ has uniformly bounded ﬁrst- and second-order partial derivatives with respect to x, and there is a constant L> 0, such that, for any s,t ∈ [0, T ], x ∈ R, |σ(t,x) − σ(s,x)|≤ L|t − s|1/2.

(h2) sup0≤t≤T {|b(t, 0)| + |σ(t, 0)|} < ∞. (h3) ϕ is twice differentiable, and there exist a constant C> 0 and a positive integer n such that T T T 2 ′ 2 ′′ 2 n ϕ Xt dt + ϕ Xt dt + ϕ Xt dt ≤ C(1 + kXk∞) , Z0 Z0 Z0 where kxk∞ = sup {| x(t)|, 0 ≤ t ≤ T } for any x ∈ C([0, T ]). (h4) f(t,x,y,z ) has continuous and uniformly bounded first- and second- T 2 order partial derivatives with respect to x,y and z and 0 f(t, 0, 0, 0) dt < ∞. T 2 Notice that in this example, Φ(X)= ϕ( 0 Xt dt) is notR necessarily globally Lipschitz in X, and the results of [16] cannot be applied directly. Under the above assumptions, (h1) andR (h4), equation (2.32) has a unique solution triple (X,Y,Z), and we have the following classical results: for any real number r> 0, there exists a constant C> 0 such that r r r/2 E sup |Xt| < ∞, E|Xt − Xs| ≤ C|t − s| 0≤t≤T for any t,s ∈ [0, T ]. For any fixed (y, z) ∈ R × R, we have Dθf(t, Xt,y,z)= ∂xf(t, Xt,y,z)DθXt. Then, under all the assumptions in this example, by Theorem 2.2.1 and Lemma 2.2.2 in [14] and the results listed above, we can verify Assumption 2.2. Therefore, Z has the Hölder continuity property (2.24). Note that in the multidimensional case we do not require the matrix σσT to be invertible. MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 23

3. An explicit scheme for BSDEs. In the remaining part of this paper, we let π = {0= t0

ti+1 ti+1

(3.1) Yt = Yti+1 + f(r, Yr,Zr) dr − Zr dWr. Zt Zt Comparing with the numerical schemes for forward stochastic diﬀerential equations, we could introduce a numerical scheme of the form 1,π π Ytn = ξ ,

ti+1 1,π 1,π 1,π 1,π 1,π Yti = Yti+1 + f(ti+1,Yti+1 ,Zti+1 )∆i − Zr dWr, Zti t ∈ [ti,ti+1), i = n − 1,n − 2,..., 0, where ξπ ∈ L2(Ω) is an approximation of the terminal condition ξ. This leads 1,π 1,π to a backward recursive formula for the sequence {Yti ,Zti }0≤i≤n. In fact, 1,π 1,π 1,π once Yti+1 and Zti+1 are deﬁned, then we can ﬁnd Yti by 1,π E 1,π 1,π 1,π Yti = (Yti+1 + f(ti+1,Yti+1 ,Zti+1 )∆i|Fti ), 1,π and {Zr }ti≤r

ti+2 π π π E 1 π (3.2) Yt = Yti+1 + f ti+1,Yti+1 , Zr dr Fti+1 ∆i ∆i+1 t Z i+1 ti+1 π − Zr dWr, t ∈ [ti,ti+1), i = n − 1,n − 2,..., 0, Zt where, by convention, E( 1 ti+2 Zπ dr|F )=0 when i = n − 1. In [16] ∆i+1 ti+1 r ti+1 the following rate of convergence is proved for this approximation scheme, R assuming that the terminal value ξ and the generator f are functionals of a forward diﬀusion associated with the BSDE, T E π 2 E π 2 (3.3) max |Yti − Yt | + |Zt − Zt | dt ≤ K|π|. 0≤i≤n i Z0 24 Y. HU, D. NUALART AND X. SONG

The main result of this section is the following, which on one hand im- proves the above rate of convergence, and on the other hand extends terminal value ξ and generator f to more general situation.

Theorem 3.1. Consider the approximation scheme (3.2). Let Assump- tion 2.2 be satisﬁed, and let the partition π satisfy max0≤i≤n−1 ∆i/∆i+1 ≤ L1, where L1 is a constant. Assume that a constant L2 > 0 exists such that 1/2 (3.4) |f(t2,y,z) − f(t1,y,z)|≤ L2|t2 − t1| for all t1,t2 ∈ [0, T ] and y, z ∈ R. Then there are positive constants K and δ, independent of the partition π, such that, if |π| < δ, then T E π 2 E π 2 E π 2 (3.5) sup |Yt − Yt | + |Zt − Zt | dt ≤ K(|π| + |ξ − ξ | ). 0≤t≤T Z0 Proof. In this proof, C> 0 will denote a constant independent of the partition π, which may vary from line to line. Inequality (2.24) in Theo- rem 2.6(b) yields the following estimate (Theorem 3.1 in [16]) with p = 2:

n−1 ti+1 E 2 2 (|Zt − Zti | + |Zt − Zti+1 | ) dt ≤ C|π|. ti Xi=0 Z Using this estimate and following the same argument as the proof of Theo- rem 5.3 in [16], we can obtain the following result: T E π 2 E π 2 E π 2 (3.6) max |Yti − Yt | + |Zt − Zt | dt ≤ C(|π| + |ξ − ξ | ). 0≤i≤n i Z0 Denote 0, if i = n; π 1 ti+1 (3.7) Zti = E π  Zr dr Fti , if i = n − 1,n − 2,..., 0.  ∆i ti Z e If ti ≤ t

We rewrite the BSDE (1.1) as follows: T T Yt = ξ + f(r, Yr,Zr) dr − Zr dWr t t (3.9) Z Z n T π = ξ + f(tk,Ytk ,Ztk )∆k−1 − Zr dWr + Rt , t kX=i+1 Z where T n π |Rt | = f(r, Yr,Zr) dr − f(tk,Ytk ,Ztk )∆k−1 t Z k=i+1 X n tk t

= [f(r, Yr,Zr) − f(tk,Ytk ,Ztk )] dr − f(r, Yr,Zr) dr tk−1 ti kX=i+1 Z Z n tk ti+1

ti+1 p E max |f(r, Yr,Zr)| dr 0≤i≤n−1 ti (3.10) Z T p/2 p/2 2 ≤ |π| E |f(r, Yr,Zr)| dr . Z0 Deﬁne a function {t(r)}0≤r≤T by T, if r = T , t(r)= t , if t ≤ r

T p/2 p−1 p/2 2 + 2 |π| E |f(r, Yr,Zr)| dr Z0 ≤ C|π|p/2, where, by convention, RT = 0. In particular, we obtain E π 2 (3.12) sup |Rt | ≤ C|π|. 0≤t≤T To simplify the notation we denote π π π π δYt = Yt − Yt , δZt = Zt − Zt for all t ∈ [0, T ] and π π Zti = Zti − Zti for i = n,n − 1,..., 0.

Then, when ti ≤ t

From (3.12), we deduce

n 2 E π 2 E π E π 2 sup |δYt | ≤ C |ftk |∆k−1 + |δξ | + |π| . ti≤t≤T ( ! ) kX=i+1 Using the Lipschitz condition on f, wee obtain

E π 2 2E π 2 sup |δYt | ≤ C (T − ti) sup |δYtk | ti≤t≤T ( i+1≤k≤n n−1 2 E π E 2 2 (3.14) + |Ztk |∆k−1 + |Ztn | ∆n−1 ! ) kX=i+1 + C(E|δξπ|2 + |πb|). b Notice that n−1 2 n−1 2 1 tk+1 E |Zπ |∆ = E Z − E(Zπ|F ) du ∆ tk k−1 tk ∆ u tk k−1 k=i+1 ! k=i+1 k Ztk ! X X b n−1 2 tk+1 E ∆k−1 E π ≤ (|Ztk − Zu ||Ftk ) du ∆k tk ! kX=i+1 Z 2 n−1 tk+1 2E E π ≤ L1 (|Ztk − Zu ||Ftk ) du k=i+1 Ztk ! (3.15) X 2 n−1 tk+1 2 E E ≤ 2L1 (|Ztk − Zu||Ftk ) du ( tk ! kX=i+1 Z 2 n−1 tk+1 E E π + (|Zu − Zu ||Ftk ) du tk ! ) kX=i+1 Z 2 = 2L1(I1 + I2). Now the Minkowski and the H¨older inequalities yield 2 n−1 tk+1 1/2 E E 2 1/2 I1 ≤ ( (|Ztk − Zu||Ftk )) du ∆k tk ! kX=i+1Z n−1 tk+1 E E 2 ≤ (T − ti) ( (|Ztk − Zu||Ftk )) du k=i+1 Ztk (3.16) X n−1 tk+1 E 2 ≤ (T − ti) |Ztk − Zu| du tk kX=i+1 Z 28 Y. HU, D. NUALART AND X. SONG

n−1 tk+1 ≤ C(T − ti) |tk − u| du ≤ C|π|. tk kX=i+1 Z In a similar way and by (3.6), we obtain

n−1 tk+1 E π 2 I2 ≤ (T − ti) |Zu − Zu | du k=i+1 Ztk (3.17) X T E π 2 = (T − ti) |δZu | du ≤ C|π|. Zti+1 On the other hand, E π 2 E 2 2 2 (3.18) (Ztn ∆n−1) = |Ztn | |∆n−1| ≤ C|π| . From (3.14)–(3.18), we have b E π 2 2E π 2 sup |δYt | ≤ C1(T − ti) sup |δYtk | ti≤t≤T i+1≤k≤n (3.19) π 2 + C2(E|δξ | + |π|), where C1 and C2 are two positive constants independent of the partition π. We can ﬁnd a constant δ> 0 independent of the partition π, such that 2 1 T C1(3δ) < 2 and T > 2δ. Denote l = [ 2δ ] ([x] means the greatest integer no larger than x). Then l ≥ 1 is an integer independent of the partition π. If |π| < δ, then for the partition π we can choose n − 1 > i1 > i2 > · · · > il ≥ 0, such that, T − 2δ ∈ (ti1−1,ti1 ], T − 4δ ∈ (ti2−1,ti2 ],...,T − 2δl ∈ [0,til ] (with t−1 = 0).

For simplicity, we denote ti0 = T and til+1 = 0. Each interval [tij+1 ,tij ], j =

0, 1,...,l, has length less than 3δ, that is, |tij − tij+1 | < 3δ. On each interval

[tij+1 ,tij ], j = 0, 1,...,l, we consider the recursive formula (3.2), and (3.19) becomes E π 2 2E π 2 sup |δYt | ≤ C1(tij − tij+1 ) sup |δYtk | tij+1 ≤t≤tij ij+1+1≤k≤ij (3.20) π 2 + C2(E|δY | + |π|). tij Using (3.20), we can obtain inductively E π 2 sup |δYt | tij+1 ≤t≤tij 2 π 2 π 2 ≤ C1(ti − ti ) E sup |δY | + C2(E|δY | + |π|) j j+1 tk tij ij+1+1≤k≤ij 2 2E π 2 ≤ C1(tij − tij ) · · · C1(tij − tij −1) |δY | +1 tij MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 29

π 2 + C2(E|δY | + |π|) tij 2 2 2 × (1 + C1(tij − tij+1 ) + C1(tij − tij+1 ) C1(tij − tij+1+1) 2 2 2 (3.21) + · · · + C1(tij − tij+1 ) C1(tij − tij+1+1) · · · C1(tij − tij −1) )

2 ij −ij+1 π 2 ≤ (C1(3δ) ) E|δY | tij π 2 + C2(E|δY | + |π|) tij

2 2 2 2 ij −ij+1 × (1 + C1(3δ) + (C1(3δ) ) + · · · + (C1(3δ) ) ) E π 2 C2 E π 2 ≤ |δYti | + 2 ( |δYti | + |π|) j 1 − C1(3δ) j π 2 π 2 ≤ E|δY | + 2C2(E|δY | + |π|) tij tij π 2 = (2C2 + 1)E|δY | + 2C2|π|. tij By recurrence, we obtain E π 2 sup |δYt | tij+1 ≤t≤tij j+1 π 2 j ≤ (2C2 + 1) E|δξ | + C2|π|(1 + (2C2 +1)+ · · · + (2C2 + 1) ) (3.22) l+1 π 2 l ≤ (2C2 + 1) E|δξ | + C2|π|(1 + (2C2 +1)+ · · · + (2C2 + 1) ) 3(2C + 1)l+1 ≤ 2 (E|δξπ|2 + |π|). 2 l+1 3(2C2+1) Therefore, taking C = 2 , we obtain E π 2 E π 2 E π 2 sup |δYt | ≤ max sup |δYt | ≤ C(|π| + |ξ − ξ | ). 0≤j≤l 0≤t≤T tij+1 ≤t≤tij Combining the above estimate with (3.6), we know that there exists a constant K> 0 independent of the partition π, such that T E π 2 E π 2 E π 2 sup |Yt − Yt | + |Zt − Zt | dt ≤ K(|π| + |ξ − ξ | ). 0≤t≤T Z0 Remark 3.2. The numerical scheme introduced before, as other similar schemes, involves the computation of conditional expectations with respect to the σ-ﬁeld Fti+1 . To implement this scheme in practice we need to ap- proximate these conditional expectations. Some work has been done to solve this problem, and we refer the reader to the references [2, 4] and [8].

4. An implicit scheme for BSDEs. In this section, we propose an implicit numerical scheme for the BSDE (1.1). Deﬁne the approximating pairs 30 Y. HU, D. NUALART AND X. SONG

(Y π,Zπ) recursively by π π Ytn = ξ , 1 ti+1 ti+1 (4.1) Y π = Y π + f t ,Y π , Zπ dr ∆ − Zπ dW , t ti+1 i+1 ti+1 ∆ r i r r i Zti Zt t ∈ [ti,ti+1), i = n − 1,n − 2,..., 0, π where the partition π and ∆i, i = n−1,..., 0, are deﬁned in Section 3, and ξ is an approximation of the terminal value ξ. In this recursive formula (4.1), on each subinterval [ti,ti+1), i = n−1,..., 0, the nonlinear “generator” f con- tains the information of Zπ on the same interval. In this sense, this formula is π π diﬀerent from formula (3.2), and (4.1) is an equation for {(Yt ,Zt )}ti≤t

Theorem 4.1. Let 0 ≤ a < b ≤ T and p ≥ 2. Let ξ be Fb-measurable and p ξ ∈ L (Ω). If there exists a constant L> 0 such that g : (Ω × R, Fb ⊗B) → (R, B) satisﬁes

|g(z1) − g(z2)|≤ L|z1 − z2| p for all z1, z2 ∈ R and g(0) ∈ L (Ω), then there is a constant δ(p,L) > 0, such that, when b − a < δ(p,L), equation (4.2) has a unique solution (Y,Z) ∈ p p SF ([a, b]) × HF ([a, b]). Proof. We shall use the ﬁxed point theorem for the mapping from p p HF ([a, b]) into HF ([a, b]) which maps z to Z, where (Y,Z) is the solution of the following BSDE: b b (4.3) Yt = ξ + g zr dr − Zr dWr, t ∈ [a, b]. Za Zt In fact, by the martingale representation theorem, there exist a progressively E b 2 measurable process Z = {Zt}a≤t≤b such that a Zt dt < ∞ and b b b R ξ + g zr dr = E ξ + g zr dr Fa + Zt dWt. a a a Z Z Z By the integrability properties of ξ, g(0) and z, one can show that Z ∈

p E b HF ([a, b]). Deﬁne Yt = (ξ + g( a zr dr)|Ft),t ∈ [a, b]. Then (Y,Z) satisﬁes equation (4.3). Notice that Y is a martingale. Then by the Lipschitz condition on g, the integrability of ξ, gR(0) and z, and Doob’s maximal inequality, p we can prove that Y ∈ SF ([a, b]). MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 31

1 2 p 1 1 Let z , z be two elements in the Banach space HF ([a, b]), and let (Y ,Z ), (Y 2,Z2) be the associated solutions, that is, b b i i i Yt = ξ + g zr dr − Zr dWr, t ∈ [a, b], i = 1, 2. Za Zt Denote Y¯ = Y 1 − Y 2, Z¯ = Z1 − Z2, z¯ = z1 − z2. Then b b b ¯ 1 2 ¯ (4.4) Yt = g zr dr − g zr dr − Zr dWr Za Za Zt for all t ∈ [a, b]. So b b ¯ E 1 2 Yt = g zr dr − g zr dr Ft a a Z Z for all t ∈ [a, b]. Thus by Doob’s maximal inequality, we have b b p E ¯ p E E 1 2 sup |Yt| = sup g zr dr − g zr dr Ft a≤t≤b a≤t≤b Za Za

b b p E 1 2 ≤ C g zr dr − g zr dr Za Za (4.5) b b p E 1 2 ≤ C zr dr − zr dr Za Za b p/2 p/2 2 ≤ C(b − a) E |z¯r| dr , Za where C> 0 is a generic constant depending on L and p, which may vary from line to line. From (4.4), it is easy to see t Y¯t = Y¯a + Z¯r dWr Za for all t ∈ [a, b]. Therefore, by the Burkholder–Davis–Gundy inequality and (4.5), we have b p/2 t p 2 E |Z¯r| dr ≤ CE sup Z¯r dWr Za a≤t≤b Za

E ¯ p E ¯ p (4.6) ≤ C |Ya| + sup |Y t| a≤t≤b h i b p/2 p/2 2 ≤ C(b − a) E |z¯r| dr , Za 32 Y. HU, D. NUALART AND X. SONG that is, 1/2 kZ¯kHp ≤ C1(b − a) kz¯kHp , where C1 is a positive constant depending only on L and p. 2 Take δ(p,L) = 1/C1 . It is obvious that the mapping is a contraction when p b − a < δ(p,L), and hence there exists a unique solution (Y,Z) ∈ SF ([a, b]) × p HF ([a, b]) to the BSDE (4.2). Now we begin to study the convergence of the scheme (4.1).

Theorem 4.2. Let Assumption 2.2 be satisﬁed, and let π be any parti- π p tion. Assume that ξ ∈ L (Ω) and there exists a constant L1 > 0 such that, for all t1,t2 ∈ [0, T ], 1/2 |f(t2,y,z) − f(t1,y,z)|≤ L1|t2 − t1| . Then, there are two positive constants δ and K independent of the partition π, such that, when |π| < δ, we have T p/2 E π p E π 2 p/2 E π p sup |Yt − Yt | + |Zt − Zt | dt ≤ K(|π| + |ξ − ξ | ). 0≤t≤T Z0 Proof. If |π| < δ(p,L), where δ(p,L) is the constant in Theorem 4.1, then Theorem 4.1 guarantees the existence and uniqueness of (Y π,Zπ). De- note, for i = n − 1,n − 2,..., 0, 1 ti+1 Zπ = Zπ dr. ti+1 t − t r i+1 ti Zti π Notice that {Zti , }i=n−1,ne −2,...,0 here is diﬀerent from that in Section 3. Then Y π = Y π + f(t ,Y π , Zπ )∆ eti ti+1 i+1 ti+1 ti+1 i ti+1 π − Zr dWr,e i = n − 1,n − 2,..., 0. Zti Recursively, we obtain n π π π π Yti = ξ + f(tk,Ytk , Ztk )∆k−1 kX=i+1 T e π − Zr dWr, i = n − 1,n − 2,..., 0. Zti Denote π π π π π π δξ = ξ − ξ , δYt = Yt − Yt , δZt = Zt − Zt , t ∈ [0, T ], and π π Zti = Zti − Zti , i = n − 1,..., 0.

b e MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 33

If t ∈ [ti,ti+1), i = n − 1,n − 2,..., 0, then by iteration, we have n π π π π δYt = δξ + [f(tk,Ytk ,Ztk ) − f(tk,Ytk , Ztk )]∆k−1 k=i+1 (4.7) X T e π π − δZr dWr + Rt , Zti π where Rt is exactly the same as that in Section 3. π π π Denote ftk = f(tk,Ytk ,Ztk )−f(tk,Ytk , Ztk ). Then for t ∈ [ti,ti+1), i = n−1, n − 2,..., 0, we have e n e π E π π π (4.8) δYt = δξ + ftk ∆k−1 + Rt Ft . ! k=i+1 X From equality (4.8) for tj ≤ t 0 denotes a generic constant independent of the partition π and may vary from line to line. On the other hand, we have, by the H¨older continuity of Z given by (2.24), n p E π |Ztk |∆k−1 ! kX=i+1 b n p 1 tk = E Z − Zπ dr ∆ tk ∆ r k−1 k=i+1 k−1 Ztk−1 ! X n n p tk tk E π ≤ |Ztk − Zr| dr + |Zr − Zr | dr tk−1 tk−1 ! kX=i+1 Z kX=i+1 Z T p p/2 p−1E π ≤ C|π| + 2 |Zr − Zr | dr Zti T p/2 p/2 p−1 p/2E π 2 ≤ C|π| + 2 (T − ti) |Zr − Zr | dr Zti T p/2 p/2 p−1 p/2E π 2 = C|π| + 2 (T − ti) |δZr | dr . Zti Hence, we obtain E π p sup |δYt | ti≤t≤T

pE p ≤ C1 (T − ti) sup |δYtk | i+1≤k≤n (4.9) T p/2 p/2E π 2 + (T − ti) |δZr | dr Zti + |π|p/2 + E|δξπ|p , where C1 is a constant independent of the partition π. By the Burkholder– Davis–Gundy inequality, we have

T p/2 T p E π 2 E π (4.10) |δZr | dr ≤ cp δZr dWr . Zti Zti

From (4.7), we obtain

T n π π π π π (4.11) δZr dWr = δξ + ftk ∆k−1 + Rti − δYti . ti Z kX=i+1 e MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 35

Thus, from (4.10) and (4.11), we obtain

T p/2 E π 2 |δZr | dr ti Z n p E π E π p E π p E π p ≤ Cp ftk ∆k−1 + |δξ | + |Rti | + |δYti | . ( ) k=i+1 X e Similar to (4.9), we have T p/2 E π 2 |δZr | dr Zti pE p ≤ C2 (T − ti) sup |δYtk | i+1≤k≤n T p/2 p/2E π 2 p/2 E π p + (T − ti) |δZr | dr + |π| + |δξ | , Zti where C2 is a constant independent of the partition π. p/2 1 If C2(T − ti) < 2 , then we have T p/2 E π 2 pE p |δZr | dr ≤ 2C2(T − ti) sup |δYtk | ti i+1≤k≤n (4.12) Z p/2 π p + 2C2(|π| + E|δξ | ). Substituting (4.12) into (4.9), we have E π p sup |δYt | ti≤t≤T p/2 pE p ≤ C1(1 + 2C2(T − ti) )(T − ti) sup |δYtk | i+1≤k≤n (4.13) p/2 p/2 π p + C1(1 + 2C2(T − ti) )(|π| + E|δξ | ) pE p p/2 E π p ≤ 2C1(T − ti) sup |δYtk | + 2C1(|π| + |δξ | ). i+1≤k≤n We can ﬁnd a positive constant δ < δ(p,L) independent of the partition π, such that, p/2 1 (4.14) C2(3δ) < 2 , p 1 (4.15) 2C1(3δ) < 2 T and T > 2δ. Denote l = [ 2δ ]. Then l ≥ 1 is an integer independent of the partition π. If |π| < δ, then for the partition π we can choose n−1 > i1 > i2 >

· · · > il ≥ 0, such that, T − 2δ ∈ (ti1−1,ti1 ], T − 4δ ∈ (ti2−1,ti2 ],...,T − 2δl ∈ 36 Y. HU, D. NUALART AND X. SONG

[0,til ] (with t−1 = 0). For simplicity, we denote ti0 = T and til+1 = 0. Each interval [tij+1 ,tij ], j = 0, 1,...,l, has length less than 3δ, that is, |tij −tij+1 | <

3δ.On[tij+1 ,tij ], we consider the recursive formula (4.1). Then (4.13)–(4.15) yield E π p sup |δYt | tij+1 ≤t≤tij pE p p/2 E π p ≤ 2C1(tij − tij ) sup |δYt | + 2C1(|π| + |δY | ) +1 k tij ij+1+1≤k≤ij (4.16) p p p/2 π p ≤ 2C1(3δ) E sup |δYt | + 2C1(|π| + E|δY | ) k tij ij+1+1≤k≤ij

1 p p/2 π p ≤ sup |δYt | + 2C1(|π| + E|δY | ). k tij 2 ij+1+1≤k≤ij As in the proof of (3.21) and (3.22), we have π p π p p/2 E sup |δY | ≤ (4C1 + 1)E|δY | + 4C1|π| t tij tij+1 ≤t≤tij and 3(4C + 1)l+1 E sup |δY π|p ≤ 1 (E|δξπ|2 + |π|p/2). t 2 tij+1 ≤t≤tij Therefore, we obtain E π p E π p sup |δYt | ≤ max sup |δYt | 0≤j≤l 0≤t≤T tij+1 ≤t≤tij (4.17) 3(4C + 1)l+1 ≤ 1 (E|δξπ|p + |π|p/2). 2

On [tij+1 ,tij ], j = 0, 1,...,l, based on the recursive formula (4.1) and (4.17), inequality (4.12) becomes

ti p/2 E j π 2 |δZr | dr t Z ij+1 pE p p/2 E π p ≤ 2C2(tij − tij+1 ) sup |δYtk | + 2C2(|π| + |δξ | ) ij+1+1≤k≤ij pE p p/2 E π p ≤ 2C2(3δ) sup |δYtk | + 2C2(|π| + |δξ | ) ij+1+1≤k≤ij

1E p p/2 E π p ≤ sup |δYtk | + 2C2(|π| + |δξ | ) 2 ij+1+1≤k≤ij 3(4C + 1)l+1 ≤ 1 + 2C (|π|p/2 + E|δξπ|p). 4 2 MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 37

Thus T p/2 E π 2 |δZt | dt Z0 p/2 l ti E j π 2 = |δZt | dt t j=0 Z ij+1 ! (4.18) X l ti p/2 p/2−1 E j π 2 ≤ (l + 1) |δZt | dt ti Xj=0 Z j+1 3(4C + 1)l+1 ≤ (l + 1)p/2 1 + 2C (|π|p/2 + E|δξπ|p). 4 2 Combining (4.17) and (4.18), we know that there exists a constant 3(4C + 1)l+1 K = (l + 1)p/2 1 + 4C 2 2 independent of the partition π, such that

T p/2 E π p E π 2 sup |Yt − Yt | + |Zt − Zt | dt 0≤t≤T Z0 ≤ K(|π|p/2 + E|ξ − ξπ|p).

Remark 4.3. The advantages of this implicit numerical scheme are: (i) we can obtain the rate of convergence in Lp sense; (ii) the partition π can be arbitrary (|π| should be small enough) without assuming max0≤i≤n−1 ∆i/∆i+1 ≤ L1.

5. A new discrete scheme. For all the numerical schemes considered in π Sections 3 and 4, one needs to evaluate processes {Zt }0≤t≤T with continuous index t. In this section, we use the representation of Z in terms of the Malliavin derivative of Y to derive a completely discrete scheme. From (2.21), {DθYt}0≤θ≤t≤T can be represented as T (5.1) DθYt = E ρt,T Dθξ + ρt,rDθf(r, Yr,Zr) dr Ft , t Z where r r 1 (5.2) ρ = exp β dW + α − β2 ds t,r s s s 2 s Zt Zt with αs = ∂yf(s,Ys,Zs) and βs = ∂zf(s,Ys,Zs). 38 Y. HU, D. NUALART AND X. SONG

Using that Zt = DtYt, µ × P a.e., from (1.1), (5.1) and (5.2), we propose the following numerical scheme. We deﬁne recursively π π Ytn = ξ, Ztn = DT ξ, π E π π π Yt = (Yt + f(ti+1,Yt ,Zt )∆i|Fti ), (5.3) i i+1 i+1 i+1 n−1 π E π π π π Zti = ρti+1,tn Dti ξ + ρti+1,tk+1 Dti f(tk+1,Ytk+1 ,Ztk+1 )∆k Fti , ! k=i X i = n − 1,n − 2,..., 0, π where ρti,ti = 1, i = 0, 1,...,n, and for 0 ≤ i < j ≤ n, j−1 tk+1 π π π ρti,tj = exp ∂zf(r, Ytk ,Ztk ) dWr (k=i Ztk (5.4) X j−1 tk+1 1 + ∂ f(r, Y π,Zπ ) − [∂ f(r, Y π,Zπ )]2 dr . y tk tk 2 z tk tk k=i Ztk ) X π An alternative expression for ρti,tj is given by the following formula: j−1 π π π ρti,tj = exp ∂zf(tk,Ytk ,Ztk )(Wtk+1 − Wtk ) (k=i (5.5) X j−1 1 + ∂ f(t ,Y π,Zπ ) − [∂ f(t ,Y π,Zπ )]2 ∆ . y k tk tk 2 z k tk tk k k=i ) X π However, we will only consider the scheme (5.3) with ρti,tj given by (5.4). We make the following assumptions:

(G1) f(t,y,z) is deterministic, which implies Dθf(t,y,z)=0. (G2) f(t,y,z) is linear with respect to y and z; namely, there are three functions g(t), h(t) and f1(t) such that

f(t,y,z)= g(t)y + h(t)z + f1(t). 2 Assume that g, h are bounded and f1 ∈ L ([0, T ]). Moreover, there exists a constant L2 > 0, such that, for all t1,t2 ∈ [0, T ], 1/2 |g(t2) − g(t1)| + |h(t2) − h(t1)| + |f1(t2) − f1(t1)|≤ L|t2 − t1| . E r (G3) sup0≤θ≤T |Dθξ| < ∞, for all r ≥ 1. Notice that (G1) and (G2) imply (ii) and (iii) in Assumption 2.2.

Remark 5.1. We propose condition (G1) in order to simplify π {Zti }i=n−1,...,0 in formula (5.3). In fact, there are some diﬃculties in general- izing the condition (G)s, especially (G1), to a forward–backward stochastic diﬀerential equation (FBSDE, for short) case. MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 39

If we consider a FBSDE t t Xt = X0 + b(r, Xr) dr + σ(r, Xr) dWr, 0 0  TZ Z T   Yt = ξ + f(r, Xr,Yr,Zr) dr − Zr dWr, Zt Zt R  where X0 ∈ , and the functions b, σ, f are deterministic, then under some π appropriate conditions [e.g., (h1)–(h4) in Example 2.11] Zti for i = n − 1,..., 0 in (5.3) is of the form

π E π Zti = ρti+1,tn Dti ξ

n−1 π π π π π + ρti+1,tk+1 ∂xf(tk+1, Xtk+1 ,Ytk+1 ,Ztk+1 )Dti Xtk+1 ∆k Fti , ! k=i X where (Xπ,Y π,Zπ) is a certain numerical scheme for (X,Y,Z). It is hard to guarantee the existence and the convergence of Malliavin derivative of Xπ, and therefore, the convergence of Zπ is diﬃcult to derive.

Theorem 5.2. Let Assumption 2.2(i) and assumptions (G1)–(G3) be satisﬁed. Then there are positive constants K and δ independent of the partition π, such that, when |π| < δ we have p/2 E π p π p p/2−p/(2 log(1/|π|)) 1 max {|Yti − Yt | + |Zti − Zt | }≤ K|π| log . 0≤i≤n i i |π| Proof. In the proof, C> 0 will denote a constant independent of the partition π, which may vary from line to line. Under the assumption (G1), we can see that π E π (5.6) Zti = (ρti+1,tn Dti ξ|Fti ), i = n − 1,n − 2,..., 0. Denote, for i = n − 1,n − 2,..., 0, π π π π δZti = Zti − Zti , δYti = Yti − Yti . Since |ex − ey|≤ (ex + ey)|x − y|, we deduce, for all i = n − 1,n − 2,..., 0, π E E π |δZti | = | (ρti,tn Dti ξ|Fti ) − (ρti+1,tn Dti ξ|Fti )| E π ≤ (|ρti,tn − ρti+1,tn ||Dti ξ||Fti )

E π ≤ |Dti ξ|(ρti,tn + ρti+1,tn ) T T 1 T × h(r) dW + g(r) dr − h(r)2 dr r 2 Zti Zti Zti

40 Y. HU, D. NUALART AND X. SONG

n−1 tk+1 n−1 tk+1 − h(r) dWr − g(r) dr tk tk kX=i+1 Z kX=i+1 Z n−1 1 tk+1 + h(r)2 dr F 2 ti k=i+1 Ztk X

E π ≤ |Dti ξ|(ρti,tn + ρti+1,tn ) ti+1 ti+1 × h(r) dWr + |g(r)| dr Zti Zti ti+1 1 2 + h(r) dr Fti . 2 t Z i

From (G2), we have π |Dti ξ|ρti+1,tn

T n−1 tk+1 T 1 2 ≤ |Dti ξ| exp h(r) dWr + g(r) dr − h(r) dr ( ti+1 tk 2 ti+1 ) Z kX=i+1 Z Z T ≤ C1 sup |Dθξ| sup exp h(r) dWr , 0≤θ≤T 0≤t≤T t Z where C1 > 0 is a constant independent of the partition π. In the same way, we obtain T

|Dti ξ|ρti,tn

≤ 2C1E sup |Dθξ| sup exp h(r) dWr 0≤θ≤T 0≤t≤T t Z tk+1 tk+1 × sup h(r) dWr + sup |g(r)| dr 0≤k≤n−1 Ztk 0≤k≤n−1 Ztk

tk+1 1 2 + sup h(r) dr Fti . 2 0≤k≤n−1 t Z k

MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 41

The right-hand side of the above inequality is a martingale as a process indexed by i = n − 1,n − 2,..., 0. t Let ηt = exp{− 0 h(u) dWu}. Then, ηt satisfies the following linear stochastic differential equation: R 1 2 dηt = −h(t)ηt dWt + 2 h(t) ηtdt, η = 1. 0 By (G1), (G2), the Hölder inequality and Lemma 2.4, it is easy to show that, for any r ≥ 0, T r E sup exp h(u) dWu 0≤t≤T Zt T t r = E exp h(u) dWu sup exp − h(u) dWu Z0 0≤t≤T Z0 T 1/2 (5.7) ≤ E exp 2r h(u) dWu Z0 t 1/2 × E sup exp −2r h(u) dWu 0≤t≤T Z0 T 1/2 2 2 E 2r = exp r h(u) dr sup ηt < ∞. 0 0≤t≤T Z ′ q For any p ∈ (p, 2 ), by Doob’s maximal inequality and the Hölder inequality, (G3) and (5.7), we have E π p sup |δZti | 0≤i≤n p T p ≤ CE sup |Dθξ| sup exp h(r) dWr 0≤θ≤T 0≤t≤T t Z tk+1 × sup h(r) dWr 0≤k≤n−1 Ztk p tk+1 1 tk+1 + sup |g(r)| dr + sup h(r)2 dr 2 0≤k≤n−1 Ztk 0≤k≤n−1 Ztk pp′/(p′−p) ≤ C E sup |Dθξ| 0≤θ≤T T pp′/(p′−p) (p′−p)/p′ × sup exp h(r) dWr 0≤t≤T Zt tk+1 tk+1 × E sup h(r) dWr + sup |g(r)| dr 0≤k≤n−1 Ztk 0≤k≤n−1 Ztk

42 Y. HU, D. NUALART AND X. SONG

′ ′ 1 tk+1 p p/p + sup h(r)2 dr 2 0≤k≤n−1 Ztk 2pp′/(p′−p) p′/(2(p′−p)) ≤ C E sup |Dθξ| 0≤θ≤T h i T 2pp′/(p′−p) p′/(2(p′−p)) × E sup exp h(r) dWr 0≤t≤T Zt ′ ′ tk+1 p tk+1 p × E sup h(r) dWr + E sup |g(r)| dr 0≤k≤n−1 Ztk 0≤k≤n−1Ztk ′ ′ tk+1 p p/p + E sup h(r)2 dr 0≤k≤n−1Ztk p/p′ = C[I1 + I2 + I3] . For any r> 1, by the H¨older inequality we can obtain ′ ′ tk+1 p tk+1 p r 1/r I1 = E sup h(r) dWr ≤ E sup h(r) dWr 0≤k≤n−1 Ztk 0≤k≤n−1 Ztk

n−1 ′ 1/r tk+1 p r ≤ E h(r) dWr . ( k=0 Ztk ) X

For any centered Gaussian variable X, and any γ ≥ 1, we know that E|X|γ ≤ C˜γγγ/2(E|X|2)γ/2, where C˜ is a constant independent of γ. Thus, we can see that

′ 1/r n−1 ti+1 p r/2 p′r ′ p′r/2 2 p′/2 p′/2−1/r I1 ≤ C˜ (p r) h(r) dr ≤ Cr |π| . ti ! Xi=0 Z 2 log(1/|π|) Take r = p′ . Assume |π| is small enough; then we have p′/2 ′ ′ 1 I ≤ C|π|p /2−p /(2 log(1/|π|)) log . 1 |π| It is easy to see that ′ tk+1 p p′ I2 = E sup |g(r)| dr ≤ C|π| 0≤k≤n−1Ztk and ′ tk+1 p 2 p′ I3 = E sup h(r) dr ≤ C|π| . 0≤k≤n−1Ztk MALLIAVIN CALCULUS, NUMERICAL SOLUTION OF BSDE 43

Consequently, we obtain 1 p/2 (5.8) E sup |δZπ |p ≤ C|π|p/2−p/(2 log(1/|π|)) log . ti |π| 0≤i≤n Applying recursively the scheme given by (5.3), we obtain n π E π π Yti = ξ + f(tk,Ytk ,Ztk )∆k−1 Fti , i = n − 1,n − 2,..., 0. ! k=i+1 X Therefore, for i = n − 1,n − 2,..., 0, n π E π π π π |δYti |≤ |f(tk,Ytk ,Ztk ) − f(tk,Ytk ,Ztk )|∆k−1 + |Rti | + |δξ | Fti , ! kX=i+1 π π where Rt is exactly the same as in Section 3 and δξ = ξ − ξ = 0. In fact, we keep the term δξπ to indicate the role it plays as the terminal value. For j = n − 1,n − 2,...,i, we have n π E π π |δYtj |≤ |f(tk,Ytk ,Ztk ) − f(tk,Ytk ,Ztk )|∆k−1

kX=i+1 π π + sup |Rt | + |δξ | Ftj . 0≤t≤T !

Acknowledgment. We appreciate the referee’s very constructive and de- tailed comments to improve the presentation of this paper.

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