Study On Numerical Methods For FBSDEs With Their Applications In PDEs

In 1990,the existence and uniqueness of the adapted solution for nonlinear back-ward stochastic differential equations(BSDEs)were first proved by Pardoux and Peng[68],which lay a foundation for the theory of BSDEs.In 1991,Peng[74]ob-tained the relationship between BSDEs and a kind of second-order parabolic PDEs,called the nonlinear Feynman-Kac formula.The theory of forward backward stochas-tic differential equations(FBSDEs)has become into a significant branch discipline of stochastic analysis and probability theory nowadays.Meanwhile,forward backward stochastic differential equations have important applications in many areas,such as stochastic optimal control,mathematical finance,nonlinear expectations,partial d-ifferential equations,risk measure,game theory,and so on.However,it is usually difficult to get the analytical solutions in an explicit closed form.Therefore,the study on numerical solutions of FBSDEs is of vital importance for FBSDEs theory and their applications in practice.In this thesis,we mainly study high accurate numerical methods for forward back-ward stochastic differential equations(FBSDEs)and forward backward stochastic d-ifferential equations with jumps(FBSDEJs),and their applications in solving partial differential equations(PDEs).Specifically speaking,based on the the structure of the solution to FBSDEs and the theory of FBSDEs and deterministic numerical method,we rigorously analyze the error estimates of the Crank-Nicolson scheme and the mul-tistep scheme for solving FBSDEs;propose the explicit prediction correction scheme for solving FBSDEJs and also analyze the scheme theoretically and numerically;give the backward stochastic representations of the Dirichlet initial boundary problem and the fractional Laplacian equation,based on which,we propose the backward stochastic algorithms for them and rigorously analyze the first order of convergence.Besides,we give the numerical simulation algorithm for simulating G-Brownian motion for the first time and analyze the stability and effectiveness of the algorithm numerically by imple-menting the numerical simulation.We expect that the simulation algorithm will play an importance role in the theory and application of(backward)stochastic differential equations driven by G-Brownian motion.The main contribution and innovation:(1)The rigorously theoretical analysis of the second-order convergence of the Crank-Nicolson(C-N)scheme for solving decoupled FBSDEs is obtained,which fills the gap between the numerical analysis and theoretical analysis of the C-N scheme.The study result has been published in Sci.China Math.[55].(2)Prove rigorously the high-order convergence of the multistep scheme,and obtain that the k-step scheme can reach k-th order of convergence.The study result has been published in East Asian J.Appl.Math.[99].(3)Propose a new explicit scheme—prediction correction scheme for solving FBSDEJs and rigorously analyze the stability and the second-order convergence of the scheme theoretically and numerically.The study result has been published in East Asian J.Appl.Math.[36].(4)Propose a backward stochastic numerical solution method for solving Dirichlet initial boundary problem.The numerical scheme is proved to be of order one theoretically and numerically.The study result will be published in J.Comput.Math.[98].(5)A backward stochastic numerical method for fractional Laplacian equations is de-veloped.Based on the stochastic presentation,the Euler scheme for solving FB-SDEs with ?-stable process is given.The numerical experiments confirm the first order of convergence.The study result has been finished[94].(6)Give a numerical simulation algorithm for the G-Brownian motion.The numerical simulations of G-Brownian motion and related others show that the proposed algo-rithms are effective and stable.The algorithms would serve as a fundamental tool for future studies,e.g.,for solving SDEs/BSDEs driven by G-Brownian motion.The study result has been published in Front.Math.China[100].The framework:The thesis is divided into six chapters.Chapter 1.IntroductionIn Chapter 1,we make a brief introduction of the background,research motivation,and development of our topic in the following chapters.Chapter 2.PreliminariesIn Chapter 2,we introduce the existing theoretical results for our topic,including that of stochastic differential equations(SDEs)and stochastic differential equations with jumps(SDEJs);the relation between the solu-tions of FBSDEs,FBSDEJs,and FBSDEs with stopping time and their corresponding kind of parabolic PDEs,i.e.,three types of Feynman-Kac formulas.Chapter 3.Error Estimates of High-Order Numerical Methods for FBSDEsIn Chapter 3,we rigorously analyze the error estimates of the Crank-Nicolson scheme and the multistep scheme for solving FBSDEs.In Part ?,based on the Taylor and Ito-Taylor expansions,the Malliavin calculus theory,and our new truncation error cancelation techniques,we rigorously prove that the strong convergence rate of the Crank-Nicolson scheme is of order two for solving decoupled FBSDEs.In Part ?,under reasonable assumptions we prove that the k-step scheme admits a k-oder convergence rate for a special kind of FBSDEs.This chapter is mainly based on the papers:· YANG Li,JIE Yang,and WEIDONG ZHAO,Error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs,Sci.China Math.,60(5),pp.923-948,2017.(SCI)· JIE YANG AND WEIDONG ZHAO,Convergence of recent multistep schemes for a forward-backward stochastic differential equation,East Asian J.Appl.Math.,5(4),pp.387-404,2015.(SCI)Chapter 4.The Prediction-Correction Method for FBSDEJsIn Chapter 4,we mainly study on the explicit prediction correction method for solving FBSDEJs.We first give the reference equations by the mar-tingale theory and the property of conditional expectations,then derive the error equations and obtain the general stability result by analyzing the error equations,and finally get the error estimate under the regularity assumptions.This chapter is mainly based on the paper:· Yu Fu,JIE YANG,AND WEIDONG ZHAO,Prediction-correction schemes for decoupled forward backward stochastic differential equa-tions with jumps,East Asian J.Appl.Math.6(3),pp.253-277,2016.(SCI)Chapter 5.Applications of FBSDEs in Solving PDEsIn Chapter 5,based on the theories of FBSDEs and PDEs,we study the application of FBSDEs in solving the Dirichlet initial boundary problem and the fractional Laplacian equation.In Part ?,we mainly study the FBSDEs numerical method for solving the Dirichlet initial boundary problem.First,we give a probabilistic representation for this kind of problem,i.e.the solution of the Dirichlet initial boundary problem can be represented by that of FBSDEs with stopping time.Based on the relation,we design an implicit Euler scheme for FBSDEs with stopping time defined on a bounded domain.Then obtain the theoretical error estimate of the scheme,and finally present some numerical experiments.In Part ?,we study the FBSDEs numerical method for solving the frac-tional Laplacian equation.First present the fractional Laplacian equation of interest and its backward stochastic probabilistic representation.By solving FBSDEs with a-stable process,we get the solution of this kind of fractional Laplacian equation.Based on the probabilistic representa-tion,we propose the Euler scheme for the problem and give numerical experiments to verify the effectiveness and accuracy of our scheme.This chapter is mainly based on the papers:· JIE YANG,GUANNAN ZHANG,AND WEIDONG ZHAO,An accu-rate numerical scheme for forward-backward stochastic differential equations in bounded domains,J.Comput.Math.,Accepted,2016.(SCI)· CLAYTON WEBSTER,JIE YANG,GUANNAN ZHANG,AND WEI-DONG ZHAO,A probabilistic scheme using Fourier-Cosine series for fractional Laplacian equations,Finished.Chapter 6.Numerical Simulation for G-Brownian MotionIn Chapter 6,we main consider numerical simulations for the G-Brownian motion defined by Peng[79].By the definition of the G-normal distri-bution,we first show that the G-Brownian motions can be simulated by solving a certain kind of H JB equations.Numerical simulation results of the G-normal distribution,the G-Brownian motion,and the correspond-ing quadratic variation process are provided,which show that our algo-rithms are stable and effective,and would be used in the theory research and applications of SDEs and BSDEs driven by G-Brownian motion.This chapter is mainly based on the paper:· JIE YANG AND WEIDONG ZHAO,Numerical simulations for the G-Brownian motion,Front.Math.China,11(6),pp.1625-1643,2016.(SCI)The main results:Chapter 3:we mainly give rigorously theoretical analysis of the er-ror estimates of the Crank-Nicolson scheme and the multistep scheme for solving FBSDEs.Part I:The Crank-Nicolson schemeConsider the following decoupled FBSDEs:We introduce the Crank-Nicolson scheme for solving decoupled FBSDEs(0.37).Scheme 0.1.Suppose that the initial condition X0 for the forward SDE in(0.37)and the terminal condition ? for the BSDE in(0.37)are given.Now we state our main error estimate results for Scheme 0.1 in Theorems 0.1 and 0.2 below.Theorem 0.1.For the weak order-2 Ito-Taylor approximation Xn+1 satisfying(0.39a),,then under Assumptions 2.1-2.2,for 0 ? n ? N-2,it holds thatwhere C is a generic positive constant depending on d,T,K',and upper bounds of derivatives of b,? and f,the error terms,and,i?n,...,N-2,j = 1,2,are defined in(3.15),(3.19),(3.24)and(3.25),respectively,inSection 3.1.Suppose the data b,?,f and ? satisfy certain regularities,by inequality(0.40)in the above theorem,we obtain the following estimation.Theorem 0.2.Suppose b,?,?Cb3,6,f?Cb3,6,6,6 and ??Cb7+? for some ?(0,1).Then for the weak order-2 Ito-Taylor approximation solution Xn+1,0 ?n?N-2,under Assumptions 2.1-3.1,it holds thatwhere C is a generic positive constant depending on d,T,K',K,L,the initial value of Xt in(0.37),and upper bounds of derivatives of b,a,f and ?.Part ?:The multistep schemeWe pay our attention to a special type of FBSDE:The multistep scheme for(0.42)is as follows.Scheme 0.2.Assume that YN-I and ZN-i,i = 0,1,…,k-1,are known.For n = N-k,1,0,with Xttn,x being the solution of the SDE in(0.42),solve Yn and Zn byAn estimate of eyn in the weak sense is given in the following theorem.Theorem 0.3.Let(Yn,Zn)be the numerical solution at t= tn obtained by Scheme 0.2.Suppose that Assumption 2.4 holds with ?i =?k,i ?t,i=0,1…,k,and f(t,X,Y)is Lipschitz continuous with respect to Y with Lipschitz constant L.Then for,it holds thatwhere C is a positive constant depending only on T,L and k,the error term is defined in(3.92)in Section 3.2.Theorem 0.4.Under certain assumptions,for 0 ? ?t ? |?0|L-1,it holds that where C is a positive constant depending on T,L and k,the error terms and are defined in(3.93)and(3.119),respectively,in Section 3.2.Theorem 0.5.Under certain assumptions, is bounded and the terminal condition satisfieswhere C is a constant depending on T and the given data b,?,? and f.Chapter 4:we propose an explicit scheme—prediction correction scheme for solving FBSDEJs and also analyze the scheme theoretically and numer-ically.We consider the following decoupled forward backward stochastic differential e-quations with jumps(FBSDEJs):To propose the prediction correction scheme,we define the following two stochastic processes:We propose the prediction correction scheme for solving the decoupled FBSDEJs(0.46)as follows.Scheme 0.3.Assume that the initial condition X0 of the forward SDEJ and the terminal condition(YN,ZN,?N)of the BSDEJ in(0.46)are known.Then,for n =N-1,...,0,solve(Yn,Zn,?n)byWe state our main result for the stability of Scheme 0.3 in the following theorem.Theorem 0.6.Let(Xt,Yt,Zu,?t)for t ?[0,T]and(Xn,Yn,Zn,?n)for n =0,...,N-1 be the exact solution of the decoupled FBSDEJs(0.46)and the numerical solution obtained by Scheme 0.3,respectively.Assume that f(t,Xt,Y,Zt,?t)is uniformly Lip-schitz with the Lipschitz constant L.Then for sufficiently small time step ?t,it holds that for n = N-1,...,1,0 where C is a positive constant depending on L and c0,C' is also a positive constant depending on c0,T and L,and the truncation errors and for i= n,...N.are defined in(4.11),(4.14)and(4.15),respectively,the error terms,and Rk=1,2,3,j=1,2,are defined in(4.21)in Section 4.1.Theorem 0.7.Under reasonable assumptions,for sufficiently small time step ?t,it holds that where ?,?,? are defined in Assumption,4.1,C>0 depends only on c0 and L,C1>0 depends on c0,T and L,and C2>0 depends on c0,T,L,X0 and the upper bounds of the derivatives of b,?,c,f and ?.The numerical experiments are shown in §4.1.3.Chapter 5:Study the applications of FBSDEs in solving the Dirichlet initial boundary problem and the fractional Laplacian equation.Part I:The FBSDEs method for the Dirichlet initial boundary problemWe are interested in numerical solution of the Dirichlet initial boundary problem:where T>0 is deterministic,x:=(x1,...,xd)T is a d-dimensional vector,? denotes the gradient operator with respect to x,K(x)is the terminal condition,and ?(t,x)represents a Dirichlet boundary condition,the nonlinear operator L0 is defined byConsider the decoupled FBSDEs with stopping time defined in(?.F,F,P):where the stopping time ?:= inf{t>0,Xt(?)D} is the first exit time of Xt from a domain D(?)Rd for an open piecewise smooth connected setD,and the initial condition X0 is in the domain D,the terminal condition ? of(0.55)is given by The solution u(t,x)of problem(0.53)and the solution of FBSDEs with stopping time(0.55)have the following relation:which is the so-called Feynman-Kac formula[74].If the PDE in(0.53)has a classical solution ,then by the Feynman-Kac formula,for each,the solution of(0.53)can be represented asBased on the relation(0.58),we get the numerical solution of problem(0.53)by solving the FBSDEs with stopping time(0.55).We summarize the Euler scheme for FBSDEs(0.55)as follows.Scheme 0.4(The implicit Euler scheme).Given the terminal condition YN =? and the condition Xn = x,for n = N-1,...,0 and × E D,the approximate solution Xsn,x and the approximate solutions Yn and Zn are obtained by where f~n:= f(tn,x,Yn,Zn).Now we give an error estimate for Scheme 0.4 in the following theorem.Theorem 0.8.Suppose that b,??Cb1,2,f?Cb1,2,2,2 and ??Cb1,2+e for some e?(0,1).Then,for sufficiently small At,it holds thatfor n = N-1,...,0,where C' and C are two positive constants independent of At.For the estimate of the two probabilities P(?tn,x?tn+1)and P(?xn<tn+1)in Theorem 0.8,we have the following theorem.Theorem 0.9.If the starting point x of Xstn,x and Xsn,x satisfiesfor any positive constant ?>0,then,for sufficiently small ?t,it holds that where the constant C>0 is independent of At.The numerical experiments are shown in §5.1.4.Part ?:Solve the fractional Laplacian equationThe fractional Laplacian problem of interest is in the following formwhere u0(x)is the initial condition,the forcing term g(l,x,u)is a function of t,x and n,and(—?)?/2 is the fractional Laplacian operatorwith the constant Cd,a being given by We remark that(0.62)is a nonlocal Cauchy problem defined on the unbounded spatial domain Rd.We give the adjoint equation of fractional Laplacian equation(0.62)as follows.where L*is the adjoint operator,defined asThe solution v,the generator f and the condition function ? of(0.64)and those of(0.62)have the following relationshipIn the rest of this paper,we focus on solving the adjoint fractional Laplacian equation by using the well known probabilistic representation of the solution to(0.64),(see[28,35,68])In(0.67),Xt is a symmetric ?-stable process.We obtain our time semidiscrete scheme to solve v(t,x)backwardly.Scheme 0.5.Assume that the terminal values vN(x)are known.For n = N-1,...,1,0,solve vn(x)byUsing the Fourier cosine expansion again,we obtain our fully discretized approx-imation to compute v(tn,x)as for n = N-1,...,1,0.The Fourier approximation coefficient vkn in(0.69)is obtained by the following procedure.Procedure 0.1(Compute approximation coefficient vkn).Step 1:For n = N,compute the terminal coefficient by a discrete Fourier cosine transform(DCT):Step 2:For n = N-1,...1,0,K=?*Nc with a small number ??(0,1),computeThe numerical experiments are given §5.2.3.Chapter 6:An algorithm for simulating G-Brownian motion is given and the numerical simulation is implemented as well.We introduce the definitions of the G-Brownian motion and the G-normal distri-bution,which are introduced by Peng[79].Definition 0.1(G-Brownian motion).A n-dimensional process{Bt}t?0 in a sublin-ear expectation space(?,H,E)is called a G-Brownian motion if the following properties are satisfied:Definition 0.2.The distribution of a G-normal distributed random vector X is char-acterized byso that ,and u = u(t,x)is the unique viscosity solution of the following nonlinear parabolic PDE defined on[0,?]×R;where In this case,the 1-dimensional G-normal distributed random variable X is denoted by.The quadratic variation process of the 1-dimensional G-Brownian motion{Bt}t?0(see Definition 0.1)is defined as[82]:where In the framework of the G-normal distribution,we are interested in the following quantitywhich can be viewed as the generalization of the classical distribution Fx(a).Based on Definitions 0.1 and 0.2 and(0.72),we provide the following algorithms to simulate the G-normal distribution FX(a)and the corresponding density p(a),the G-Brownian motion,and it.s quadratic variation process {B}t.Algorithm 0.1(Simulation of Fx(a)and p(a)).Algorithm 0.2(Simulation of the G-Brownian motion Bt).Algorithm 0.3(Simulation of the quadratic variation<B>t).The numerical simulation results of the G-normal distribution,the G-Brownian motion and its quadratic variation are presented in § 6.3.