倒向随机微分方程及其应用

The Backward Stochastic Differential Equations and Its Application

本文将介绍一类新的方程：倒向随机微分方程．为便于理解，我们将首先通过与常微分方程和经典的随机微分方程（Ｉｔ．ｏ方程）的对比．并通过数理经济和数学金融学中的一个典型的例子来引入倒向随机微分方程．然后给出解的存在唯一性定理和比较定理．并介绍非线性Ｆｅｙｎｍａｎ－Ｋａｃ公式，它给出了倒向随机微分方程的解与一大类常见的非线性偏微分方程（组）的解之间的对应关系，从而为将来利用Ｍｏｎｔé－Ｃａｒｌｏ型的随机计算方法计算大量的偏微分方程开辟了新的途径．

This survey presents a new type of equations: backward stochastic differential equations (BSDE). It points out the essetial differences between the classical notions of ordinary differential equations, Itós (forward) stochastic differential equations and that of BSDE. An existence and uniqueness of BSDE is given. This paper also presents the comparison theorem, nonlinear Feynman Kac formula which gives one to one correspondence between a large kind of solutions of (systems of) nonlinear partial differential equations and those of BSDE. As a remarkable example in applications, the relations of BSDE and mathematical finance are emphasized.