Numerical algorithms for backward stochastic differential equations with 1-d brownian motion: Convergence and simulations

DOI:10.1051/m2an/2010059 www.esaim-m2an.org

NUMERICAL ALGORITHMS FOR BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH 1-D BROWNIAN MOTION:

CONVERGENCE AND SIMULATIONS

Shige Peng

and Mingyu Xu

Abstract. In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are introduced. Then we prove the convergence of different algorithms and present simulation results for different types of BSDEs.

Mathematics Subject Classification. 60H10, 34K28.

Received November 13, 2008. Revised November 26, 2009.

Published online August 24, 2010.

1. Introduction

Non-linear backward stochastic differential equations (BSDEs in short) were firstly studied by Pardoux and Peng [18], who proved the existence and uniqueness of the adapted solution, under smooth square integrability assumptions on the coefficient and the terminal condition, and when the coefficientg(t, ω, y, z) is Lipschitz in (y, z) uniformly in (t, ω). From then on, the theory of backward stochastic differential equations (BSDE) has been widely and rapidly developed. And many problems in mathematical finance can be treated as BSDEs.

The natural connection between BSDE and partial differential equations (PDE) of parabolic and elliptic types is also important applications. It is known that only a limited number of BSDEs can be solved explicitly. To develop numerical methods and numerical algorithms is very helpful, both theoretically and practically.

Keywords and phrases.Backward stochastic differential equations, reflected stochastic differential equations with one barrier, numerical algorithm, numerical simulation.

∗ This work is partly supported by the National Basic Research Program of China (973 Program), No. 2007CB814902 and No. 2007CB814906.

∗∗Second author is also partly supported by Youth Grant of National Science Foundation (No. 10901154/A0110) and financial support of president fund of AMSS, CAS.

1 School of Mathematics and System Science, Shandong University, 250100 Jinan, P.R. China.[email protected].

2 Key Laboratory of Random Complex Structures and Data Science, Academy of Mathematics and Systems Science, CAS (No. 2008DP173182), P.R. China. [email protected].

3 Department of Financial Mathematics and Control science, School of Mathematical Science, Fudan University, 200433 Shanghai, P.R. China.

Article published by EDP Sciences c EDP Sciences, SMAI 2010

The solution of a BSDE is a couple of progressive measurable processes (Y, Z), which satisfies Yt=ξ+

T t

g(s, Ys, Zs)ds− T

t

ZsdBs, (1.1)

where B is a Brownian motion. Here ξ is terminal condition and g is a generator. From [13,18], we know that when ξis a square integrable random variable, and g satisfies Lipschitz condition and some integrability condition, BSDE (1.1) admits a unique solution.

The calculation and simulation of BSDEs is essentially different from those of SDEs (see [15]). Wheng is linear in y and z, we may solve the solution of BSDE by considering its dual equation, which is a forward SDE. However for many nonlinear cases of g, we can not find the solution explicitly. Here we describe a software package that compute our numerical solutions for BSDEs with a convenient user-machine interface⁴. This package computes solutions of BSDEs, reflected BSDEs with one or two barriers as well as BSDEs with constraints. One for significant advantage of this package is that users have a very convenient interface. Any users who know the ABC of BSDE can use this package very easily. The input-output interface was also carefully designed.

This paper is organized as follows. In Section 2, we introduce the discretization of BSDEs, then present implicit and explicit schemes for numerical calculation and consider their convergence. In Section 3, we continue to consider reflected BSDEs with one barrier which is an Itˆo process, by implicit reflected scheme, explicit reflected scheme, penalized explicit-implicit scheme and penalized explicit scheme, then we prove the convergence of these schemes. In Section 4, we show some numerical simulations for standard BSDE and reflected BSDE.

In Section 5, we apply penalized schemes to BSDEs with constraint onz and BSDE with solutiony reflecting on a function ofz.

We should point out that there have been many recent different algorithms for computing solutions of BSDEs and the related results in numerical analysis, for example [1–8,10,11,14,16,17,23–25]. In contrast to these results, our method uses very simple method.

2. Numerical schemes for standard BSDEs

Let (Ω,F, P) be a complete probability space, (Bt)t≥0 be a 1-dimensional Brownian motion defined on a fixed interval [0, T]. We denote by{Ft}_0≤t≤T the natural filtration generated by the Brownian motionB,i.e., Ft=σ{Bs; 0≤s≤t}augmented with allP-null sets ofF. We consider for a fixed n∈N,

B_tⁿ:=√ δ

[t/δ] j=1

εⁿ_j, for all 0≤t≤T, δ= T n,

where {εⁿj}ⁿj=1 is a {1,−1}-valued i.i.d. sequence with P{εⁿj = 1} = P{εⁿj = −1} = 0.5, i.e., a Bernoulli sequence. We setGjⁿ:=σ{εⁿ₁, . . . , εⁿ_j}andtj=δj.

Letg: [0, T]×R^d×R^d→R^dbe a Lipschitz function in (y, z) uniformly oft,i.e.,g satisfies for a fixedμ >0

|g(t, y₁, z₁)−g(t, y₂, z₂)| ≤μ(|y₁−y₂|+|z₁−z₂|) ∀t ∈[0, T], ∀(y₁, z₁),(y₂, z₂)∈R^d×R^d. (2.1) Andg(·,0,0) is square integrable.

4The study of simulations of BSDE has been started since 1996 in Shandong University, Mathematical Finance Laboratory directed by Peng Shige. First simulation was done by Zhou Haibin, then following his works Xu Mingyu worked on this software package since her master program (from 2000). This paper is a summary of algorithms for BSDE and reflected BSDE with one barrier that have been used in the package.

We will approximate a pair ofR^d-valued (Ft)-progressively measurable processes (Y, Z) defined on [0, T] such that E[sup_0≤t≤T|Yt|²] +E[T

0 |Zt|²dt]<∞, which satisfies

−dYt=g(t, Yt, Zt)dt−ZtdBt (2.2) with given terminal condition YT =ξ ∈ L²(FT,R^d), where L²(FT) is the space of FT measurable R^d-valued random variable satisfyingE|ξ|² <∞. It is clear that Y has continuous paths. An existence and uniqueness theorem for equation (2.2) was established in [18], when the generatorg satisfies (2.1) andg(·,0,0) is a square integrable. In many situations we are also interested in BSDEs of the following form ford= 1:

−dYt=g(t, Yt, Zt)dt+ dAt−ZtdBt, t∈[0, T], (2.3) where (At)t∈[0,T] is an (Ft)-predictable RCLL (right continuous with left limits) process with almost surely bounded variation such that A₀ = 0 and E[sup_0≤t≤T|At|²]< ∞. By the standard existence and uniqueness theorem for solutions of BSDE, for each given A andYT =ξ∈ L²(FT), there exists a unique pair (Y, Z) for equation (2.3). HereY has RCLL paths. We call the triple (Y, Z, A) a g-supersolution (resp. g-subsolution), if Ais an increasing process (resp. decreasing process). It is called a g-solution ifA≡0. It is easy to check that, if both (Y, Z, A) and (Y,Z,¯ A) are¯ g-supersolutions on [0, T], then (Z, A)≡( ¯Z,A). Thus we often call¯ Y a g-super(sub)solution (org-solution when A≡0) without specifying the related (Z, A).

2.1. Implicit and explicit schemes for BSDEs

We first give an assumption for discrete terminal conditionξⁿ∈L²(FT,R^d) andg.

Assumption 2.1. Considerξ which isFT-measurable and ξⁿ which isGnⁿ-measurable, such that E[|ξ|²] + sup

n E[|ξⁿ|²]<∞ and

nlim→∞E[|ξ−ξⁿ|²] = 0.

Moreover we assumen

j=0|g(tj,0,0)|²δ is uniformly bounded inn.

Example 2.1. Setξ= Φ((Bt)_0≤t≤T), where Φ :D_[0,T] :→R^d and satisfies Lipschitz condition. By Donsker’s theorem and Skorokhod representation theorem, there exists a probability space, such that sup_0≤t≤T|Bⁿt −Bt| → 0, asn→ ∞, inL²(FT), sinceεⁿ_j is inL^2+δ. Soξⁿ:= Φ((Bⁿ_t)_0≤t≤T), withξ, satisfies Assumption2.1.

The numerical solution of (2.2) is obtained by (Y_tⁿ, Z_tⁿ)≡(yⁿ_j, z_jⁿ),t∈[jδ,(j+ 1)δ), δn=T. (yⁿ_j, z_jⁿ)_0≤j≤n

is the solution of discrete BSDE which starts fromy_nⁿ=ξⁿ. Our discrete BSDE on the small interval is yⁿ_j =yⁿ_j+1+g(tj, yⁿ_j, z_jⁿ)δ−z_jⁿεⁿ_j+1√

δ. (2.4)

Then for giveny_jⁿ₊₁, we want to findGjⁿ-measurable (y_jⁿ, z_jⁿ). The feasibility of this scheme for smallδis due to the following easy lemma.

Lemma 2.1. Let yⁿ_j+1 be a givenGjⁿ+1-measurable random variable. Then, whenδ <1/μ, there exists a unique Gjⁿ-measurable pair(y_jⁿ, z_jⁿ)satisfying(2.4).

Proof. We set Y₊ =y_jⁿ₊₁|εⁿ_j+1=1 andY₋ =y_jⁿ₊₁|εⁿ_j+1=−1. BothY₊ and Y₋ areGjⁿ-measurable. Equation (2.4) is then equivalent to the following algebraic equation:

yjⁿ = Y₊+g(tj, yjⁿ, zⁿj)δ−zⁿj

√δ,

y_jⁿ = Y₋+g(tj, y_jⁿ, zⁿ_j)δ+zⁿ_j√ δ.

This is equivalent to

z_jⁿ= 1 2√

δ(Y₊−Y₋) = 1

√δE[yⁿ_j+1εj+1|Gjⁿ]. (2.5) and

yⁿ_j −g(tj, yⁿ_j, z_jⁿ)δ=1

2(Y₊+Y₋) =E[y_jⁿ₊₁|Gⁿj]. (2.6) Becausegis assumed to be Lipschitz, the mapping Θ(y) =y−g(tj, y, zⁿ_j)δis strictly monotonic: whenδμ <1,

Θ(y)−Θ(y), y−y ≥(1−δμ)|y−y|²>0.

So there exists a unique valuey_jⁿ satisfying (2.6).

This lemma shows a way to solve (2.4), and we named this algorithm as ‘implicit scheme’. In many cases, Θ⁻¹ cannot be solved explicitly. Thus we introduce the following explicit scheme by usingE[yⁿ_j+1|Gjⁿ] to approximate yⁿ_j in gof (2.4). We set ¯Y_Tⁿ= ¯y_nⁿ=ξⁿ and, starting fromj =n−1, solve in following reverse order,

¯

y_jⁿ= ¯y_jⁿ₊₁+g(tj, E[¯y_jⁿ₊₁|Gjⁿ],z¯_jⁿ)δ−z¯_jⁿεⁿ_j+1√

δ. (2.7)

Then we get,

yⁿ_j = E[¯y_jⁿ₊₁|Gjⁿ] +g(tj, E[¯y_jⁿ₊₁|Gⁿj], zⁿ_j)δ, zⁿ_j = 1

√δE[¯y_jⁿ₊₁εⁿ_j+1|Gjⁿ] = y¯_jⁿ₊₁|εⁿ_j+1=1−y¯_jⁿ₊₁|εⁿ_j+1=−1

2√

δ ·

This explicit scheme is useful wheng is not linear iny, like g(t, y, z) = sin(y) and the following example.

Example 2.2. In dynamic risk measure (cf.[21]), for a positionX in the market, defineρ(X) :=Y₀, whereY₀ is the solution of BSDE associated with (g,−X). Ifgis a convex function, thenρ:L²(FT)→Rgives a convex dynamic risk measure.

Another advantage of explicit scheme is in the case of d >1. If such case, we consider a multi-dimensional BSDE driven by 1-d Brownian motion. So to solve Θ⁻¹ means to solve a d-dimensional equation group with dvariables. Even in linear case, it is not easy, since we need to solve a d-dimensional linear equation. In such situation, explicit scheme can reduce many calculations, so we can get the result quicker.

Remark 2.1. To find g-super(sub)solution with an increasing process A as in (2.3), we need to consider the discretization ofA, settingAⁿ₀ = 0,Aⁿ_j :=j−1

i=0E[At_i+1−At_i|Giⁿ]. SinceAis an increasing process,Aⁿ_j is also increasing. Then instead of (2.4), we get

yⁿ_j =yⁿ_j+1+g(tj, y_jⁿ, zⁿ_j)δ+ (Aⁿ_j+1−Aⁿ_j)−z_jⁿεⁿ_j+1√ δ, whereAⁿ_j+1−Aⁿ_j isGjⁿ-measurable. Then from implicit scheme we get

yⁿ_j = Θ⁻¹(E[y_jⁿ₊₁|Gjⁿ] + (Aⁿ_j+1−Aⁿ_j)), z_jⁿ = 1

√δE[y_jⁿ₊₁εⁿ_j+1|Gjⁿ] = y_jⁿ₊₁|εⁿ_j+1=1−y_jⁿ₊₁|εⁿ_j+1=−1

2√

δ ·

And from explicit scheme, we get

yⁿ_j = E[¯yⁿ_j+1|Gjⁿ] +g(tj, E[¯yⁿ_j+1|Gjⁿ], zⁿ_j)δ+ (Aⁿ_j+1−Aⁿ_j), zⁿ_j = 1

√δE[¯yⁿ_j+1εⁿ_j+1|Gjⁿ] = y¯ⁿ_j+1|εⁿ_j+1=1−y¯_jⁿ₊₁|εⁿ_j+1=−1

2√

δ ·

In this paper, we will not make special efforts to study the convergence of discreteg-super(sub)solution. Indeed, if we set

yjⁿ=yjⁿ+Aⁿj,zⁿj =zjⁿ, 0≤j≤n,

then (yⁿ,zⁿ) is discrete solution of discrete BSDE with coefficientg(t, y, z) =g(t, y−At, z). WhenAⁿ→Ain certain sense, then we can get the convergence of (yⁿ, zⁿ) by (yⁿ,zⁿ), which is discrete solution of a classical BSDE.

However in many cases, the increasing process A is not given, it is associated with (Y, Z) in order to keep (Y, Z) to satisfying certain condition, like reflected BSDE and constraint BSDE. We will discuss them later in this paper.

2.2. Convergence results for numerical schemes for BSDEs

Y_tⁿ =yⁿ_[t/δ], Z_tⁿ=zⁿ_[t/δ], Y¯_tⁿ= ¯y_[ⁿ_t/δ], Zⁿ_t =zⁿ_[t/δ], 0≤t≤T,

where (yⁿ_j, z_jⁿ)_0≤j≤nand (yⁿ_j, zⁿ_j)_0≤j≤nare discrete solutions of (2.4) by implicit and explicit schemes, respectively.

By Donsker’s theorem and Skorokhod representation theorem, there exists a probability space, such that sup_0≤t≤T|Btⁿ−Bt| → 0, as n → ∞, in L²(FT), since εk is in L^2+δ. Here L^2+δ is the space of random variableφsatisfyingE[(φ)^2+δ]<+∞. Then we have:

Theorem 2.1. We suppose that Assumption 2.1 hold and that g is Lipschitz in y and z. Then the discrete solutions {(Yⁿ, Zⁿ)}^∞n=1 under the implicit scheme and {( ¯Yⁿ,Z¯ⁿ)}^∞n=1 under the explicit scheme converge to the solution (Y, Z)of(2.2)in the following senses: as n→ ∞,

E

sup

0≤t≤T|Y_tⁿ−Yt|²+ T

0 |Z_sⁿ−Zs|²ds

→0, (2.8)

and

E

0≤supt≤T

Y¯_tⁿ−Yt ²+ T

0

Z¯_sⁿ−Zs ²ds

→0. (2.9)

The convergence (2.8) for this implicit scheme was obtained in 2001 by a profound result of Briandet al.[5], which can also be found in [6]. From these results, convergence (2.9) can be derived. Before proving (2.9), we first present following lemmas.

Lemma 2.2. Let a,b andαbe positive constants,δb <1 and a sequence(vj)j=1,...,n of positive numbers such that, for every j

vj+α≤a+bδ j i=1

vi. Then

sup

j≤n

vj+α≤ae^bT.

This is a type of Gronwall lemma for discrete cases. The proof can be found in [17], so we omit it.

Lemma 2.3. We assume thatδ is small enough such that(1 + 2μ+ 2μ²)δ <1. Then E

sup

j

yⁿj ²+

n−1

j=0

zⁿj ²δ

≤Cξⁿ,ge^{(1+2μ+2μ2)T} (2.10)

whereCξⁿ,g = (1 +δμ)E[|ξⁿ|²] +n−1

j=0 g²(tj,0,0)δ.

Proof. From explicit scheme

yⁿ_j =yⁿ_j+1+g(tj, E[yⁿ_j+1|Gjⁿ], zⁿ_j)δ−zⁿ_j√ δεⁿ_j+1. We have

|¯y_jⁿ|²− |yⁿj+1|² = − |¯z_jⁿ|²δ+ 2[¯y_jⁿ·g(tj, E[yⁿ_j+1|Gjⁿ], zⁿ_j)]δ− |g(tj, E[yⁿ_j+1|Gjⁿ], zⁿ_j)|²δ²

−2¯y_jⁿz¯_jⁿ√

δεj+1+ 2¯z_jⁿg(tj, E[yⁿ_j+1|Gjⁿ], zⁿ_j)δ√

δεⁿ_j+1 (2.11)

Taking expectation and the sum forj=i, . . . , n−1 yields E|y¯ⁿ_i|²≤E|ξⁿ|²−

n−1

j=i

E|¯z_jⁿ|²δ+ 2δE

n−1 j=i

{|¯y_jⁿ| ·(|g(tj,0,0)|+μ|E[yⁿj+1|Gjⁿ]|+μ|¯z_jⁿ|)}.

Since the last term is dominated by δE

n−1

j=i

|y¯_jⁿ|²(1 +μ+ 2μ²) +|g(tj,0,0)|²+μ|E[yⁿ_j+1|Gjⁿ]|²+1 2|z¯_jⁿ|²

≤δE

n−1 j=i

|y¯_jⁿ|²(1 + 2μ+ 2μ²) +|g(tj,0,0)|²+1 2|z¯_jⁿ|²

+μδE|ξⁿ|²,

we thus have

E|¯y_iⁿ|²+1 2

n−1 j=i

E|¯zⁿ_j|²δ≤

n−1 j=i

|g(tj,0,0)|²δ+ (1 +μδ)E|ξⁿ|²+δ(1 + 2μ+ 2μ²)

n−1 j=i

E|y¯ⁿ_j|²

Then by Lemma2.2, we obtain sup

i

E|y¯_iⁿ|²+1 2

n−1

j=0

E|z¯_jⁿ|²δ≤Cξⁿ,ge^{(1+2μ+2μ2)T}.

For (2.10), we recall (2.11), and take the sum forj=i, . . . , n−1 and sup overj, then take expectation. Notice that i

j=0y¯ⁿ_jz¯_jⁿ√

δεⁿ_j+1 andi

j=0g(tj, E[yⁿ_j+1|Gjⁿ], zⁿ_j)¯zⁿ_jδ√

δεⁿ_j+1 are both martingales with respect toGiⁿ, we apply Burkholder-Davis-Gundy inequality for them with similar techniques as before, then get

E[sup

i |¯yⁿ_i|²]≤cCξⁿ,gⁿ+Cμδ

n−1

j=0

E|¯y_jⁿ|²≤cCξⁿ,gⁿ+CμTsup

j E|¯yⁿ_j|².

With previous results, we obtain (2.10).

Proof of Theorem 2.1. The convergence of (Yⁿ, Zⁿ) to (Y, Z) is proved in [5]. To prove (2.9), the result for (Yⁿ, Zⁿ), it suffices to prove asn→ ∞,

E

0≤supt≤T

Y_tⁿ−Y¯_t^{n 2}+ T

0

Z_sⁿ−Z¯_s^{n 2}ds

→0. (2.12)

From (2.4) and (2.7), we have

|yⁿ_i −yⁿ_i|² = y_iⁿ₊₁−yⁿ_i+1 ²− |zⁿ_i −z¯ⁿ_i|²δ+ 2[(yⁿ_i −y¯ⁿ_i)·(g(tj, y_iⁿ, z_iⁿ)−g(tj, E[yⁿ_i+1|Giⁿ], zⁿ_i))]δ

− |g(tj, y_iⁿ, zⁿ_i)−g(tj, E[yⁿ_i+1|Giⁿ], zⁿ_i)|²δ²−2(y_iⁿ−yⁿ_i)(zⁿ_i −z¯_iⁿ)√ δεⁿ_j+1 + 2(g(tj, yiⁿ, zⁿi)−g(tj, E[yⁿi+1|Giⁿ], zⁿi))(ziⁿ−z¯iⁿ)δ√

δεⁿj+1. (2.13)

Then we take expectation and the sum overifromj ton−1. Withξⁿ−ξⁿ= 0, we get E y_jⁿ−yⁿ_j ² ≤ −E

δ

n−1 i=j

|ziⁿ−zⁿ_i|²

+ 2

n−1

i=j

E[|yⁿi −yⁿ_i| · |g(tj, yⁿ_i, z_iⁿ)−g(tj, E[yⁿ_i+1|Giⁿ], zⁿ_i)|]δ

≤ −1 2E

δ

n−1

i=j

|z_iⁿ−zⁿ_i|²

+ 2μ²δE _n−1

i=j

|y_iⁿ−yⁿ_i|²

+ 2μδE

n−1 i=j

|yⁿ_i −yⁿ_i| · |y_iⁿ−E[yⁿ_i+1|Giⁿ]|.

Sinceyⁿi −E[yⁿi+1|Gⁿj] =g(tj, E[yⁿj+1|Gjⁿ], zⁿj)δ, the last term is dominated by

δ

n−1

i=j

(2μ+ 1)E|yⁿ_i −y¯_iⁿ|²+

n−1

i=j

μ²E|g(tj, E[yⁿ_j+1|Gjⁿ], zⁿ_j)|²δ³.

But with (2.10), the second term is bounded by Cδ². We thus have E y_jⁿ−yⁿ_j ²+δ

2E _n−1

i=j

|z_iⁿ−zⁿ_i|²

≤(1 + 2μ+ 2μ²)δ _n−1

i=j

E[|y_iⁿ−yⁿ_i|²

+Cδ².

By Lemma2.2, we get

sup

j≤n

E y_jⁿ−yⁿ_j ²≤Cδ²e^{(2μ+2μ2+1)T}.

Then we reconsider square of the difference between the discrete solutions of implicit scheme and explicit scheme shown in (2.13). This time we first take the sum and sup_j, then take expectation. Using Burkholder-Davis- Gundy inequality and similar techniques, we get

E

sup

j |yⁿj −y¯_jⁿ|²

≤ CμE

δ

n−1 i=j

|yⁿi −yⁿ_i|²+δ

n−1 i=j

|zⁿi −zⁿ_i|²

≤CμTsup

j≤n

E y_jⁿ−yⁿ_j ²,

with previous results, (2.9) follows.

We now prove a more general result which will be useful in proving convergence results for schemes of reflected BSDEs. Consider the following BSDE

−dYt = [g₁(t, Yt, Zt) +g₂(t, Yt, Zt)]dt−ZtdBt, (2.14) YT = ξ.

Hereg₁andg₂are both Lipschitz functions. Then we have the following implicit-explicit scheme to only replace yⁿ_j byE[y_jⁿ₊₁|Gjⁿ] ing₂,

¯

y_jⁿ= ¯y_jⁿ₊₁+g₁(tj, yⁿ_j,¯zⁿ_j)δ+g₂(tj,[¯y_jⁿ₊₁|Gjⁿ],z¯_jⁿ)δ−z¯_jⁿεⁿ_j+1√

δ, (2.15)

or, equivalently,

yⁿ_j = E[¯y_jⁿ₊₁|Gⁿj] +g₁(tj, yⁿ_j,z¯ⁿ_j)δ+g₂(tj, E[¯y_jⁿ₊₁|Gjⁿ], zⁿ_j)δ, zⁿ_j = 1

√δE[¯y_jⁿ₊₁εⁿ_j+1|Gjⁿ] = y¯_jⁿ₊₁|εⁿ_j+1=1−y¯ⁿ_j+1|εⁿ_j+1=−1

2√

δ ·

We also set ¯Y_tⁿ = ¯yⁿ_[t/δ], Z¯_tⁿ= ¯zⁿ_[t/δ], 0≤t≤T. Meanwhile we consider the fully implicit scheme yⁿ_j =yⁿ_j+1+g₁(tj, yⁿ_j, z_jⁿ)δ+g₂(tj, yⁿ_j, zⁿ_j)δ−zⁿ_jεⁿ_j+1√

δ, and letY_tⁿ=y_[ⁿ_t/δ], Z_tⁿ=z_[ⁿ_t/δ], 0≤t≤1.

Proposition 2.1. Under same assumptions of Theorem 2.1, assume g₁ and g₂ are Lipschitz functions. Let (Y, Z) be the solution of BSDE(2.14). Then asn→ ∞,

E

0≤supt≤T

Y¯_tⁿ−Yt ²+ T

0

Z¯_sⁿ−Zs ²ds

→0. (2.16)

Moreover there exist a constant C₂ depending on T, μ₁ and μ₂ which are Lipschitz constants of g₁, g₂, such that

E

sup

0≤t≤T

Y¯_tⁿ−Y_t^{n 2}+ T

0

Z¯_sⁿ−Z_s^{n 2}ds]

≤C₂δ².

The proof is similar to that of Theorem2.1and we omit it.

Remark 2.2. This scheme is very useful. For example, we will use it for penalization BSDE, which will be discusses in Section 4.1.

3. Algorithms for reflected BSDEs with one barrier

In this section, we discuss the algorithms for reflected BSDEs with one continuous lower barrierL. A solution of such equation is a triple (Y, Z, K) on [0, T] satisfyingE[sup_0≤t≤T|Yt|²+T

0 |Zs|²ds+|KT|²]<∞and Yt = ξ+

T t

g(s, Ys, Zs)ds+KT −Kt− T

t

ZsdBs, (3.1)

Yt ≥ Lt, dKt≥0, 0≤t≤T, with T

0 (Yt−Lt)dKt= 0.

In [12], existence and uniqueness of the solution of this equation is proved when g satisfies Lipschitz condi- tion (2.1) and E[|ξ|²+T

0 g²(t,0,0)dt+ sup_0≤t≤T(L⁺_t)²] <∞. Here we consider the case when Lt is an Itˆo process,i.e. Lt=L₀+t

0lsds+t

0σsdBs, 0≤t≤T andξ= Φ((Bs)_0≤s≤T) satisfying requires of integrability, for convenience of discretization of processes.

Remark 3.1. We call a progressively measurable processφtis in spaceS²(0, T), if it satisfiesE[sup_0≤t≤T|φt|²]<

∞. If a predictable processφtis in spaceL²_F(0, T), then it satisfiesE[T

0 |φs|²ds]<∞. And we define a space ofFt-measurable random variablesξ, which satisfiesE[|ξ|^β]<∞, asL^β(Ft), forβ∈R⁺.