Application Of Localized Radial Basis Function Method And Low-discrepancy Sequences In Solving BSDEs

Backward stochastic differential equations(BSDEs)have both theoretical and practical importance in many fields.However,there are few BSDEs that can be solved explicitly.Therefore,efficient numerical methods for backward stochastic differential equations is very important.Through unremitting efforts,now we have been able to solve nonlinear backward stochastic differential equations with high-precision.According to properties of BSDEs and Ito integral,0 scheme of integral discrete method for backward stochastic differential equations[46]could solve BSDEs efficiently.In practical applications,we tend to encounter multidimensional backward stochastic differential equations.However,because of the complexity of multi-dimensional problems,many methods that numerically solve high-dimensional BSDEs will become very complex,making calculations very time-consuming or even impossible to calculate.Based on ? scheme,this thesis aims at solving multidimensional backward stochastic differential equations efficiently.Low-discrepancy sequences are often used as integration points in quasi-Monte Carlo methods.According to discrepancy theory of point set,low-discrepancy point set is more evenly distributed in the space than random point set and reg-ular lattices which are generated by the tensor product.As we know,existence and be uniqueness for polynomial function interpolation can not be guaranteed in a multidimensional space.Meanwhile,globally supported radial basis function is conditional positive definite and its shape parameters is hard to determine.To use the data of low-discrepancy point set,we introduce compactly supported radial basis functions.According to its construction method theorem,we have positive coefficient matrix for interpolation when compactly supported radial ba-sis functions are used.In the first chapter,we review the literature of backward stochastic differential equations and their numerical methods,low-discrepancy sequences and radial ba-sis functions.The second chapter briefly introduces the relevant conclusions of Ito integral and properties of backward stochastic differential equations.The third chapter analyzes the theoretical structure and theorems of ? scheme for BSDEs,and specifies skills to use this scheme,such as the approximate of conditional expectation.In the fourth chapter,we first introduce discrepancy theory of point set and propose to use low-discrepancy point sets as grids rather than regular lattice.We then talk about compactly supported positive definite radial basis functions and explain the necessity and importance.At the last of chapter 4,we combine low-discrepancy point set and localized radial basis function method and use the method for interpolation in a series of numerical experiments.The fifth chapter demonstrates results of numerically solving BSDEs by the method we propose.Finally,we concludes in the last chapter.Referring previous research,the highlights of this thesis are mainly the follow-ing aspects:First,Base on the evenly-distributed property of low-discrepancy point sets,we use low-discrepancy point sets as grid points rather than integration points.In multidimensional space,compared with regular lattices which are usually used,low-discrepancy point sets could reduce the number of grid points to some extent.Second,we use compactly supported positive definite radial basis functions for interpolation in multidimensional space.Considering the data structure when solving BSDEs,we further introduce the localization method and use nearby grid points for estimation.In this way,we can reduce the computation cost while maintaining a certain degree of accuracy.By a series of numerical experiments,this thesis discusses the detail of localized radial basis function method base on low-discrepancy sequence.Compared with results of cubic spline interpolation,we know how to interpolate well by radial basis functions.As for error control,we put forward that we ought to make interpolation tests on terminal condition to determine the number of grid points in space.Numerical experiments in this thesis show that based on low-discrepancy point sets,localized radial basis function method can effectively solve backward stochastic differential equations by ? scheme of BSDEs.