Mean square rate of convergence for random walk approximation of forward-backward SDEs

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approximation of forward-backward SDEs

Christel Geiss, Céline Labart, Antti Luoto

To cite this version:

Christel Geiss, Céline Labart, Antti Luoto. Mean square rate of convergence for random walk approx-

imation of forward-backward SDEs. 2020. �hal-01838449v2�

MEAN SQUARE RATE OF CONVERGENCE FOR RANDOM WALK AP- PROXIMATION OF FORWARD-BACKWARD SDES

CHRISTEL GEISS,^∗Department of Mathematics and Statistics, University of Jyvaskyla C´ELINE LABART,^∗∗ Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA ANTTI LUOTO,^∗Department of Mathematics and Statistics, University of Jyvaskyla

Abstract

Let (Y, Z) denote the solution to a forward-backward SDE. If one constructs a random walk Bⁿ from the underlying Brownian motion B by Skorohod embedding, one can show L₂-convergence of the corresponding solutions (Yⁿ, Zⁿ) to (Y, Z). We estimate the rate of convergence in dependence of smoothness properties, especially for a terminal condition function inC^2,α. The proof relies on an approximative representation ofZⁿand uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the PDE associated to the FBSDE as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

Keywords: Backward stochastic differential equations; approximation scheme;

finite difference equation; convergence rate; random walk approximation 2010 Mathematics Subject Classification: Primary 60H10; 60H35; 60G50

Secondary 60H30

1. Introduction

Let (Ω,F,P) be a complete probability space carrying the standard Brownian motion B = (Bt)t≥0 and assume that (F^t)t≥0 is the augmented natural filtration. Let (Y, Z) be the solution of the forward-backward SDE (FBSDE)

Xs=x+ Z s

0

b(r, Xr)dr+ Z s

0

σ(r, Xr)dBr, Y_s=g(XT) +

Z T s

f(r, Xr, Y_r, Z_r)dr− Z T

s

Z_rdB_r, 0≤s≤T. (1)

∗Postal address: P.O.Box 35 (MaD) FI-40014 University of Jyvaskyla, Finland

∗∗Postal address: Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chamb´ery, France

1

Let (Yⁿ, Zⁿ) be the solution of the FBSDE if the Brownian motion B is replaced by a scaled random walkBⁿ given by

Bⁿ_t =√ h

[t/h]

X

i=1

εi, 0≤t≤T, (2)

whereh=^T_n and (εi)i=1,2,... is a sequence of i.i.d. Rademacher random variables. Then (Yⁿ, Zⁿ) solves the discretized FBSDE

X_sⁿ=x+ Z

(0,s]

b(r, X_rⁿ₋)d[Bⁿ]r+ Z

(0,s]

σ(r, X_rⁿ₋)dB_rⁿ, Y_sⁿ =g(X_Tⁿ) +

Z

(s,T]

f(r, X_rⁿ₋Y_rⁿ₋, Z_rⁿ₋)d[Bⁿ]r− Z

(s,T]

Z_rⁿ₋dBⁿ_r, 0≤s≤T. (3) The approximation of BSDEs using random walk has been investigated by many authors, also numerically (see, for example, [5], [25], [29], [31], [32], [33], [16]). In 2001, Briand et al. [5]

have shown weak convergence of (Yⁿ, Zⁿ) to (Y, Z) for a Lipschitz continuous generatorf and a terminal condition in L2. The rate of convergence of this method remained an open problem.

Bouchard and Touzi in [7] and Zhang in [41] proposed instead of random walk an approach based on the dynamic programming equation, for which they established a rate of convergence. But this approach involves conditional expectations. Various methods to approximate these conditional expectations have been developed ([23], [17], [14]). Also forward methods have been introduced to approximate (1): a branching diffusion method ([26]), a multilevel Picard approximation ([40]) and Wiener chaos expansion ([6]). Many extensions of (1) have been considered, among them schemes for reflected BSDEs ([3], [13]), high order schemes ([10], [9]), fully-coupled BSDEs ([18], [8]), quadratic BSDEs ([12]), BSDEs with jumps ([22]) and McKean-Vlasov BSDEs ([1], [15], [11]).

The aim of this paper is to study the rate of the L2-approximation of (Y_tⁿ, Z_tⁿ) to (Yt, Zt) when X satisfies (1). For this, we generate the random walk Bⁿ by Skorohod embedding from the Brownian motionB. In this case theLp-convergence ofBⁿ toBis of orderh¹⁴ for anyp >0.

The special caseX=B has already been studied in [21], assuming a locallyα-H¨older continuous terminal functiongand a Lipschitz continuous generator. An estimate for the rate of convergence was obtained which is of orderh^α4 for theL2-norm ofY_tⁿ−Yt,and of order ^h

α

√ 4

T−t for theL2-norm ofZ_tⁿ−Zt.

In the present paper, where we assume that X is a solution of the SDE in (1), rather strong conditions on the smoothness and boundedness onf and g and also on b and σ are needed. In Theorem 3.1, the main result of the paper, we show that the convergence rate for (Y_tⁿ, Z_tⁿ) to

(Yt, Zt) inL2is of orderh^14∧α2 provided thatg^′′is locallyα-H¨older continuous. To the best of our knowledge, these are the first cases a convergence rate for the approximation of forward-backward SDEs using random walk has been obtained.

Remark 1.1. For the diffusion setting – in contrast to the case X = B – we can derive the convergence rate for (Y_tⁿ, Z_tⁿ) to (Yt, Zt) in L2 only under strong smoothness conditions on the coefficients which include also thatg^′′is locallyα-H¨older continuous (see Assumption2.3below).

These requirements appear to be necessary. This becomes visible in Subsection 2.2.2 where we introduce a discretized Malliavin weight to obtain a representation ˆZⁿforZⁿ.While it holds that Zˆⁿ =Zⁿ when X =B, in our case ˆZⁿ does not coincide with Zⁿ.However, one can show that the difference ˆZ_tⁿ−Z_tⁿ converges to 0 inL2 as n → ∞ using a H¨older continuity property (see (62) in Remark 4.1) for the space derivative of the generator in (3). For this H¨older continuity property to hold one needs enough smoothness in space from the solutionuⁿto the finite difference equation associated to the discretized FBSDE (3). Provided that Assumption 2.3holds we show the smoothness properties for uⁿ in Proposition 4.2 applying methods known for L´evy driven BSDEs.

The paper is organized as follows: Section 2 contains the setting, main assumptions and the approximative representation ˆZⁿ of Zⁿ. Our main results about the approximation rate for the case of no generator (i.e. f = 0) and for the general case are in Section 3. One can see that in contrast to what is known for time discretization schemes, for random walk schemes the Lipschitz generator seems to cause more difficulties than the terminal condition: while in the casef = 0 we need thatg^′is locallyα-H¨older continuous, in the casef 6= 0 is this property is required forg^′′.In Section 4 we recall some needed facts about Malliavin weights, about the regularity of solutions to BSDEs and properties of the associated PDEs. Finally, we sketch how to prove growth and smoothness properties of solutions to the finite difference equation associated to the discretized FBSDE. Section5contains technical results which mainly arise from the fact that the construction of the random walk by Skorohod embedding forces us to compare our processes on different ’time lines’, one coming from the stopping times of the Skorohod embedding, and the other one is ruled by the equidistant deterministic times due to the quadratic variation process [Bⁿ].

2. Preliminaries 2.1. The SDE and its approximation scheme

We introduce

Xt=x+ Z t

0

b(s, Xs)ds+ Z t

0

σ(s, Xs)dBs, 0≤t≤T and its discretized counterpart

X_tⁿ_k =x+h Xk j=1

b(tj, X_tⁿ_j−1) +√ h

Xk j=1

σ(tj, X_tⁿ_j−1)εj, tj:=j^T_n, j= 0, ..., n, (4) where (εi)i=1,2,... is a sequence of i.i.d. Rademacher random variables. Letting G^k :=σ(εi : 1 ≤ i ≤ k) with G⁰ := {∅,Ω}, it follows that the associated discrete-time random walk (B_tⁿ_k)ⁿ_k=0 is (G^k)ⁿ_k=0-adapted. Recall (2) and h = ^T_n. If we extend the sequence (X_tⁿ_k)k≥0 to a process in continuous time by definingX_tⁿ:=X_tⁿ_kfort∈[tk, tk+1),it is the solution of the forward SDE (3).

We formulate our first assumptions. Assumption 2.1 (ii) will be not used explicitely for our estimates but it is required for Theorem4.1below.

Assumption 2.1.

(i) b, σ∈C_b^0,2([0, T]×R), in the sense that the derivatives of orderk = 0,1,2 w.r.t. the space variable are continuous and bounded on[0, T]×R,

(ii) the first and second derivatives ofbandσw.r.t. the space variable are assumed to beγ-H¨older continuous (for someγ∈(0,1],w.r.t. the parabolic metricd((t, x),(¯t,x)) = (¯ |t−¯t|+|x−x¯|²)¹²) on all compact subsets of[0, T]×R.

(iii) b, σare ¹₂-H¨older continuous in time, uniformly in space, (iv) σ(t, x)≥δ >0for all(t, x).

Assumption 2.2.

(i) g is locally H¨older continuous with orderα∈(0,1]and polynomially bounded (p0≥0, Cg>

0) in the following sense

∀(x,x)¯ ∈R², |g(x)−g(¯x)| ≤Cg(1 +|x|^p0+|x¯|^p0)|x−x¯|^α. (5)

(ii) The function [0, T]×R³: (t, x, y, z)7→f(t, x, y, z)satisfies

|f(t, x, y, z)−f(¯t,x,¯ y,¯ ¯z)| ≤Lf(p

t−¯t+|x−x¯|+|y−y¯|+|z−z¯|). (6)

Notice that (5) implies

|g(x)| ≤K(1 +|x|^p0+1) =: Ψ(x), x∈R, (7) for someK >0.From the continuity off we conclude that

Kf := sup

0≤t≤T|f(t,0,0,0)|<∞. Notation:

• k · k^p:=k · kLp(P) forp≥1 and forp= 2 simplyk · k.

• If ais a function,C(a) represents a generic constant which depends ona and possibly also on its derivatives.

• E0,x:=E(·|X0=x).

• Letφbe aC^0,1([0, T]×R) function. φxdenotes∂xφ, the partial derivative ofφw.r.t. x.

2.2. The FBSDE and its approximation scheme

Recall the FBSDE (1) and its approximation (3). The backward equation in (3) can equivalently be written in the form

Y_tⁿ_k=g(X_Tⁿ) +h

n−1

X

m=k

f(tm+1, X_tⁿ_m, Y_tⁿ_m, Z_tⁿ_m)−√ h

nX−1 m=k

Z_tⁿ_mεm+1, 0≤k≤n, (8) if one putsX_rⁿ:=X_tⁿ_m, Y_rⁿ :=Y_tⁿ_m andZ_rⁿ:=Z_tⁿ_m forr∈[tm, tm+1).

Remark 2.1. Equations (3) and (8) do not contain any orthogonal part to the random walkBⁿ since we are in a special case where the orthogonal part is zero. Indeed, for (εk)k=1,···,nfollowing the Rademacher law assume (G^k:=σ(εi, i= 1,· · ·, k)) as filtration, and let for theGⁿ-measurable random variableF(ε1, ..., εn) hold the representation

F(ε1, ..., εn) =c+ Xn m=1

hmεm+Nn,

where (hm)ⁿ_m=1is predictable and (Nm)ⁿ_m=1a martingale orthogonal to (Bⁿ_tm)ⁿ_m=1given byBⁿ_tm =

√h(ε1+...+εm). By definition, orthogonality of the martingales N and Bⁿ means that their product is a martingale, i.e. we have

E[Nk+1B_tⁿ_k+1|G^k] =NkB_tⁿ_k,

and sinceNkB_tⁿ_k =E[Nk+1B_tⁿ_k|G^k], this implies especially thatENk+1εk+1 = 0. Assume Nk+1 is given byNk+1=H(ε1, ..., εk+1).Then 0 =EH(ε1, ..., εk+1)εk+1=¹₂[H(ε1, ..., εk,1)−H(ε1, ..., εk,−1)]

implying that Nk+1 is G^k-measurable since H(ε1, ..., εk,1) =H(ε1, ..., εk,−1), and therefore the martingale (Nm)ⁿ_m=1 is identically zero. (See also [5, page 3] or [34, Proposition1.7.5].)

One can derive an equation for Zⁿ = (Z_tⁿ_k)ⁿ_k=0⁻¹ if one multiplies (8) by εk+1 and takes the conditional expectation w.r.t. G^k, so that

Z_tⁿ_k = E^G_k(g(X_Tⁿ)εk+1)

√h +E^G_k √ h

n−1

X

m=k+1

f(tm+1, X_tⁿ_m, Y_tⁿ_m, Z_tⁿ_m)εk+1

!

, 0≤k≤n−1,(9)

whereE^Gk :=E(·|G^k).

Remark 2.2. Fornlarge enough, the BSDE (3) has a unique solution (Yⁿ, Zⁿ) (see [36, Propo- sition 1.2]), and (Y_tⁿ_k, Z_tⁿ_k)ⁿ_k=0⁻¹ is adapted to the filtration (G^k)ⁿ_k=0⁻¹.

2.2.1. Representation forZ We will use the following representation forZ, due to Ma and Zhang (see [30, Theorem 4.2])

Zt=Et g(XT)N_T^t + Z T

t

f(s, Xs, Ys, Zs)N_s^tds

!

σ(t, Xt), 0≤t≤T (10) whereEt:=E(·|F^t), and for alls∈(t, T], we have (cf. Lemma4.1)

N_s^t= 1 s−t

Z s t

∇Xr

σ(r, Xr)∇Xt

dBr, (11)

where∇X= (∇Xs)s∈[0,T] is the variational process i.e. it solves

∇Xs= 1 + Z s

0

bx(r, Xr)∇Xrdr+ Z s

0

σx(r, Xr)∇XrdBr, (12) with (Xs)s∈[0,T] given in (1).

Remark 2.3. In the following we will assume thatg^′′exists. In such a case we have the following representation forZ:

Zt=Et g^′(XT)∇XT + Z T

t

f(s, Xs, Ys, Zs)N_s^tds

!

σ(t, Xt), 0≤t≤T. (13) 2.2.2. Approximation for Zⁿ In this section we state the discrete counterpart to (10), which, in the general case of a forward processX, does not coincide withZⁿ (given by (9)). In contrast to the continuous-time case, where the variational process and the Malliavin derivative are connected by ^∇_∇^X_Xs^t = _σ(s,X^DsXt_s) (s≤t), we can not expect equality for the corresponding expressions if we use the discretized version of the processes (∇Xt)tand (DsXt)s≤tintroduced in (15). This counterpart Zˆⁿ toZ is a key tool in the proof of the convergence ofZⁿ to Z. As we will see in the proof of

Theorem3.1, the study ofkZ_tⁿ_k−Ztkkgoes through the study ofkZ_tⁿ_k−Zˆ_tⁿ_kk andkZˆ_tⁿ_k−Ztkk.

Before defining the discretized version of (∇Xt)t and (DsXt)s≤t, we shortly introduce the discretized Malliavin derivative and refer the reader to [4] for more information on this topic.

Definition 2.1. (Definition of T_m,+, T_m,− and Dⁿm.) For any function F : {−1,1}ⁿ → R, the mappingsT_m,+ andT_m,− are defined by

T_m,±F(ε1, . . . , εn) :=F(ε1, . . . , εm−1,±1, εm+1, . . . , εn), 1≤m≤n.

For anyξ=F(ε1, . . . , εn), the discretized Malliavin derivative is defined by Dmⁿξ:= E[ξεm|σ((εl)l∈{1,...,n}\{m})]

√h = T_m,+ξ−T_m,−ξ 2√

h , 1≤m≤n. (14)

Definition 2.2. (Definition of φ^(k,l)x .) Letφbe aC^0,1([0, T]×R)function. We denote φ^(k,l)_x := Dⁿkφ(tl, X_tⁿ_l−1)

DkⁿX_tⁿ_l−1 :=

Z 1 0

φx(tl, ϑT_k,+X_tⁿ_l−1+ (1−ϑ)T_k,−X_tⁿ_l−1)dϑ.

IfDkⁿX_tⁿ_ℓ−1 6= 0the second^′ :=^′ holds as an identity.

We are now able to define the discretized version of (∇Xt)tand (DsXt)s≤t.

Definition 2.3. (Discretized processes (∇X_t^n,t_m^k,x)m∈{k,...,n} and (DkⁿX_tⁿ_m)m∈{k,...,n}.) For allm in{k, . . . , n} we define

∇X_t^n,t_m^k,x= 1 +h Xm l=k+1

bx(tl, X_t^n,t_l−1^k,x)∇X_t^n,t_l−1^k,x+√ h

Xm l=k+1

σx(tl, X_t^n,t_l−1^k,x)∇X_t^n,t_l−1^k,xεl, 0≤k≤n,

DⁿkX_tⁿ_m=σ(tk, X_tⁿ_k−1) +h Xm l=k+1

b^(k,l)_x DkⁿX_tⁿ_l−1+√ h

Xm l=k+1

σ_x^(k,l)(DkⁿX_tⁿ_l−1)εl, 0< k≤n. (15)

Remark 2.4. (i) Although∇X^n,tk,X

n tk

tm is not equal to _σ(t^Dnk+1Xtmn

k+1,X_tkⁿ), we can show that the differ- ence of these terms converges inLp (see Lemma5.4).

(ii) With the notation introduced above, (9) rewrites to Z_tⁿ_k = E^G_k Dⁿk+1g(X_Tⁿ)

+E^G_k h

nX−1 m=k+1

Dk+1ⁿ f(tm+1, X_tⁿ_m, Y_tⁿ_m, Z_tⁿ_m)

!

. (16) In order to define the discrete counterpart to (10), we first define the discrete counterpart to (N_s^t)s∈[t,T] given in (11):

N_t^n,t_ℓ ^k:=√ h

Xℓ m=k+1

∇X^n,tk,X

n tk

tm−1

σ(tm, X_tⁿ_m−1) εm

tℓ−tk

, k < ℓ≤n. (17)

Notice that there is some constantbκ2>0 depending onb, σ, T, δ such that

E^G_k|N_t^n,t_ℓ ^k|²¹2

≤ bκ2

(tℓ−tk)¹², 0≤k < ℓ≤n. (18) Definition 2.4. (Discrete counterpart to (13).) Let the processZˆⁿ= ( ˆZ_tⁿ_k)ⁿ_k=0⁻¹ be defined by

Zˆ_tⁿ_k:=E^G_k Dⁿk+1g(X_Tⁿ)

+E^G_k h

n−1

X

m=k+1

f(tm+1, X_tⁿ_m, Y_tⁿ_m, Z_tⁿ_m)N_t^n,t_m^k

!

σ(tk+1, X_tⁿ_k), (19)

Remark 2.5. In (19) We could have used also the approximate expressionE^G_k(g(X_Tⁿ)N_t^n,t_n ^kσ(tk+1, X_tⁿ_k)), but since we will assume thatg^′′ exists, we work with the correct term.

The study of the convergenceE^G0,x|Z_tⁿ_k−Zˆ_tⁿ_k|²requires stronger assumptions on the coefficients b,σ, f andg.

Assumption 2.3. Assumptions 2.1 and 2.2 hold. Additionally, we assume that all first and second derivatives w.r.t. the variablesx, y, zofb(t, x), σ(t, x)andf(t, x, y, z)exist and are bounded Lipschitz functions w.r.t. these variables, uniformly in time. Moreover,g^′′ satisfies (5).

Proposition 2.1. If Assumption 2.3holds, then

E^G0,x|Z_tⁿ_k−Zˆ_tⁿ_k|²≤C2.1Ψˆ²(x)h^α,

where E^G0,x := E^G(·|X0 = x), the function Ψˆ is defined in (61) below, and C2.1 depends on b, σ, f, g, T, p0 andδ.

Proof. According to [5, Proposition 5.1] one has the representations

Y_tⁿ_m =uⁿ(tm, X_tⁿ_m), and Z_tⁿ_m =Dⁿm+1uⁿ(tm+1, X_tⁿ_m+1), (20) whereuⁿ is the solution of the finite difference equation (43) with terminal conditionuⁿ(tn, x) = g(x).Notice that by the definition ofDm+1ⁿ in (14) the expressionDm+1ⁿ uⁿ(tm+1, X_tⁿ_m+1) depends in fact onX_tⁿ_m.Hence we can put

f(tm+1, X_tⁿ_m, Y_tⁿ_m, Z_tⁿ_m) = f(tm+1, X_tⁿ_m, uⁿ(tm, X_tⁿ_m),Dm+1ⁿ uⁿ(tm+1, X_tⁿ_m+1))

=: Fⁿ(tm+1, X_tⁿ_m).

From (19) and (16) we conclude that (we useE:=E^G_0,x fork · k) kZ_tⁿ_k−Zˆ_tⁿ_kk

= E^Gk h

n−1

X

m=k+1

Dⁿk+1f(tm+1, X_tⁿ_m, Y_tⁿ_m, Z_tⁿ_m)

!

−E^G_k h

nX−1 m=k+1

f(tm+1, X_tⁿ_m, Y_tⁿ_m, Z_tⁿ_m)N_t^n,t_m^kσ(tk+1, X_tⁿ_k)!

≤

n−1

X

m=k+1

h m−k

Xm ℓ=k+1

E^G_k

"

Dⁿk+1Fⁿ(tm+1, X_tⁿ_m)− DⁿℓFⁿ(tm+1, X_tⁿ_m)σ(tk+1, X_tⁿ_k)∇X^n,tk,X

n tk

tℓ−1

σ(tℓ, X_tⁿ_ℓ−1)

#.

With the notation introduced in Definition2.2applied to Fⁿ,

Dⁿk+1Fⁿ(tm+1, X_tⁿ_m)− DℓⁿFⁿ(tm+1, X_tⁿ_m)σ(tk+1, X_tⁿ_k)∇X^n,tk,X

n tk

tℓ−1

σ(tℓ, X_tⁿ_ℓ−1)

≤ k(Dk+1ⁿ X_tⁿ_m)(Fn,(k+1,m+1)

x −F_x^n,(ℓ,m+1))k +

F_x^n,(ℓ,m+1)

(Dk+1ⁿ X_tⁿ_m)−(DⁿℓX_tⁿ_m)σ(tk+1, X_tⁿ_k)∇X^n,tk,X

n tk

tℓ−1

σ(tℓ, X_tⁿ_ℓ−1)

=: A1+A2.

ForA1 we use Definition2.2again and exploit the fact that

x7→F_xⁿ(tm+1, x) :=∂xf(tm+1, x, uⁿ(tm, x),Dⁿm+1uⁿ(tm+1, X_t^n,t_m+1^m,x))

is locally α-H¨older continuous according to (62). By Hlder’s inequality and Lemma 5.4 (i) and (iii),

A1≤ kDk+1ⁿ X_tⁿ_mk⁴ Z 1

0 kF_xⁿ(tm+1, ϑT_k+1,+X_tⁿ_m+ (1−ϑ)T_k+1,−X_tⁿ_m)

−F_xⁿ(tm+1, ϑT_ℓ,+X_tⁿ_m+ (1−ϑ)T_ℓ,−X_tⁿ_m)k⁴dϑ≤C(b, σ, f, g, T, p0) ˆΨ(x)h^α2. For the estimate of A2 we notice that by our assumptions theL4-norm ofFx^n,(ℓ,m+1) is bounded byCΨ²(x),so that it suffices to estimate

(Dk+1ⁿ X_tⁿ_m)−(DⁿℓX_tⁿ_m)σ(tk+1, X_tⁿ_k)∇X^n,tk,X

n tk

tℓ−1

σ(tℓ, X_tⁿ_ℓ−1)

4

≤

(Dk+1ⁿ X_tⁿ_m)−σ(tk+1, X_tⁿ_k)DⁿℓX_tⁿ_m σ(tℓ, X_tⁿ_ℓ−1)

Dⁿk+1X_tⁿ_ℓ−1 σ(tk+1, X_tⁿ_k)

4

+

σ(tk+1, X_tⁿ_k)DℓⁿX_tⁿ_m

σ(tℓ, X_tⁿ_ℓ−1) ∇X^n,tk,X

n

t_ℓ−1 tk − Dⁿk+1X_tⁿ_ℓ−1 σ(tk+1, X_tⁿ_k)

!

4

. (21)

The second expression on the r.h.s. of (21) is bounded by C(b, σ, T, δ)h¹² as a consequence of Lemma 5.4 (ii)-(iii). To show that also the first expression is bounded by C(b, σ, T, δ)h¹², we rewrite it using (15) and get

DℓⁿX_tⁿ_m

σ(tℓ, X_tⁿ_ℓ−1)Dⁿk+1X_tⁿ_ℓ−1− Dk+1ⁿ X_tⁿ_m

= 1 +

Xm l=ℓ+1

DⁿℓX_tⁿ_l−1

σ(tℓ, X_tⁿ_ℓ−1)(b^(ℓ,l)_x h+σ_x^(ℓ,l)√ hεl)

!

× σ(tk+1, X_tⁿ_k) +

ℓ−1

X

l=k+2

Dk+1ⁿ X_tⁿ_l−1(b^(k+1,l)_x h+σ^(k+1,l)_x √ hεl)

!

− σ(tk+1, X_tⁿ_k) + ^ℓX−1

l=k+2

+ Xm

l=ℓ

Dⁿk+1X_tⁿ_l−1(b^(k+1,l)_x h+σ_x^(k+1,l)√ hεl)

!

≤Dk+1ⁿ X_tⁿ_ℓ−1(b^(k+1,ℓ)_x h+σ^(k+1,ℓ)_x √ hεℓ) +

Xm l=ℓ+1

DⁿℓX_tⁿ_l−1

σ(tℓ, X_tⁿ_ℓ−1)Dⁿk+1X_tⁿ_ℓ−1− Dⁿk+1X_tⁿ_l−1

b^(ℓ,l)_x h+σ^(ℓ,l)_x √ hεl

+

Xm l=ℓ+1

Dk+1ⁿ X_tⁿ_l−1

b^(ℓ,l)_x h+σ_x^(ℓ,l)√ hεl−

b^(k+1,l)_x h+σ^(k+1,l)_x √ hεl

. (22)

We take theL4-norm of (22) and apply the BDG inequality and H¨older’s inequality. The second term on the r.h.s. of (22) will be used for Gronwall’s lemma, while the first and the last one can be bounded byC(b, σ, T)h¹²,by using Lemma5.4-(iii). For the last term we also use the Lipschitz

continuity ofbx andσx in space and Lemma5.4-(i).

3. Main results

In order to compute the mean square distance between the solution to (1) and the solution to (3) we construct the random walk Bⁿfrom the Brownian motionB by Skorohod embedding. Let

τ0:= 0 and τk:= inf{t > τk−1 :|Bt−Bτk−1|=√

h}, k≥1. (23)

Then (Bτk−Bτk−1)^∞_k=1 is a sequence of i.i.d. random variables with P(Bτk−Bτk−1 =±√

h) = ¹₂,

which means that √ hεk

=d Bτk −Bτk−1. We will denote by Eτk the conditional expectation w.r.t.F^τk:=G^k.In this case we also use the notationX^τk :=X_tⁿ_k for allk= 0, . . . , n,so that (4)

turns into

X^τk=x+ Xk j=1

b(tj,X^τj−1)h+ Xk j=1

σ(tj,X^τj−1)(Bτj −Bτj−1), 0≤k≤n.

Assumption 3.1. We assume that the random walk Bⁿ in(3)is given by B_tⁿ=

[t/h]

X

k=1

(Bτk−Bτk−1), 0≤t≤T, where theτk, k= 1, ..., nare taken from(23).

Remark 3.1. Note that forp >0 there exists a C(p)>0 such that for allk= 1, . . . , n it holds

1

C(p)(tkh)¹⁴ ≤(E|Bτ_k−Bt_k|^p)^1p ≤C(p)(tkh)¹⁴.

The upper estimate is given in Lemma 5.1. For p ∈ [4,∞) the lower estimate follows from [2, Proposition 5.3]. We get the lower estimate forp∈(0,4) by choosing 0< θ <1 and 0< p < p1

such that ¹₄ = ¹⁻_p^θ+ _p^θ

1. Then it holds by the log-convexity of Lp norms (see, for example [35, Lemma 1.11.5]) that

kB_τk−B_tkk¹p^−θ≥ kBτ_k−Bt_kk⁴ kBτk−Btkk^θp1

≥ C(4)⁻¹(tkh)¹⁴ C(p1)(tkh)¹⁴θ ≥

C(p)(tkh)¹⁴1−θ

.

Since fort∈[tk, tk+1) it holdsB_tⁿ=Bτk andkBt−Btkk^p≤C(p)h¹²,we have for anyp >0 that sup

0≤t≤TkB_tⁿ−Btk^p=O(h¹⁴). (24) Proposition 3.1 states the convergence rate of (Yv, Zv) to (Y_vⁿ, Z_vⁿ) in L2 when f = 0 and Theorem3.1generalizes this result for anyf which satisfies Assumption2.3.

Proposition 3.1. Let Assumptions 2.1 and3.1 hold. If f = 0and g ∈ C¹ is such that g^′ is a locally α-H¨older continuous function in the sense of (5), then for all 0 ≤ v < T, we have (for sufficiently largen) that

E0,x|Yv−Y_vⁿ|²≤C3.1^y Ψ(x)²h¹², and E0,x|Zv−Z_vⁿ|²≤C3.1^z Ψ(x)²h^α2, whereC3.1^y =C(Cg, b, σ, T, p0, δ)andC3.1^z =C(Cg^′, b, σ, T, p0, δ).

Theorem 3.1. Let Assumptions2.3and3.1be satisfied. Then for allv∈[0, T)and large enough n, we have

E0,x|Yv−Y_vⁿ|²+E0,x|Zv−Z_vⁿ|²≤C3.1Ψ(x)ˆ ²h^12∧α withC3.1=C(b, σ, f, g, T, p0, δ)andΨˆ is given in (61).

Remark 3.2. As noticed above, the filtrationG^k coincides with F^τk, for all k = 0, . . . , n. The expectationE0,x appearing in Proposition 3.1 and in Theorem 3.1is defined on the probability space (Ω,F,P).

Remark 3.3. In order to avoid too much notation for the dependencies of the constants, if for example onlyg is mentioned and notCg,this means that the estimate might depend also on the bounds of the derivatives ofg.

From (24) one can see that the convergence rates stated in Proposition3.1 and Theorem3.1 are the natural ones for this approach. The results are proved in the next two sections. In both proofs, we will use the following remark.

Remark 3.4. Since the process (Xt)t≥0is strong Markov we can express conditional expectations with the help of an independent copy ofB denoted by ˜B, for exampleEτkg(X_Tⁿ) = ˜Eg( ˜X^ττnk,Xτk) for 0≤k≤n, where

X˜^ττn^k,Xτk =X^τk+ Xn j=k+1

b(tj,X˜^ττj^k−1^,Xτk)h+ Xn j=k+1

σ(tj,X˜^ττj^k−1^,Xτk)( ˜B˜τj−k−B˜τ˜j−k−1), (25) (we define ˜τk := 0 and ˜τj := inf{t >τ˜j−1 :|B˜t−B˜τ˜j−1|=√

h}forj ≥1 andτn:=τk+ ˜τn−k for n≥k). In fact, to represent the conditional expectationsEtk andEτk we work here with ˜Eand the Brownian motionsB^′ andB^′′,respectively, given by

B^′_t=Bt∧tk+ ˜B(t−tk)⁺ and B^′′_t =Bt∧τk+ ˜B(t−τk)⁺, t≥0. (26) 3.1. Proof of Proposition3.1: the approximation rates for the zero generator case

To shorten the notation, we useE:=E0,x.Let us first deal with the error ofY. Ifv belongs to [tk, tk+1) we haveY_vⁿ=Y_tⁿ_k. Then

E|Yv−Y_vⁿ|²≤2(E|Yv−Ytk|²+E|Ytk−Y_tⁿ_k|²).

Using Theorem4.1we boundkYv−Ytkk by

C4.1^y Ψ(x)(v−tk)¹² =C(Cg, b, σ, T, p0, δ)Ψ(x)(v−tk)¹²

(sinceα= 1 can be chosen wheng is locally Lipschitz continuous). It remains to bound E|Ytk−Y_tⁿ_k|² = E|Etkg(XT)−Eτkg(X_Tⁿ)|²=E|Eg( ˜˜ X_t^t_n^k,Xtk)−Eg( ˜˜ X^ττnk,Xτk)|². By (5) and the Cauchy-Schwarz inequality (Ψ1:=Cg(1 +|X˜_t^t_n^k,Xtk|^p0+|X˜^ττn^k,Xτk|^p0)),

|Eg( ˜˜ X_t^t_n^k,Xtk)−Eg( ˜˜ X^ττnk,Xτk)|² ≤ (˜E(Ψ1|X˜_t^t_n^k,Xtk−X˜^ττkn,Xτk|))²≤E(Ψ˜ ²₁)˜E|X˜_t^t_n^k,Xtk−X˜^ττnk,Xτk|².

Finally, we get by Lemma5.2-(v) that

E|Ytk−Y_tⁿ_k|² ≤

EE(Ψ˜ ⁴₁)¹2

EE˜|X˜_t^t_n^k,Xtk −X˜^ττnk,Xτk|⁴¹2

≤C(Cg, b, σ, T, p0)Ψ(x)²h¹².

Let us now deal with the error ofZ. We usekZv−Z_vⁿk ≤ kZv−Ztkk+kZtk−Z_tⁿ_kkand the representation

Zt=σ(t, Xt)˜E(g^′( ˜X_T^t,Xt)∇X˜_T^t,Xt) (see Theorem4.2), where

X˜_s^t,x = x+ Z s

t

b(r,X˜_r^t,x)dr+ Z s

t

σ(r,X˜_r^t,x)dB˜r−t, (27)

∇X˜_s^t,x = 1 + Z s

t

bx(r,X˜_r^t,x)∇X˜_r^t,xdr+ Z s

t

σx(r,X˜_r^t,x)∇X˜_r^t,xdB˜r−t, 0≤t≤s≤T.

For the first term we get by the assumption ong and Lemma5.2-(i) and (iii)

kZv−Ztkk = kσ(v, Xv)˜E(g^′( ˜X_T^v,Xv)∇X˜_T^v,Xv)−σ(tk, Xtk)˜E(g^′( ˜X_T^tk,Xtk)∇X˜_T^tk,Xtk)k

≤ kσ(v, Xv)−σ(tk, Xtk)k⁴kE(g˜ ^′( ˜X_T^v,Xv)∇X˜_T^v,Xv)k⁴ +kσk∞kE(g˜ ^′( ˜X_T^v,Xv)∇X˜_T^v,Xv)−E(g˜ ^′( ˜X_T^tk,Xtk)∇X˜_T^v,Xv)k +kσk∞kE(g˜ ^′( ˜X_T^tk,Xtk)∇X˜_T^v,Xv)−E(g˜ ^′( ˜X_T^tk,Xtk)∇X˜_T^tk,Xtk)k

≤ C(Cg^′, b, σ, T, p0)Ψ(x)h

h¹²+kXv−Xt_kk⁴+

EE˜|X˜_T^v,Xv−X˜_T^tk,Xtk|^4α1₄ +

EE˜|∇X˜_T^v,Xv− ∇X˜_T^tk,Xtk|⁴¹₄i

≤ C(Cg^′, b, σ, T, p0)Ψ(x)h^α2.

We compute the second term usingZ_tⁿ_kas given in (16). Hence, with the notation from Definition 2.2,

kZtk−Z_tⁿ_kk² = Eσ(t_k, Xtk)˜Eg^′( ˜X_t^t_n^k,Xtk)∇X˜_t^t_n^k,Xtk−ED˜ ⁿk+1g( ˜X^ττn^k,Xτk)²

≤ kσk²_∞E

E(g˜ ^′( ˜X_t^t_n^k,Xtk)∇X˜_t^t_n^k,Xtk)−E˜Dk+1ⁿ g( ˜X^ττnk,Xτk) σ(tk, Xt_k)

2

= kσk²∞E

E(g˜ ^′( ˜X_t^t_n^k,Xtk)∇X˜_t^t_n^k,Xtk)−E˜

g^(k+1,n+1)_x Dk+1ⁿ X˜^ττn^k,Xτk

σ(tk, Xtk)

2

.