随机递归最优控制及其在金融中的应用研究

本项目主要研究随机递归最优控制理论及其在金融中的应用研究。主要内容分为以下两个部分：第一部分研究随机递归控制问题中推广的验证定理，其特点是生成元中含有扩散项系数，并且生成元关于变量满足单调性条件，同时我们将推广的验证定理运用到具有推广的随机微分效用偏好的投资者的最优投资-消费问题中，给出最优策略的显示解和合理的经济意义。第二部分研究随机递归最优控制系统中生成元关于变量满足单调性条件下的动态规划原理及相应的粘性解理论。本项目的研究内容直接来源于倒向随机偏微分方程和随机控制中富有挑战性的热点问题，具有重要的理论和金融应用意义。

The project is concerned with optimal stochastic recursive optimal control theory and its applications in finance..Part 1 studies a generalized verification theorem, which formulate the Hamilton-Jacobi-Bellman equation that contained the diffusion under a weaker non-Lipschitz condition than what Kraft et al. (2013) considered. As applications, we consider a continuous time intertemporal consumption and portfolio choice problem with a generalized stochastic differential utility (GSDU) preference for an investor. With the generalized verification theorem, we shall show the explicit closed-form optimal consumption and portfolio solutions..Part 2 is concerned with dynamic programming principle and viscosity solutions of the stochastic recursive optimal control problem. When the aggregator is monotonic in the variable, we give the corresponding dynamic programming principle and show that the value function is the viscosity solution to the corresponding Hamilton-Jacobi-Bellman equation with suitable conditions. .The content of this project not only comes from the hot topics in the theory of backward stochastic partial differential equations (BSPDE) and stochastic control problems, but also involves more and more finance applications.