High Accurate Numerical Methods And Their Error Estimates For Solving Forward Backward Stochastic Differential Equations

The existence and uniqueness of the solution for nonlinear backward s-tochastic differential equations (BSDEs) were first proved by Pardoux and Peng [52] in1990. Since then, BSDEs have been extensively studied by many researchers. In [56], Peng first obtained the relation between the backward stochastic differential equation and the parabolic partial differential equa-tion (PDE), and then the stochastic maximum principle for optimal control problems based on BSDEs was derived in [55]. Since then, BSDEs have been extensively studied by many researchers. In the survey article [32] of El Karoui, Peng and Quenez and the book [46] of Ma and Yong, many inter-esting properties and applications (especially in mathematical finance and s-tochastic control) were presented and discussed. The nonlinear g-expectation via a particular nonlinear BSDE was introduced in [57], and in [35] it was found that a dynamic coherent risk measure can be represented by a properly defined g-expectation. At the same time, along with the fast development in the theory of stochastic differential equations (SDEs) and BSDEs, signif-icant progresses also have been made in the study of the forward-backward stochastic differential equation (FBSDEs in short). FBSDEs have very im-portant applications in many scientific fields such as randomized control, chemistry, physics, financial risk analysis and partial differential equations, but it is often very difficult to find analytic solutions of FBSDEs, thus it becomes more and more important to study numerical schemes for solving FBSDEs. However, there currently exit only few methods for numerically solving FBSDEs; most of them have low accuracy and thus are practically not very useful in real applications.There have been many efforts devoted to numerically solving BSDEs. By using the relation between the BSDE and PDE, a four step scheme was proposed in [45]. Following the spirit of the four step scheme, a characteristic finite difference method for solving parabolic PDEs was used to solve BSDEs in [29]. Based on similar ideas, the authors in [47,49,50,78] proposed al-gorithms to compute solutions of forward-backward SDEs. Another family of numerical methods was developed directly from BSDE instead of the re-lated PDE. Bally in [6] developed a special discretization scheme, which was performed on a random net, namely the jump times of a Poisson process. Chevance in [23] proposed a numerical method for BSDEs by using binomial approach. Weak convergence of these numerical solutions for BSDEs was obtained in [12,44]. In Zhang's doctoral dissertation [75], Zhang studied the properties of the solutions of FBSDEs, proposed stochastic Euler scheme and obtained its half order convergence rate. In [76], Zhang proposed a numerical scheme and studied its convergence rate under weak regularity assumptions. E. Gobet et al. generalized Zhang's result in the strong Lp-sense (p≥1)[36] and proposed a numerical scheme based on iterative regressions on function bases [37]. Bouchard and Touzi developed a backward simulation scheme in [16]. Different from above backward schemes, Bender and Denk introduced a forward scheme for solving BSDEs in [11]. Delarue and Menozzi proposed a time-space discretization scheme for solving quasi-linear parabolic PDEs in [27] and then improved this scheme through a special interpolation proce-dure [28]. In [58], Peng proposed an iterative linear approximation algorithm which converges under reasonable assumptions, and then Memin, Peng and Xu studied discrete reflected BSDEs in [48] using this algorithm. Cvitanic and Zhang transformed the forward-backward SDE to a control problem and proposed a steepest descent method to solve it [26]. In2006, Delarue and S. Menozzi proposed random algorithms for solving quasi-linear PDEs, and got its half convergence rate. Gobet and Labart [36] gave error expansion for the discretization of decoupled FBSDEs, and obtained weak convergence rate and strong half convergence rate of Euler scheme. In2008, based on [15], Bouchard and Elie studied Euler scheme for solving decoupled FBSDEs with jumps, and got its half convergence rates. Bouchard and Chassagneux [14] obtained half convergence rate of Euler scheme for solving decoupled FBSDEs with reflection.In2006, the θ-scheme for numerical solutions of general BSDEs was proposed by Zhao, Chen and Peng [79]. In the θ-scheme, deterministic time and space partitions were taken to discretize BSDEs, and a Monte Carlo method with some interpolation approximation procedures was used to cal-culate conditional mathematical expectations. The θ-scheme solves the BS-DE on a time-space grid and it is developed by approximating integrals with deterministic integrand functions, which are obtained by taking conditional mathematical expectations in the BSDEs.In the thesis, we study high accurate numerical methods and their error estimates for solving forward backward stochastic differential equations. The thesis is organized by eight chapters. In the following, we list the main results of this thesis.Chapter1:We will give a brief introduction to BSDEs and FBSDEs. We will review the Feynman-Kac formula, introduction to Malliavin Calculus, and summarize some knowledge for the numerical solution of SDEs, SDEs with jumps, FBSDEs, numerical solution of FBSDEs, numerical solution of FBSDEs with jumps or reflections. Finally we give our main innovation results.Chapter2:In this chapter, we will study the error estimates of θ scheme proposed by [79] for BSDEs. We will prove that this scheme is of second-order convergence for solving y and first-order convergence for solving z in LP space. And by using the variational equation, we will also prove that the Crank-Ninolson scheme proposed by [69] is second-order convergence in both solving y and z in LP space.For n=N-1,...,1,0, we will introduce the following θ scheme for solving BSDEs (see [79]): where (yn, zn) is a numerical solution of (yt, zt) at the time tn.Theorem2.2.1Suppose f∈Cb2,4and if φ∈Cb4. Let yt be the solution of the BSDE (2.2) and (yn,zn) be the solution of Scheme2.1.1(θ1=1/2). Then for sufficiently small time step Δtn and p≥1, we have where C is a constant depending only on T, upper bounds of derivatives of φ and f.Theorem2.2.2Suppose that f∈Cb2,4and φ∈Cb4. Let (yt, zt)(0≤t≤T) be the exact solution of the BSDEs (2.2), and (yn,zn)(n=N,...,0) be the solution of Scheme2.1.1. Then for sufficiently small time step Δtn and p≥1, we have where C is a constant depending only on T, upper bounds of derivatives of φ and f.Given random variable yN, solve random variables yn and zn, n=N-1,N-2,...,1,0,byTheorem2.3.1Suppose that f f∈Cb2,4and φ∈Cb4. Let (yt, zt) be the solution of (2.2), and let (yn,zn) be the solution of Scheme2.1.1. Then for sufficiently small time step Δtn and p≥1we have where C is a constant depending only on T, upper bounds of derivatives of φ and f.Theorem2.4.1Suppose f∈Cb2,4, h∈Cb2and φ∈Cb4. Let(yt,zt) be the solution of the BSDEs (2.2) and (yn,zn) be the solution of Scheme2.1.1. Assume E[yT-yN|2]≤C(Δt)4and E[|zT-zN|2]=E[φx'(WT)-zN|2)≤C(Δt)2. Then for sufficiently small time step At, we have where C is a constant depending on c0, T, and upper bounds of functions h, φ and f and their derivatives.For the general BSDEs with f=f(s,ys,zs), we will get second order convergence rate of Crank-Nicolson scheme (θ1=θ2=θ3=1/2).Theorem2.5.1Let (yt, zt) be the solution of the BSDEs (2.65) with the terminal condition yT=φ(WT), and (yn,zn) be its approximate solution produced by using the Crank-Nicolson scheme. Suppose that φ∈Cb3, f∈Cb3,6,6, and the initial error satisfies for some constant C>0. Then we have the following L2error estimate: when At is sufficiently small, it holds that for0≤n≤N-1, where CT,φ,f0is a generic constant depending only on T, upper bounds of derivatives of φ and f.And we will give second order strong convergence rate of Crank-Nicolson scheme in Lp norm.Theorem2.5.2Under the conditions of Theorem2.5.1, we have the follow-ing Lp(p≥1) error estimate:when At is sufficiently small, it holds that for0≤n≤N-1, where Cp,T,φ,f>0is a generic constant depending only on p, T, upper bounds of derivatives of φ and f.Chapter3: Based on the idea for solving BSDEs in [79], we will propose a generalized scheme for solving BSDEs by using Girsanov transformations, and analysis the error estimates of our schemes.Theorem3.4.1Suppose f∈Cb1,3,3. Let(yt, zt) and (yn, zn)(n=N....,0) be the solutions of BSDEs (3.1) and Scheme3.2.1(9∈[0,1],), respectively. Suppose E[|ytN-yN|2+|φx(WT)-zN|2]≤(Δt)2. Then for sufficiently small time step Δt, we have where C is a constant, which depends only on T, upper bound of functions φ, f and their derivatives.Theorem3.4.2Suppose f∈Cb2,3and h∈Cb3. Let (yt,zt) be the solution of (3.1) with the generator f=f(t,yt)+h(t)zt, and (yn,zn) is the solution of Scheme3.2.2. Suppose E[|ytN-yN|2]≤(Δt)4. Then for sufficiently small time step Δt, we have the following estimate where C is a constant, which depends only on T, upper bound of functions φ,h,f and their derivatives.Theorem3.4.3Suppose f∈Cb2,3and h∈Cb3. Let (yt,zt) be the solution of (3.1) with the generator f∈f(t,yt)+h(t)zt, and (yn,zn) is the solution of Scheme3.2.3. Suppose E[|ytN-yN|2+|φx(WT)-zN|2]≤(Δt)4.Then for sufficiently small time step Δt, we have where C is a constant, which depends only on T, upper bound of functions φ, f and their derivatives.Chapter4: Based on the idea for solving BSDEs in [79], we will propose a generalized θ scheme for solving BSDEs. Scheme4.1.1Given random variables yN and zN, solve random variables yn and zn, satisfying the equations and for n=N-1,...,0with the deterministic parameters θi∈[0,1](i=1,2), θ3∈(0,1], and θ4∈[-1,1] constrained by|θ4|≤θ3.Moreover we will theoretically obtain the error estimates for the scheme, and prove that the convergence rates of the scheme varies from half order to second order.Theorem4.3.1Let yt and yn be the solutions of BSDEs (4.1) and Scheme4.2.1, respectively. Then for sufficiently small time step Δtn, we have the following conclusions.1. For parameters θi∈[0,1](i=1,2), θ3∈(0,1], and θ4∈[-1,1) with constraint|θ4|<θ3, if E[|ytN-yN|2]≤CΔt, E[|ztN-zN|2]≤CΔt, φ∈Cb1, and f∈Cb1/2,1,1, then we have2. In particular, for θi=1/2(i=1,2,3), θ4∈(-1/2,1/2), if E[|ytN-yN|2]≤C(Δt)4,E[|ztN-zN|2]≤C(Δt)4,φ∈Cb3, and f∈Cb2,4,4, then we have Here C is a constant depending only on c0, T, the bounds of derivatives of φ and f, and the parameters θi(i=1,...,4). Using variational equation, we will get the optimal estimate for solving z.Theorem4.3.2Let (yt, zt) and (yn, zn) be the solutions of BSDEs (4.1) and Scheme.4.2.1, respectively. Then for sufficiently small time step Δtn, we have the following conclusions.easel: For θi∈[0,1](i=1.2), θ3∈(0,1] and θ4∈(-1,1) constrained by|θ4|<θ2, if and f∈Cb1,3,3, we have estimatescase2: For θi∈[0,1](i=1,2), θ3∈(0,1] and θ4∈(-1,1) constrained by|θ4|<θ3,if if and f∈Cb1,3,3, we havecase3: In particular, for θi=1/2(i=1,2,3) and θ4∈(-1/2,1/2), if and f∈Cb2,5,5, we have Here C is a constant which depends only on C0, the bound of derivatives of functions f and ip, and the parameters θi(i=1,...,4).At last, we will perform various numerical experiments to demonstrate the effectiveness and accuracy of the proposed generalized θ-scheme. Chapter5: In this chapter, we will consider the decoupled FBSDEs. Then we will propose numerical θ scheme5.1.1for solving decoupled FBSDEs and obtain its error estimates.Theorem5.3.1Let (Xttn,Xn,Yttn,Xn,Zttn,Xn) and (Xn,Yn,Zn) be the so-lutions of FBSDEs (5.2) and Scheme5.1.1, respectively. Then for n=N-1,...,0and sufficiently small time step Δt, we have where CL is a constant, which depends on the Lipschitz constant L of f(t, X, Y, Z) with respect to Y and Z, R%is defined by (5.9), RY1n=EtnXn[Ytn+1tn,Xn-Ytn+1tn+1,Xn+1Theorem5.3.2Let (Xttn,Xn,Yttn,Xn,Zttn,Xn) and (Xn,Yn,Zn) be the so-lutions of FBSDEs (5.2) and Scheme5.1.1. Let BYn be defined by (5.9), Then for n=N-1,...,0and sufficiently small time step At, we have E[|Ytntn,Xn-Yn|2] where CL is a constant which depends only on Lipschitz constant L of f[t, X, Y) with respect to YTheorem5.3.3Let YN=φ(XN) and ZN=φx(XN)σ(tN,XN). Let (Xt,Yt,Zt) be the solution of FBSDEs (5.1). Let the functions b,σ∈Cb2,4, f∈Cb2,4,4and φ∈Cb4+α(α∈(0,1)). Then under Hypothesis5.1.1, we have Here C is a positive constant which depends only on c0, T, K, and upper bound of the derivatives of b,σ f and φ.Theorem5.3.4Let YN=φ(XN) and f=f(t, Xt,Yt). Let (Xt, Yt, Zt) be the solution of FBSDEs (5.1). Let the functions b,σ∈Cb3,6,f∈Cb3,6,6and φ∈Cb6+α(α∈(0,1)). Then under the conditions of Hypothesis5.1.1and Theorem5.3.2, we have Here C is a positive constant which depends only on c0, T, K, and upper bound of the derivatives of b,σ, f and φ.Chapter6:We will study the accurate numerical methods for solving decoupled FBSDEs (6.1). Combining the numerical schemes for solving the forward SDEs, we will propose a new kind of numerical scheme (Scheme6.2.1) for solving the decoupled FBSDEs (6.1). Then we will rigorously analyze Scheme6.2.1, and theoretically obtain its error estimate. Theorem6.3.1Suppose f and φ are Lipschitz continuous functions. Let eYn=Yttn,Xn-Yn,eZn=Ztntn,Xn-Zn,and efn=f(tn,Xn,Ytntn,Xn,Ztntn,Xn)-f(tn,Xn,Yn,Zn), where Xn is the numerical approximation of the diffusion process Xt at the time tn. Let (Yt,Zt) and (Yn,Zn) be the solutions of the BSDEs (6.1) and Scheme6.2.1, respectively. Than for sufficiently small time step Δt, we have where θ1,θ3∈[0.1],θ2∈(0,1],0≤n≤N, C is a constant depending on T, and upper bounds of X0, functions φ and f and their derivatives, RYn, RZn, and RZ2n are defined by (6.6),(6.9),(6.14) and (6.18) respectively, andTheorem6.4.1Let Hypothesis6.5.1hold. Let YN=φ(XN), ZN=φx(XN)σ(tN,XN), eYn=Ytntn,Xn-Yn and eZn=Ztntn,Xn-Zn.If Xn+1converges strongly with order a to Xtn+1tn,Xn as Δt→0,φ∈Cb2+δ for some δ∈[0,1],b,σ∈Cb1,2and f∈Cb1,2,2,2. For θi∈[0,1](i=1,2,3),α=1/2or1, we nave the following estimate If Xn+1converges strongly with order α+1to Xtn+1tn,Xn as Δt→0,φ∈Cb2+δ for some δ∈[0,1],b,σ∈Cb3,6and f∈Cb1,2,2,2. For θi=1/2(i=1,2,3) and α=1/2,1,3/2,2, we have the following estimateTheorem6.4.2Lei Hypothesis6.5.1hold. Let YN=φ(XN), ZN=φx(XN)σ(tN,XN), eYn=Ytntn,Xn-Yn and eZn=Ztntn,Xn-Zn. If Xn+1converges weakly with order β+1to Xtn+1tn,Xn as Δt→0, φ∈Cb2+δ for some δ∈[0,1],b,σ∈Cbβ,2β and f∈Cb1,2,2,2For θi=[0,1](i=1,2,3),β=1,2, γ=1,2, we have the following estimate If Xn+1converges weakly with order β+1to Xtn+1tn,Xn as Δt→0, φ∈b6+δ for some δ∈[0,1], b,σ∈Cb3,6and f∈C3,6,6,6. For θi=1/2(i=1,2,3), β=1,2,γ=1,2, we have the following estimate Here C is a constant depending on T, and upper bounds of X0, functions φ and f and their derivatives.Chapter7: In this chapter, we will study high accurate numerical schemes for solving decoupled forward-backward stochastic differential equa-tions with jumps (7.1). In Section7.4, we will study error estimate of Euler-scheme7.1.1for solving FBSDEs with jumps (7.1), and rigorously and the-oretically prove first order convergence rate of our scheme. And in Section 7.5, we will propose Crank-Nicolson scheme (Scheme7.1.1) for solving the de-coupled FBSDEs with jumps (7.1), and theoretically obtain its second order convergence rate.Theorem7.4.1For0≤n≤N, Let eYn:=Ytntn,Xn-Yn and Let (Xttn,Xn,Yttn,Xn,Zttn,Xn) and (Xn,Yn,Zn) be the solutions of FBSDEs (7.3) and Scheme7.1.1, respectively. Let Ry be defined by (7.9), RY1n=Then for sufficiently small time step Δt, we have where n=N-1,...,0, C is a constant which depends only on c0,and the Lipschitz constant L of f(t, X, Y, Z, Γ) with respect to Y, Z. and Γ.Theorem7.4.2Let YN=φ(XN), ZtNtN,XN=ZN and ΓtNtN,XN=ΓN. Under Hypothesis7.1.1, we have Here C is a positive constant which depends only on T, upper bound of the derivatives of a,b,c and φ.Theorem7.5.2Let (Xttn,Xn,Yttn,Xn, Zttn,Xn,Γttn,Xn) and (Xn, Yn, Zn,Γn) be the solutions of FBSDEs (7.3)(f=f(s,Xs,Ys)) and Scheme7.5.1, respec tively. Let eYn:=Ytntn,Xn-Yn and Let RYn be defined by (7.9), Then for sufficiently small time step Δt, we have where n=N-1,...,0, CL is a constant which depends only on the Lipschitz constant L of f(t,X,Y) with respect to Y.Theorem7.5.3Let YN=φ(XN). Assume a,b,c∈Cb2,4and f∈Cb2,4,4. When θ1=1/2,θ2, θ3∈[0,1], under Hypothesis7.1.1we have and Here C is a positive constant which depends only on T, and upper bound of derivatives of a,b,c, f and φ. Chapter8: In this chapter we will study the accurate numerical meth-ods for solving decoupled forward-backward stochastic differential equation-s with reflection. Firstly we will rigorously prove the first order conver gence rate of the scheme proposed in [14], which generalize the result for the scheme with half-order convergence rate. Secondly we will propose the Crank-Nicolson scheme of FBSDEs with reflections, rigorously and theoreti-cally analyze our scheme, and obtain its second order convergence rate.Theorem8.1.1Let (Xttn,Xn,Yttn,Xn,Zttn,Xn) and (Xn,Yn,Zn) be the solu tions of FBSDEs (8.2) and Scheme8.1.1, respectively. Let eYn=Ytntn,Xn-Yn and eZn:=Ztntn,Xn-Zn(0≤n≤N).Then for sufficiently small time step Δt, we have where n=N-1,...,0, CL is a positive constant, which depends only on Co, the Lipschitz constant L of f(t,X,Y,Z) with respect to Y and Z, is defined by (8.5), RZ2n=Theorem8.1.2Let YN=φ(XN) and ZtNtN,Xn=ZN. Assume φ∈Cb3+α(α∈(0,1]), a,b∈Cb1,2,f∈Cb1,2,2,2, and a,b,h,f satisfy Hypothesis8.1.1. Then we have Here C is a positive constant which depends only on T, upper bound of the derivatives of a,b and φ. Theorem8.2.1When f=f(s,Xs,Ys), let (Xttn,Xn,Yttn,Xn,Zttn,Xn) and (Xn,Yn, Zn) be the solutions of FBSDEs (8.1) and Scheme8.2.1, respective ly. Let eYn:=Ytntn,Xn-Yn and Let RYn be defined by (8.3), and Then for sufficiently small time step Δt, we have where n=N-1,...,0, CL is a constant which depends only on the Lipschitz constant L of f(t, X, Y) with respect to YTheorem8.2.2Let YN=φ(XN). Assume φ∈Cb4+α(α∈(0,1]) and a,b∈Cb2,4,f∈Cb2,4,4,4, and a, b, h, f satisfy Hypothesis8.1.1. Then we have and Here C is a positive number, which depends only on T, upper bound of the derivatives of a.b,φ.