正倒向随机微分方程组的数值解法

doi:10.12286/jssx.2015.4.337

1990年, Pardoux和Peng(彭实戈)解决了非线性倒向随机微分方程(backward stochastic differential equation, BSDE)解的存在唯一性问题, 从而建立了正倒向随机微分方程组(forward backward stochastic differential equations, FBSDEs)的理论基础;之后, 正倒向随机微分方程组得到了广泛研究, 并被应用于众多研究领域中, 如随机最优控制、偏微分方程、金融数学、风险度量、非线性期望等.近年来, 正倒向随机微分方程组的数值求解研究获得了越来越多的关注, 本文旨在基于正倒向随机微分方程组的特性, 介绍正倒向随机微分方程组的主要数值求解方法.我们将重点介绍讨论求解FBSDEs的积分离散法和微分近似法, 包括一步法和多步法, 以及相应的数值分析和理论分析结果.微分近似法能构造出求解全耦合FBSDEs的高效高精度并行数值方法, 并且该方法采用最简单的Euler方法求解正向随机微分方程, 极大地简化了问题求解的复杂度.文章最后, 我们尝试提出关于FBSDEs数值求解研究面临的一些亟待解决和具有挑战性的问题.

关键词： 正倒向随机微分方程组, 数值解法, 积分逼近, 微分逼近

In 1990, Pardoux and Peng obtained the existence and uniqueness result of the adapted solution for nonlinear backward stochastic differential equations. This result lays the foundation of the theory of forward backward stochastic differential equations. Since then, FBSDEs have been extensively studied, and have been found applications in many important fields, such as stochastic optimal control, partial differential equations, mathematical finance, risk measure, nonlinear mathematical expectation and so on. In this paper, we will review recent progresses for numerical methods for FBSDEs. We shall mainly introduce the integral and differential based numerical approximation methods, including both one-step and multi-step methods, and the corresponding numerical analysis and theoretical analysis will also be presented. It is worth to note that, by using the differential approximation method, one can propose strongly stable, highly accurate, and highly parallelized methods for solving fully coupled FBSDEs with the forward SDE solved by the Euler scheme. At the end of the paper, we briefly introduce some challenging problems on solving FBSDEs and some possible related applications.

Key words: Forward backward stochastic differential equations, numerical method, integral approximation, differential approximation

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正倒向随机微分方程组的数值解法

Numerical methods for forward backward stochastic differential equations

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