金融数学中与风险控制有关的自由边界问题

本项目以金融公司的投资理财问题为背景，抽象出一类带有扩散项控制的随机最优控制模型，并利用动态规划原理将之转化为偏微分方程或相应的变分不等式的自由边界问题。自由边界为线性（或拟线性）方程和完全非线性方程对应区域的交界线。本项目的研究成果不仅可为金融公司的决策提供定性定量的依据，同时也将极大地丰富完全非线性方程的自由边界理论，创造出许多解决此类问题的一般方法。

In this project, a stochastic optimal control model with diffusion control is abstracted from the background of assets management problems for financial companies, and it is transformed into a free boundary problem of partial differential equations or corresponding variational inequalities by using the principle of dynamic programming. The free boundary is the boundary between the corresponding region of linear (or quasilinear) equation and fully nonlinear equation. The research results in this project can not only provide qualitative and quantitative basis for financial companies' decision-making, but also greatly enrich the free boundary theory of fully nonlinear equations and create many general methods to solve such problems.