【作者】 胡盈；

【导师】 胡良剑；

【作者基本信息】 东华大学 ， 应用数学， 2005， 硕士

【摘要】 近年来,倒向随机微分方程的理论有了很快的发展,并且在金融中有着越来越广泛的应用。与此同时,倒向随机微分方程数值方法的研究进展相对滞后。由于有相当多的倒向随机微分方程不能解析求解,因此对其数值方法的研究具有重要的理论和应用意义。 本文使用离散流的方法,提出了一些求解具有路径依赖终值条件的倒向随机微分方程的数值方法,阐述了这些数值方法之间的区别和联系,并且通过对金融模型的求解,揭示了这些数值方法隐含的金融意义以及它们与金融中某些原有的随机计算方法的联系。基于Ma等(见"Ma J, Protter P, San Martin J, and Torres S. Numerical Method for Backward Stochastic Differential Equations[J]. Ann. Appl. Probab. 2002, 12: 302-316.")采用左节点差分格式的数值方法,本文提出了采用右节点差分格式的数值方法,以及同时采用左右节点差分格式的数值方法。通过对几种数值方法误差来源的分析,指出右节点差分格式的数值方法可以避免近似求解隐式差分方程而产生的误差。同时,为了使数值方法能够处理金融模型,本文的数值方法采用了更一般的随机游动对更多还原

【Abstract】 In recent years, the theory of Backward Stochastic Differential Equations (BSDEs) has been developed rapidly and is applied to finance extensively. At the same time, the research on numerical methods of BSDEs lags behind. Since the solutions of BSDEs can rarely be solved analytically, the numerical methods for BSDEs are indispensable and significant in theory and applications.This master’s thesis proposes some new numerical methods for BSDEs with path-dependent terminal values, via discretization of the filtration. Furthermore, the difference and relationship among these numerical methods are investigated. By solving some financial models, the underlying financial meanings of these numerical methods are revealed and the relationship between the numerical methods and some known stochastic simulation methods in finance is exposed.Based on the idea of Ma et al (in "Ma J, Protter P., Martin J S,and Torres S. Numerical Method for Backward Stochastic Differential Equations [J]. Ann. Appl. Probab. 2002, 12: 302-316."), which introduces a numerical method making use of the left-node discrete version of BSDE, this thesis proposes two numerical methods by using different discrete versions with the right-node and the left-and-right-nodes, respectively. After discussing the error sources of these methods, we claim that the right-node discrete version may avoid the errors of solving implicit difference equation approximately. Meanwhile, we utilize a more general random walk for discretization of Brownian motion to fulfill the requirement of financial applications.In light of its successful applications in option pricing, the trinomial model is employed in the thesis for approximation of Brownian motion to accelerate the convergence rates of the above numerical methods of BSDEs. In some difference schemes, the trinomial approximation will produce the same consequences as the binomial approximation, by using merely half nodes of the latter. The result implies that the trinomial approximation may be faster than the binomial approximation. Then, we introduce some more general numerical methods by use of polynomial approximations.Furthermore, the proposed numerical methods are applied to discuss financial models. By pricing contingent claims in completemarket, the numerical methods are compared with no-arbitrage equilibrium analysis. In this case, the different discrete versions of these numerical methods imply different discount factors. By applying these methods to price Europe option, we demonstrate the equality of the numerical methods and the conventional binomial or trinomial method for option pricing. Finally, the proposed methods and conclusions are illustrated by numerical examples. In addition, these methods are applied to price some path-dependent options, such as barrier options.更多还原