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。拟议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。还应注意有效实施这些模型所需的基础和分析工具。要考虑的随机方程主要是正倒向随机微分方程(简称FBSDE)。FBSDE是一个耦合的随机方程系统，其中某些分量从初始条件随时间向前演化，而其他分量从随机终止条件随时间向后演化。文献中的几个问题已经通过用FBSDE刻画金融模型的方式得到了解决。

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.