正倒向随机微分方程的两类新的单步方法

华中科技大学硕士学位论文

Abstract

Forward backward stochastic differential equations(FBSDEs)composed of forward stochastic differential equation given initial conditions in probability space and backward s-tochastic differential equation given?nal conditions.This type of equations aimed at reach-ing?nal conditions from the initial conditions with random noisy in the environment.Nat-ural phenomenon can be described more reasonable and approximated with better precision using forward backward stochastic differential equations.It has been applied to multiple ?elds,including?nancial mathematics,game theory and stochastic control,etc.However, an analytical solution is hard to obtained for most of the problems we’re interested.The importance of analysis and construction of numerical methods revealed itself under such circumstances.In this thesis,we present two iterative methods based on one step multi-derivative method and compoundedθ-method.

There are?ve sections in total.In the?rst chapter,we give a brief introduction to the background of theoretical analysis of FBSDEs,numerous applications of FBSDEs, and a survey of current research progress on FBSDEs from domestic to overseas.In the meantime,contribution of this thesis will also be presented.The fundamental results regarding stochastic differential equations and backward differential equations,and integral formula of Gauss-Hermite are included in the second chapter.In the third chapter,novel iterative method based on one step multi-derivative method will be analyzed.Through the comparison between numerical results from linearθ-method and this method,the superiority of this method in precision and effectiveness can be proofed.Another method based on compoundedθ-method is also analyzed in the forth chapter.Numerical results along with theoretical analysis are listed in this chapter.In the last chapter,summarization of this thesis and future work for FBSDEs.Possible directions for further study are also discussed.

Key words:Backward stochastic differential equations Forward backward stochas-tic differential equations Numerical method One step multi-derivative

method Compoundedθ-method

华中科技大学硕士学位论文

摘要.......................................................................I Abstract.....................................................................II 1绪论

1.1研究背景 (1)

1.2国内外研究现状 (2)

1.3本文研究内容与意义 (5)

2预备知识

2.1随机微分方程 (6)

2.2倒向随机微分方程 (10)

2.3Gauss-Hermit求积公式 (12)

3正倒向随机微分方程的单步多导数方法

3.1引言 (13)

3.2倒向随机微分方程的单步多导数方法 (13)

3.3正倒向随机微分方程的单步多导数方法 (21)

3.4数值实验 (25)

4倒向随机微分方程的混合θ-方法

4.1引言 (35)

4.2倒向随机微分方程的混合θ-方法 (35)

4.3局部截断误差分析 (36)

4.4全离散混合θ-方法 (38)

4.5数值实验 (39)

5总结与展望

致谢 (45)

参考文献 (46)

附录1科研项目 (51)