SINC Methods For Backward Stochastic Differential Equations

In 1990,the existence and uniqueness of the adapted solution of nonlinear backward stochastic differential equations(BSDEs)were first proved by Pardoux and Peng[1].This result lays the foundation for the theoretical study of back-ward stochastic differential equations.Since then,BSDEs have been extensively studied by many researchers.It has been found that backward stochastic dif-ferential equations have important applications in many fields,such as financial mathematics,stochastic control,biology,financial futures market,risk measure,partial differential equation theory and stochastic games,and so on.However,it is often quite difficult to obtain the analytic solutions of backward stochastic differential equations.Therefore,the study on on numerical solutions of BSDEs is of great,significance to BSDEs theory and their applications in practice.The Sine method is a numerical method with exponential order accuracy,which has high precision in interpolation approximation,differential approximation and integral approximation.Recently,the application results of the Sine method show the distinct advantages of the exponential order convergence on the integral d-ifferential approximation and the interpolation approximation.For this reason,we study the applications of the Sine method in solving backward stochastic differential equations and explore its advantages.The main innovation of this thesis is to solve the backward stochastic differ-ential equations by using the Sine method.When solving backward stochastic differential equations,we propose using the Sine integral formula to approximate theconditional expectation to solve the backward stochastic differential equation and propose a Sine full-discrete scheme.By selecting the time space step size and the parameters in the Sine integral reasonably,a Sine full-discrete scheme that does not require space interpolation is proposed.In addition,we obtain the theoretical error estimates of the two Sine full-discrete scheme correspondingly.Finally,some numerical tests are presented to demonstrate our theoretical re-sults,and the numerical tests results are consistent with the theoretical analysis results.We list the main results of this article below:Chapter 1:we make a brief introduction of the background,the research status and development of our topic in the following chapters.Chapter 2:we introduce some basic knowledge of stochastic analysis and back-ward stochastic differential equations,including probability space,conditional expectation,Ito integral and Feynman-Kac formula.Chapter 3:we introduce the Sine method,including the Sine function approx-imation,the integral approximation and give some numerical examples.Chapter 4:we use the Sine integral formula to approximate the condition mathematical expectation and propose a full-discrete scheme to solve backward stochastic differential equations.Scheme 0.1.Suppose that the terminal condition(yiN,ZiN)are given,for n=N-1,…,0,we solve yin and zin by the following equations:Here Etnxi[·]means using Sine integral to approximate conditional mathematical expectation Etnxi[·];yn+1 and zn+1 are corresponding interpolation at the point xi+Wtn+1-Wtn(i?Z)by using the grid values of yln+1 and zln+1,l?Z,respectively.Then,by reasonable selection of parameters,let the parameter h in the Sine integral formula,the time step and space step satisfy the condition of h(?)=?x.At this time,the points used in calculating the condition mathematical expectation fall on the grid points of the upper layer,and the calculation of condition mathematical expectation does not require interpolation,and the Sine full-discrete scheme without space interpolation is given under this parameter selection.Scheme 0.2.Suppose that the terminal condition(yiN,ziN)are given,under the condition of h·(?)=?x,for n=N-1,…,0,we solve yin and zin by the following equations:Here Etnxi[·]means using Sine integral to approximate conditional mathematical expectation Etnxi[·]under the condition of h·(?)=?x;yn+1 and n+1 are the values at the grid point xi+k(k=-M,…,M)respectively.Chapter 5:we give rigorously theoretical analysis of the error estimates of the two Sine full-discrete scheme proposed in Chapter 4,and the error estimate for the Sine full-discrete scheme of the backward stochastic differential equations in which the generator is independent of z is given by the following theorem.Theorem 0.1.Let(yt,zt)be the analytic solution of the BSDE(5.1)and(yin,zin)the solution of the fully-discrete scheme 0.1 with the linear polynomial interpola-tion used to calculate yln+1 and zn+1.Then it holds for the Sine fully-discrete scheme with general ??[0,1].In particular,we have that for the Sine fully-discrete scheme with ?=1/2,and for the Sine fully-discrete scheme with ?=1.Theorem 0.2.Let(yt,zt)be the analytic solution of the BSDE(5.1)and(Yin,zin)the solution of the fully-discrete scheme 0.2 under the condition of h·(?)=?x.Then it holds for the Sine fully-discrete scheme with general ??[0,1].In particular,we have that for the Sine fully-discrete scheme with ?=1/2,and for the Sine fully-discrete scheme with ?= 1.Chapter 6:We consider the accuracy of the Sine method for solving BSDEs.Some numerical tests are presented to demonstrate our theoretical results,and the numerical tests results are consistent with the theoretical analysis results.Chapter 7:We summarize the full text and look forward to it.