Study On Numerical Methods For FBSDEs Based On Stochastic Characteristic Approximation

In 1990,Pardoux and Peng[49]put forward the problem of nonlinear backward stochastic differential equations(BSDEs)and originally proved the existence and u-niqueness of the solution to nonlinear BSDEs.Then,in 1991,Peng[52]gave the nonlinear Feynman-Kac formula,and established the relationship between BSDEs and quasilinear partial differential equations(PDEs).After that,a series of excellent the-ories about forward backward stochastic differential equations(FBSDEs)have sprung up continuously,promoting the expansion of human cognition and the development of science and technology.Now,the relevant theoretical achievements of FBSDEs have played a pivotal role in the fields of finance,biomedicine,physics,mechanical engineer-ing and so on.However,under normal conditions,the analytical solution of FBSDEs has a complex solution structure,which is difficult to obtain directly,so the study of numerical schemes of FBSDEs has full necessity and application value.This paper mainly studies the numerical solution of FBSDEs based on stochastic characteristic approximation methods,and the main research contents are:First of all,based on the property of the solution of forward backward stochastic differential equations(FBS-DEs)and the extrapolation method,we creatively put forward the explicit high-order multistep scheme for solving FBSDEs,and under the condition of the stability of the method,the error estimations are given to verify the convergence rate of the scheme.And the comparison of running time confirms that the explicit iterative scheme has better computation efficiency than the old one;Then,based on the explicit multistep scheme,we originally proposed the stochastic characteristic approximation method.By using the characteristics of the diffusion process,the scheme could trace the pathes of the direction of the diffusion process to get the interpolation values.Then we gave the numerical tests to draw the conclusion that,compared with the traditional ap-proximate method,the stochastic characteristic approximation approximation method has higher accuracy and is more efficient;Then,based on the stochastic characteristic approximation method,the finite difference scheme for solving FBSDEs is proposed,which simplifies the scheme structure and further enhances the accuracy of numerical results.The main contribution and innovation of the paperThe main contribution and innovation points of this paper are as follows.(1)Based on the traditional high-order multistep schemes,the author replaces the numerical iteration method by the extrapolation method on the time levels.We put forward the high-order multistep schemes to solve FBSDEs explicitly.Based on the stability,the error estimate is given,and numerical results verify that the new scheme has higher precision.Compared with the old methods,the explicit multistep scheme can greatly shorten the running time and get the higher efficiency.(2)Based on the explicit high-order multistep scheme and the characteristics of diffusion process,a high-order scheme based on stochastic characteristic approx-imation method is proposed.This method can trace the real pathes of diffusion process effectively,which makes the interpolation results more consistent with the actual situation.And it improves the accuracy of the numerical results effectively,and can better simulate the actual motion of FBSDEs.(3)On the basis of the stochastic characteristic approximation methods,the relation between Yt and Zt can be obtained by using Feynman-Kac formula.Thus the difference method can be used to replace the differential part to calculate,which simplifies the scheme steps,improves the error accuracy,and makes the calculation result more accurate and efficient.The framework of the paperThis paper has six chapters.Chapter 1 IntroductionThis chapter briefly introduces the research background,motivation and devel-opment status of the numerical methods for solving FBSDEs,and sorts out the context and outline of the fall text.Chapter 2 PreliminaryThis chapter introduces the related basic knowledge of the forward backward s-tochastic differential equation,and introduces the differential multistep method for solving FBSDEs.At the same time,it gives the basic theory of stochastic characteristics approximation methods,which lays the foundation of stochastic characteristic approximation methods in the following paper.Then,a brief intro-duction of finite difference approximation is given,which can enrich and improve the theoretical system of FBSDEs numerical solution.Chapter 3 A Fully Discrete Explicit Multistep Scheme for Solving Coupled FBS-DEsThis chapter,the main researchs are the explicit high-order multistep scheme for solving forward backward stochastic differential equations.First of all,replacing the twice iterations by using Lagrange extrapolation method,we use the values of other time levels to interpolate the values on the time level which we need.Then we give error estimates and list some numerical examples to verify the high efficiency and high precision of the scheme.And we learn that the result can greatly shorten the running time and improve the efficiency.This chapter is mainly from the paper:·Y.Liu,Y.Sun,and W.Zhao,A fully discrete explicit multistep scheme for solving coupled forward backward stochastic differential equations,Adv.Appl.Math.Mech.,12(2020),pp.643-663.(SCI,Q1)Chapter 4 Explicit Multistep Stochastic Characteristic Approximation Methods for Solving FBSDEsThis chapter mainly studies the numerical method for solving forward backward stochastic differential equations based on stochastic characteristic approximation methods.Firstly,based on the property of the diffusion process,we design a scheme to track the pathes of the diffusion process.And then use the points in the direction of these pathes to interpolate the values of numerical solution.Then,numerical examples are given to verify the high precision and accuracy of the stochastic characteristic approximation methods.This chapter is mainly from the paper:·Y.Liu,Y.Sun,and W.Zhao,Explicit Multistep Stochastic Characteristic Approximation Methods for Solving Forward Backward Stochastic Differ-ential Equations,Discret.Contin.Dyn.Syst.-Ser.S,doi:10.3934/d-cdss.2021044.(SCI,Q2).Chapter 5 The Differences Between Stochastic Characteristic Approximation Meth-ods and Traditional Multistep MethodsThe advantages and disadvantages between the stochastic characteristic approx-imation methods and the traditional differential multistep method for solving FBSDEs,this chapter will focus on accuracy,convergence rate,running time.Through the figures,we could have a full perspective to know the actual pro-cess.This chapter analyzes the limitations of traditional differential multistep schemes,and verify the superiority of the proposed new scheme.Chapter 6 Stochastic Difference Approximation Methods for Solving FBSDEsMainly studies in this chapter are the stochastic characteristic difference meth-ods for solving forward backward stochastic differential equations.In order to further improve the numerical results,based on the stochastic characteristic ap-proximation methods,we put forward the finite difference approximation method.This method can instead of the differential approximation,and then we propose the stochastic characteristic difference methods.The new schemes have clearer structure and the operation step of the new scheme is more concise.Then the nu-merical example is given to verify that the new scheme can improve the accuracy,also has good computing results.This chapter is mainly from the paper:·Y.Liu,Y.Sun,and W.Zhao,Stochastic Characteristic Difference Approx-imation Methods for Solving Forward Backward Stochastic Differential E-quations,Finished.The main results of this paperIn Chapter 3,we propose a fully discrete explicit multistep scheme for solving coupled FBSDEs,and then test the high accuracy and high efficiency of the scheme.Let(?,F,F,P)be a filtered complete probability space with the natural filtration F={Ft}0?t?T,generated by the m-dimensional standard Brownian motion W=(Wt)0?t?T.We are concerned with the numerical solutions of fully coupled FBSDEs on(?,F,F,P)in the form(?)(0.3)for t ?(0,T],where T>0 is the deterministic terminal time,X0 ?F0 is the initial condition of SDE,b:[0,T]×Rd×Rp×Rp×m?and ?:[0,T]×Rd×Rp×m?Rd×m?Rd×m are the drift and diffusion coefficients,respectively,?:Rd?Rp is the terminal function of XT for BSDE,and f:[0,T]×Rd×Rp×Rp×m?Rp is the generator of the BSDE.The two stochastic integrals in(0.3)with respect to Ws are of the Ito type.For equation(0.3),if we want to get the time semidiscrete scheme,we need to discrete the time interval.For the time interval[0,T],we introduce a regular time partition as 0=t0<t1<t2<…<tN=T.For simple representation,we denote tn+k-tn by ?tn,k and Wtn+k-Wtn by ?Wn,k,and use the notations ?ttn,t=t-tn,and ?Wtn,t=Wt-Wtn for t?tn.Without losing generality,in the sequel,the uniform time partition ti=i?t(i=0,1,…,N)with ?t=T/N is used.Let Xn,Yn and Zn be the numerical approximations for the solutions Xt,Yt and Zt of the fully coupled FBSDEs(0.3)at time tn,respectively.And we get the following time semidiscrete numerical scheme for the fully coupled FBSDEs(0.3).?? 0.7.Let K=max{k,l},and assume YN-i and ZN-i,i=0,…,K-1,are known.For n=N-K,…,0,j=1,…,k,solve Xn,j,Yn=Yn(Xn)and Zn=Zn(Xn)by(?)where Yn+j stand for the values of Yn+j at the space points Xn,j,and YLn=YLn(Xn)and ZLn=ZLn(Xn)are defined as#12 with Ll,n,i(tn)given as#12Then,in order to get the fully discrete explicit scheme for solving the coupled FBSDEs(0.3),we need to have the following the time-space partition.For the space Rd we introduce a space partition Dhn on each time level t=tn with parameter hn?0.The space partition Dhn is a set of discrete grid points in Rd,i.e.Dhn={xi|xi?Rd}.We define the density hn of the girds in Dhn by#12 where dist(x,Dhn)is the distance from x to Dhn.For each x ?Rd,we define a local subset Dh,xn of Dhn satisfying·dist(x,Dh,xn)<dist(x,Dhn\Dh,xn);·the number of elements in Dh,xn is finite and uniformly bounded,that is,there exists a positive integer Ne such that#Dh,xn?Ne.We call Dh,xn the neighbor grid set in Dhn at x.Then,we propose the following fully discrete explicit scheme for solving the fully coupled FBSDEs(0.3).?? 0.8.Let K=max{k,l},assume that YN-i and ZN-i,i=0,…,K-1,are known.For n=N-K,…,0,j=1,…,k,and for x?Dhn,solve Xn,j,Yn=Yn(x)and Zn=Zn(x)by(?)where YLn=YLn(x)and ZLn?ZLn(x)are defined as#12 with Ll,n,i(tn)given in(0.7).The scheme 0.8 creatively uses Lagrange interpolation method instead of the im-plicit iterative methods,which makes the implicit scheme change into the explicit scheme,and the structure of the scheme is simple and intuitive.At the same time,using the points on other time levels to interpolate the approximate points on the time layer can greatly improve the efficiency of the solution,shorten the running time and improve the accuracy.In Chapter 4,we propose an explicit multistep stochastic characteristic approximation method for solving FBSDEs,and then test the high accuracy of the scheme.We propose the following semidiscrete explicit multistep scheme for solving the fully coupled FBSDEs(0.3).??0.9.Let K=max{k,l},and assume YN-i and ZN-i,i=0,…,K-1,are known.Then for n=N-K,…,0 and j=1,…,k,we solve Xn+j,Xn,j,Yn=Yn(Xn)and Zn=Zn(Xn)by#12 where Yn+j stand for the values of Yn+j at the space points Xn,j,and(?)with Yn+i=Yn+i(Xn+i)and Zn+i=Zn+i(Xn+i).Then,we propose the following time-space fully discrete explicit scheme for solving the fully coupled FBSDEs(0.3)?? 0.10.Let K=max{k,l},and assume YN-i and ZN-i,i=0,…,K-1,are known.Then for n=N-K,…,0,j=1,…,k,and for x ? Dhn,we solve Xn+j,Xn,j,Yn=Yn(x)and Zn=Zn(x)by#12 where and#12 with Yn+i=Yn+i(Xn+i),Zn+i=Zn+i(Xn+i).In scheme 0.10,the explicit multistep stochastic characteristic approximation meth-ods can solve Yt=Y(t,Xt)and Zt=Z(t,Xt)in b,? and f.According to the property of the diffusion process Xt,when the time point changes,Xt will have a rapid and obvious change.Therefore,in order to improve the accuracy of the prediction,we choose the points of Xt,which is after drifting and diffusing,to approximate the values of Yt,Zt,that is,the space points traces the pathes of Xt.Therefore,the stochastic characteristic approximation method is introduced to obtain the numerical solution of FBSDEs with high precision according to the properties of diffusion process.?0.3.As for the explicit multi-step scheme of solving FBSDEs based on stochastic characteristic approximation methods,we give a detailed introduction in the fifth chap-ter.This chapter mainly elaborates the limitations of the traditional implicit multistep method,analyzes and explores the advantages of the stochastic characteristic approxi-mation methods.In Chapter 6,we propose a stochastic difference approximation method for solving FBSDEs,and then test the high efficiency of the scheme.We propose the following semidiscrete explicit scheme for solving the fully coupled FBSDEs(0.3).?? 0.11.Let K=max{k,l},and and ZN-i,j=0,…,K-1,are known.Then for n=N-K,…,0 and j=1,…,k,we solve Xn+j,Xn,j,Yn=Yn(Xn)and Zn=Zn(Xn)by#12 where Yn+j stand for the values of Yn+j at the space points Xn,j,and#12 with Yn+i=Yn+i(Xn+i)and Zn+i=Zn+i(Xn+i).Then,we propose the following fully discrete explicit scheme for solving the fully coupled FBSDEs(0.3)?? 0.12.Let K=max{k,l},and assume YN-i and ZN-i,i=0,…,K-1,are known.Then for n=N-K,…,0 and j=1,...,k,we solve Xn+j,Xn,j,Yn=Yn(Xn)and Zn=Zn(Xn)by#12 where#12 with Yn+i=Yn+i(Xn+i)and Zn+i=Zn+i(Xn+i).In this chapter,based on the stochastic difference approximation method to solve FBSDES and the relation between Yt and Zt is deduced from Feynman-Kac formula.When the finite difference method is used to replace the differential approximation method,the scheme can be simplified.And the efficiency can be improved.Therefore,the stochastic difference approximation method can further improve the overall effect of calculation.?0.4.To sum up,the explicit multistep stochastic characteristic approximation meth-ods for solving FBSDEs(0.3)and its related theories are put forward in this paper,which has a vital originality and practical values.The numerical experiments show that the stochastic characteristic approximation method has a great prospect in the numeri-cal computation theory for solving forward backward stochastic differential equations.Based on the stochastic characteristic approximation method,we can further improve the efficiency and accuracy of solutions of FBSDEs?BSDEs?PDEs and other relat-ed equations,which is conducive to our continuous research in the field of stochastic computing.