Multilevel Monte Carlo Method For Backward Stochastic Differential Equations With Uncertainty

Among the many kinds of improvements about Monte Carlo method,mul-tilevel Monte Carlo method was proposed by Stefan Heinrich[i and Michael Giles[2].The time or space division problems which can be solved by Monte Carlo method could also be solved by multilevel Monte Carlo method in low computation cost while keeping the order of error.In 2006,Weidong Zhao,Lifeng Chen and Shige Peng[3]gave a way to get the numerical solution of backward stochastic differential equations in the following form:This numerical method improves the convergence rate of error.Considering backward stochastic differential equations with uncertainty,there is a random variable ω independent with Brownian motion in the back-ward stochastic differential equations:When t = 0,WO = 0,y0(ω)should be a random variable.We want to know the expectation of y0(ω),that is,E[y0(ω)],when t =0.Normally,we can use Monte Carlo method to get the approximation of expectation E[y0(ω)],and in this thesis we will use multilevel Monte Carlo method instead to reduce the computation cost.Numerical experiment results show that this improvement can reduce the computation cost while maintaining the order of error.Because of the advan-tages of low computation cost,this improved method,the multilevel Monte Carlo method for backward stochastic differential equations with uncertainty should be of practical value in the field of financial engineering.This thesis is divided into six chapters.Chapter I is the introduction,and in this part we will roughly introduce the methods we reference.In Chap-ter Ⅱ,we will introduce Monte Carlo method in detail.Chapter Ⅲ is about the multilevel Monte Carlo method.Chapter IV is about the numerical solu-tion of backward stochastic differential equations,we will focus on how to get numerical solution of backward stochastic differential equations by θ-scheme.In Chapter V we will propose multilevel Monte Carlo method for backward stochastic differential equations with uncertainty,then try to complete the analysis of error and computation cost.Chapter VI is the results of numerical experiment and the conclusion.