High Order Numerical Methods And Error Estimates For Solving Decoupled And Weakly Coupled Forward-Backward Stochastic Differential Equations

The existence and uniqueness of the solution for nonlinear backward stochastic differential equations (BSDE) were first proved by Pardoux and Peng [46]. In [49], Peng first obtained the relation between the BSDE and the partial differential equation (PDE), and then the stochastic maximum principle for optimal control problems based on BSDE was derived in [48]. Since then, BSDE have been extensively studied by many researchers. Accordingly, also with the study of forward stochastic differential equation and BSDE, forward backward stochastic differential equations (FBSDEs) have also been developed in theoretical and applied research internationally. And the theory of FBSDEs has been studied by a lot of researchers attracted by its various applications, namely in finance, stochastic control, physics, chemistry, aerospace, partial differential equations and so on.By now, the theory of FBSDEs has been a significant branch of probability theory and stochastic analysis. However, there currently exist only few methods for numerically solving FBSDEs, most of them have low accuracy and thus are practically not very useful in real applications. With further research about FBSDEs scientific computing, we need to study new numerical methods for solving FBSDEs. Combining theories of FBSDEs and scientific computing, proposing new numerical methods for solving FBSDEs with innovative thinking, studying the convergence and stability of proposed methods, and making theoretical numerical analysis and error estimates are in need. These studies have important significance to make us understand FBSDEs’characteristics, develop the theory of FBSDEs’scientific computing richly, and accelerate practical applications of FBSDEs. The aim of this paper is to study high-order numerical methods, error estimates and algorithm implementation for solving FBSDEs. So far, the numerical methods of solving FBSDEs are roughly divided into two categories:One is based on the relationship FBSDEs and corresponding parabolic PDE, by solving the PDE viscosity solution to achieve the purpose of solving FBSDEs; The other is to get corresponding numerical schemes through directly numerical discrete FBSDEs. our aim is to study high-order numerical scheme of solving FBSDEs through the second technology.This paper mainly first study the characteristics of forward-backward stochastic dif-ferential equations and combine with the deterministic theory of scientific computing, we study of the decoupled forward-backward stochastic differential equations (decoupled F-BSDEs), weakly coupled forward-backward stochastic differential equations (weakly cou-pled FBSDEs), backward stochastic differential equations with jump (BSDE with jump) and decoupled forward-backward stochastic differential equations with jumps (decoupled FBSDEs with jumps) systematic. In what follows, we get high-order numerical schemes for solving the upper equations. Then, we give numerical analysis for these numerical schemes and obtain stability and convergent. At last, we give numerical results which carried out in each chapter of this thesis are consistent with the theoretical results.Let us introduce the main content and the organization of the thesis.In Chapter1, the Introduction gives an overview of our topics in Chapter3to Chapter6.In Chapter2, we introduce the existence and uniqueness of the solution of stochastic differential equation (with jumps). Then we give some properties of the analysis solution and approximation solution of stochastic differential equation (with jumps).In Chapter3, we mainly study one-step numerical method for solving decoupled FBSDEs. The technical is:(1) Using the theory of stochastic analysis and FBSDEs, FBSDEs will become determinate integral equations through mathematical expectation condition.(2) Using numerical integration theory of scientific computing, we make use of nu-merical integration to approximate the integral of deterministic equations in the time domain;(3) Approximated the conditional expectation in (2) accurately by Gauss numerical integration;(4) We solve eqnarray in (3) using numerical algebra;(5) Use the theory of stochastic analysis, the theory of FBSDEs and the theory of scientific computing, give stability, convergence and error estimate analysis.The innovation of this chapter:stochastic analysis, FBSDEs’theory and scientific computing combined with a deterministic theory:proposing high-order numerical scheme for solving FBSDEs.(1) Construct a special new standard Brownian motion;(2) By the new Brownian motion, propose high-order numerical method for decou-pled forward-backward stochastic differential equations. And then, we get error esti-mates, stability analysis and convergent;(3) Algorithm processing for numerical schemes.This chapter is mainly based on the paper:Zhao, W., Zhang, W., and Ju, L., A Numerical Method and its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations, Commun. Comput. Phys.15(2014), pp.618-646.In Chapter4, we mainly study multi-step numerical method for solving decoupled FBSDEs. First, we propose the concept of a new backward orthogonal polynomials. Based on standard backward orthogonal polynomials, we construct a new kind of Brow-nian motion. Then we propose numerical scheme for solving decoupled FBSDEs using the new Brownian motion. In what follows, we prove that the stability of this new multi-step scheme. Finally, our numerical results carried out in this chapter are consistent with the theoretical results.This chapter is mainly based on the paper:Zhao, W., Zhang, W., and Ju, L., A multistep scheme for decoupled forward-backward stochastic differential equations, Submitted.In Chapter5, we consider the problem of weakly coupled forward-backward stochas-tic differential equations, where the weak coupling means that coefficient functions is dependent the solution of backward stochastic differential equation Y, but it does not depend on the process Z. We use two different approaches to study the issue and get the same order of convergence.(1) Improved numerical schemes for solving weakly coupled forward-backward s-tochastic differential equations, which we call improved Euler scheme and improved Markovian iterative numerical scheme. Then, we analysis the error estimate and prove that the original schemes and improved schemes convergence rate are of first-order.(2) Use the method in (2), we give the other method to prove that the convergence rate of the original scheme and the proposed improved scheme are of first-order.Innovations:(1) Improved Euler numeric scheme and improved Markovian iterative numerical scheme significantly reduced in the calculation of solving conditions mathematic expec-tation.(2) Convergence rate is first-order which improved the convergence rate is of half-order.This chapter is mainly based on the following papers:Zhang, W., Zhao, W., On Euler-type schemes for weakly coupled FBSDEs and the optimal convergence analysis, Frontiers of Mathematics in China, DOI:10.1007/s11464-014-0366-6,2014.Error estimates of the numerical scheme for solving weakly coupled forward-backward stochastic differential equations, finished.In Chapter6, we study the theory of forward-backward stochastic differential e-quations with jumps, propose numerical algorithm and take related theoretical analysis. Consider the different between FBSDEs with compensated Poisson process and FBSDEs with Poisson random measure. And then, we consider the problems of BSDE with jump and decoupled FBSDEs with jumps. First we study Ito-Taylor expansion of forward d-ifferential equations, and then propose a new operator to give reference equations for proposing high-order numerical schemes to solve the equations. And we give error esti-mates and numerical calculations to verify the convergence of the proposed high-order scheme.This chapter is mainly based on the following papers:A new kind of accurate discrete-time approximation of backward stochastic differen-tial equation with jumps, finished.Second order numerical scheme and error estimates of solving decoupled forward backward stochastic differential equations with jumps, finished.This paper contains five chapters, we give an overview of the structure and the main results of this dissertation.Chapter1Introduction;Chapter2Stochastic differential equation (with jumps);Chapter3One-step numerical scheme, error estimates and algorithm processing for solving decoupled forward-backward stochastic differential equations;Chapter4Multi-step numerical scheme, error estimates and algorithm processing for solving decoupled forward-backward stochastic differential equations;Chapter5Improved numerical methods and error estimates for weakly coupled forward-backward stochastic differential equations;Chapter6High order numerical analysis for solving backward stochastic differen-tial equation with jump and decoupled forward-backward stochastic differential equations with jumps.Chapter3:we first construct a new Brownian motion△Ws, then a new numerical method for solving the decoupled forward-backward stochastic dif-ferential equations is proposed based on the new Brownian motion and some specially derived reference equations. We rigorously analyze errors of the proposed method under general situations. In the following, we present er-ror estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method; in particular, we prove that the proposed method is second-order accurate if used together with the order-2.0weak Taylor scheme for the SDE. Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.We consider the following decoupled forward-backward stochastic differential equa-By introducing a new Brownian motion: we get one-step numerical scheme for solving decoupled FBSDEs:Scheme3.1.1.(Xn, Yn, Zn) is numerical solution of (Xt, Yt, Zt) at t=tn, we obtain (Xn,Yn,Zn),n=N-1,N-2,…,1,0through the following equations,For one-step numerical scheme,we have the following error estimate. Theorem3.3.1. Let(Xt,Yt,Zt),t∈[0,T] and (Xn,Yn,Zn),n=0,1,…,N,be the exact solution of the decoupled FBSDEs (3.1) and the approximate solution obtained by Scheme3.1.1,respectively. Assume that the function f(t,X,Y,Z)is Lipschitz continuous with respect to X,Y and Z and the Lipschitz constant is L.Let c0be the time partition regularity parameter defined in(3.2).Then for sufficiently small time step△tn,it holds that for n=N-1,…,1,0,where C is a positive constant depending on co and L,C’is also a positive constants depending on c0,丁and L,Riy；and Riz are defined in(3.9)and (3.16),andLemma3.4.1. If f(t,x,y,z)∈C2,4,4,4,b(t,x),σ(t,x)∈C2,4b,φ∈C4+αb,α∈(0,1)and|b(t,x)|2≤K(1+|x|2),|σ(t,x)|2≤K(1+|x|2),then for sufficiently small time step△tn, we have that for any0≤n≤Ⅳ-1, where C is a positive constant depending only on T,K,and upper bounds of the deriva-tives of b,σ,f and φ. Lemma3.4.2. Assume that the conditions of Lemma3.4.1hold, then for sufficiently small time step△tn, we have that for any0≤n≤N-1, where C is a positive constant depending only on T,K, and the upper bounds of the derivatives of b,σ,f and φ.Lemma3.4.3. Assume the conditions of Lemma3.4.1hold, then for sufficiently small time step△tn, we have that for any0≤n≤N-1, where C is a positive constant depending only on T, K, and upper bounds of the deriva-tives of b,σ,f and φ.Lemma3.4.4. Assume the conditions of Lemma3.4.1hold, then for sufficiently small time step△tn, we have that for any0≤n≤N-1, where C is a positive constant depending only on T and K, and upper bounds of the derivatives of b,σ, f and φ.Theorem3.4.1. Suppose that Assumption2.3.2and the conditions of Lemma3.4.1holds. Then for sufficiently small time step△tn,we have that for any0≤n≤N-1, where C1is a positive constant depending on c0, T and L, C2is also a positive constant depending on c0, T, L, K, the initial value of Xt in (6.1), and the upper bounds of the derivatives of b,σ,/and φ.Chapter4:we first introduce a new set of orthogonal polynomials, which we call backward orthogonal polynomials and study some of their simple properties. Based on the theory of numerical integrals and polynomial ap-proximation, we develop the new multi-step scheme by using the backward orthogonal polynomials and a special Gaussian process. Then, we get a gen-eral error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are also p-resented to numerically demonstrate the accuracy of the proposed multi-step scheme and to verify the theoretical results.In this chapter, we still consider the decoupled forward-backward stochastic differ-ential equations appear in chapter3. We give a multi-step numerical scheme, a definition and some main results.Definition4.1.1. We call a set of polynomials{Qi(s)}Li=0defined on the interval [0,1] the backward orthogonal polynomials, if for each i=0,1,…,L, it holds that Qi(s)∈Pi[0,1] andLemma4.1.1. There is a unique set of polynomials{Qi(s)}Li=0on [0,1] satisfying the conditions stated in (4.3). Furthermore, let Qi(s)=∑ij=0αi,jsj, then for each i=0,1,…,L, the αi,j is determined byLemma4.1.2. Let{Pi(s)}Li=0be the backward orthogonal polynomials on [a,b]. Then we have P0(α)=1and Pi(α)>1if1≤i≤L.Lemma4.1.3. Let {Pi(s)}Li=0be the backward orthogonal polynomials on [a,b]. As-sume that the function g(s) is differentiable with g(i+1) bounded for some i with0≤i≤L. Then we haveScheme4.2.1. Let K=max{Ky,Kz,Kf}. Assume random variables X0, YN-i and ZN-i,i=0,1,…, K-1, are known. Let{Xn}Nn=0be the numerical solution of the forward SDE in the decoupled FBSDEs by a numerical method for solving the SDE. For n=N-K,…,0,1.solve the random variable Zn by:2.solve Yn by:Lemma4.3.2. Suppose that N and K are two nonnegative integers with N≥K and△t is any positive number.Assume{an),{bn},and{Rn},n=0,1,…,N,are nonnegative,and satisfy the inequality for0≤n≤N-K,where C1,C2and C3are positive constants.Let Mα=(?)αi Mb=(?) bi and T=N△t.If C1-C3K2≥C4for some constant C4>0,then for n=N-K,N-K-1,…,0.it holds that where C is a constant depending on C1,C2,C3,K and T.Theorem4.3.1. Let(Xt,Yt,Zt),t∈[0,T] and (Xn,Yn,Zn),n=0,1,…,N be the exact solutions of the decoupled FBSDEs(4.1) and (6.1)and the approximate solution obtained by the multi-step semi-discrete Scheme4.2.1,respectively.Assume that the function f(t,Xt,Yt,Zt)is Lipschitz continuous with respect to XT,Yt and Zt and the Lipschitz constant is L.Let c0be the time partition regularity parameter defined in (2.2).Let K=max{Ky,Kz,Kf},B=max{BKy,BKf}and PKz=(?) QKz(s).Let Mey=(?) E[|eiy|2] and Mez=(?) E[|eiz|2]. Then if the time step△t is sufficiently small,it holds that for n=N-K,N-K-1,…,0, where C is a constant depending on c0,T,L,B,K, QKz(0) and PKz, Ry and Riz are defined in (4.14) and (4.19), respectively, andLemma4.4.2. Let Rny and Rnz be the local truncation errors defined in the reference equation (4.14) and (4.19). Then under Assumption4.4.1, we have the following local estimates: where C≥0is a generic constant depending only on T, the upper bounds of derivatives of b,σ,φand f.Theorem4.4.1. Suppose that Assumption4.4.1and Assumption2.3.2hold, and suppose the initial values satisfy Mey O((At)Ky+1) and Mez=O((At)Kf+1-Kz+If the order-β weak Taylor Scheme is used to solve the SDE (4.1) with β=γ=K+1in Scheme4.2.1, then for0≤n≤N-K, we have the following error estimate: where C is a constant depending on c0, T, L, K, B, QKz(0), PKz, the initial value of Xt in (4.1), and the upper bounds of the derivatives of b, σ, f and φ.Chapter4:we introduce a new Euler-type scheme and its iterative al-gorithm for solving weakly coupled forward backward stochastic differential equations. Although the schemes share some common features with the ones proposed in [6], less computational work is needed for our method. We use two different methods to prove that the error estimate of proposed scheme is of one-order which improve the half-order error estimate in [6]. Numerical tests are given to demonstrate the first-order accuracy of the schemes.In this chapter, we consider the following weakly coupled forward-backward stochas- tic differential equations:Scheme5.1.3.(Improved Euler-type scheme) Given random variables XO, YN and ZN. For n=N-1, N-2,…,0, solve random variables Xn, Yn and Zn byScheme5.1.4.(Improved Markovian iterative Euler-type scheme) Given X0,m, YN,m=φ(XN,m) and let un,0(X0,m)=0. For n=N-1,N-2,…,0, solve random variables Xn,m, Yn,m and Zn,m byTheorem5.3.1.(i) Let (Xt,Yt,Zt), t∈[0,T) and (Xn, Yn, Zn), n=0,1,…,N, be the exact solutions of the weakly coupled FBSDEs (5.1) and the approximation solution ofScheme5.1.1or Scheme5.1.3, respectively. Suppose that Assumption5.2.3and Assumption5.2.4hold true. Then for sufficiently small time step At, it holds that for n=N-1,N-2,…,0,(ii) And further, if (Xn,m, Yn,m, Zn,m), n=0,1,…,N and m=0,1,2,…, are the approximation solutions of Scheme5.1.2or Scheme5.1.4, respectively, then there are constants C>0and0<c<1such that, for sufficiently small At, we have the estimate Here the C is a positive constant which does not depend on the time partition and the approximation solution.Lemma5.3.2. Under Assumption5.2.4., for sufficiently small time step At, we have the estimates for any0≤n≤N-1, where C is a positive constant which does not depend on the time partition and the approximation solution.Lemma5.3.4. Let(Xt,Yt,Zt), t∈[0,T] and (Xn,Yn,Zn), n=0,1,…,N, be the exact solution of the weakly coupled FBSDEs (5.1) and the approximate solution ob-tained by equations (5.17),(5.18) and (5.19), respectively. Assume that the function f(t,X,Y,Z) is uniformly Lipschitz continuous with respect to X, Y and Z. Then for sufficiently small time step At, for n=N-1,N-2,…,0, it holds that Riy and Riz are defined in (5.14) and (5.15), and And if Assumption5.2.4. holds, we have the following estimates: which, by using (5.24), implies Here the C is a positive constant which does not depends on time partition and the solution of (5.17)-(5.19).Theorem5.3.2. Let (Xt,Yt,Zt),t∈[0,T] and (Xn,Yn,Zn), n=0,1,…,N, be the exact solution of the weakly coupled FBSDEs (5.1) and the approximation solution obtained by Scheme5.1.3, respectively. Suppose that Assumptions5.2.3and5.2.4hold. Then for sufficiently small time step At, it holds that for any0≤n≤N-1|Ytn(x)-Yn(x)|2≤C(1+|x|2)(△t)2, where C is a positive constant.Chapter6:we first study the Ito-Taylor expansion for forward stochas-tic differential equation, then give a new Brownian motion, a new Poisson process and reference equations for proposing high-order numerical method to solve backward stochastic differential equation with jump and decoupled forward backward stochastic differential equation with jumps. Last, we make error analysis and give numerical experiments to show that the scheme is of high-order.In this chapter, we consider the following decoupled forward-backward stochastic differential equations with jumps:where ξ=φ(XT), Γ:=∫Eρp(e)U(e)λ(de), sup(?)|p(e)|<K. Here W is a d-dimensional Brownian motion and μ is an independent compensated Poisson measure μ,(de,ds)=μ(de,ds)-λ(de)ds.Based on the new Brownian motion in Chapter3in order to solve Γ, we construct a similar compensation Poisson process First, we give a high-order numerical scheme for solving (X, Y, Z, Γ):Scheme6.3.1. Given YN,ZN,ZN,ΓN和X°=X°. We get the random variables Xn, Yn, Zn and Γn, n=N-1, N-2,…,1,0, through the following equations, where Xn+l is solved byLemma6.4.2. If f(t, x, y, z)∈C2,4,4,4b,b(t,x), σ{t, x)∈C2,4b,φ€C4+αb,α∈(0,1) andAssumption2.4.1. holds, then for sufficiently small time step△tn, we have that for any0≤n≤N-1, E[|Rny|2]≤C(1＋E[|Xn|8])(△t)6, where C is a positive constant depending only on T,K and upper bounds of the deriva-tives of b, σ, f and φ.Lemma6.4.3. Assume that the conditions of Lemma6.4.2. hold, then for suffi-ciently small time step△tn, we have that for any0≤n≤N-1, where C is a positive constant depending only on T, K, and the upper bounds of the derivatives of b, σ, f and φ.Lemma6.4.4. Assume the conditions of Lemma6.4.2. hold, then for sufficiently small time step△tn, we have that for any0<n<N-1, where C is a positive constant depending only on T, K and upper bounds of the deriva-tives of b,σ, f and φ.Lemma6.4.5. Assume the conditions of Lemma6.4.2. hold, then for sufficiently small time step Atn, we have that for any0<n<N-1, where C is a positive constant depending only on T and K, and upper bounds of the derivatives of b, σ, f and φ.Lemma6.4.6. Assume the conditions of Lemma6.4.2. hold, then for sufficiently small time step△tn, we have that for any0≤n≤N-1, where C is a positive constant depending only on T and K, and upper bounds of the derivatives of b,σ, f and φ.Lemma6.4.7. Assume the conditions of Lemma6.4.2. hold, then for sufficiently small time step△tn, we have that for any0≤n≤N-1, where C is a positive constant depending only on T and K, and upper bounds of the derivatives of b,σ, f and φ.Lemma6.4.8. Assume the conditions of Lemma6.4.2. hold, then for sufficiently small time step△tn, we have that for any0≤n≤N-1, where C is a positive constant depending only on T and K, and upper bounds of the derivatives of b,σ, f and φ. Theorem6.4.1. Let (Xt,Yt,Zt,Γt), t∈[0, T] and (Xn,Yn,Zn,Γn), n=0,1,…,N, be the exact solution of the decoupled FBSDEs (6.1) and the approximate solution obtained by the numerical Scheme6.3.1, respectively. Assume that the function f(t,Xt,Y, Zt,Γt) is Lipschitz continuous with respect to Xt, Yt, Zt and Γt and the Lip-schitz constant is L. Let c0be the time partition regularity parameter defined in (6.4). Then for sufficiently small time step△t, it holds that for n=N-1,…,1,0, where C is a positive constant depending on c0and L, C is also a positive constants depending on c0, T and L, Riy and Riz are defined in (6.9) and (6.16), andTheorem6.4.2. Suppose that Assumption4.4.1. and the conditions of Lemma6.4.2hold true. Then for sufficiently small time step△tn, we have that for any0≤n≤N-1, where C is a positive constant depending only on c0and L, C1is a positive constant depending on c0, T and L,C2is also a positive constant depending on c0, T, L, K, the initial value of Xt in (6.1), and the upper bounds of the derivatives of b, a, f and φ.