Stochastic dynamics in financial modeling

在对金融变量进行数学建模时，重要的是要考虑导致易于处理的解决方案的动态，同时捕捉经验数据的定性特征。提出的研究计划包括研究各种随机方程，它们的性质，以及它们在金融建模中的应用。我们将特别关注各种期限结构衍生品、商品合约、违约（信用）风险和保险产品的定价和对冲。还应注意有效实施模型所需的基础和分析工具。要考虑的随机方程的主要类别是前向倒向随机微分方程（简称FBSDEs）。FBSDE是一个随机方程的耦合系统，其中某些分量从一个初始条件在时间上向前发展，而其他分量从一个随机的终端条件在时间上向后发展。通过用FBSDEs来描述金融模型，已经解决了文献中的几个问题。

In modeling financial variables mathematically it is important to consider dynamics which lead to tractable solutions while at the same time capture qualitative features of empirical data. The proposed research program involves the study of various stochastic equations, their properties, and their application to modeling in finance. In particular we shall focus on the pricing and hedging of various term-structure derivatives, commodity contracts, default (credit) risk, and insurance products. Attention shall also be given to the foundations and analytic tools necessary to implement the models effectively. The main class of stochastic equations to be considered are forward-backward stochastic differential equations (FBSDEs, for short). A FBSDE is a coupled system of stochastic equations of which certain components evolve forward in time from an initial condition and other components evolve backward in time from a random terminal condition. Several problems in the literature have been solved by characterizing financial models in terms of FBSDEs.