Stochastic optimal control problems in risk management

随机控制理论起源于不确定条件下的决策问题，它研究如何根据预定的性能指标对随机动力系统进行优化。基于Bellman最优性原理，在(连续时间)马尔可夫框架下的标准随机控制问题可以通过遵循动态规划原理来求解，从而得到了Hamilton-Jacobi-Bellman方程。随机控制作为一种优雅而强大的技术，在经济、金融、保险、风险管理等领域有着广泛的应用。这些领域的最新发展给研究人员发展理论研究和数值计算的新方法带来了许多新的挑战和机遇。这项研究的主要目的是以随机控制理论为主要技术手段，研究保险学和经济学中的一些关键问题。具体地说，我们将考虑(1)年化之谜；(2)保险市场中的逆向选择和优势选择；(3)可变年金的最优收费结构；(4)极端风险缓解；(5)一般保险的均衡定价。预计这项拟议的研究可以改善风险管理，并为从业人员和学术研究人员提供关键的见解。在这项研究拨款的过程中，预计将有大量的论文发表在相关领域的主要期刊上。本科生、研究生和博士后研究员将在这个拟议的研究计划中进行密集的参与和培训。

Stochastic control theory arises from decision-making problems under uncertainty; it studies how to optimize stochastic dynamical systems according to some predetermined performance criterion. Based on Bellman's principle of optimality, standard stochastic control problems in the (continuous-time) Markovian framework can be solved by following the dynamic programming principle, which leads to the Hamilton-Jacobi-Bellman equations. As an elegant and powerful technique, stochastic control has found numerous applications in economics, finance, insurance, and risk management. Recent development in these areas have brought many new challenges and opportunities to researchers to develop new methodologies for theoretical studies and numerical computation. The primary objective of this proposed research is to study some crucial problems in insurance and economics by utilizing stochastic control theory as the main technique. Specifically, we will consider (1) Annuitization puzzle; (2) Adverse selection and advantage selection in insurance markets; (3) Optimal fee structures of variable annuities; (4) Extreme risk mitigation; (5) Equilibrium pricing of general insurance. It is anticipated that this proposed research can improve risk management and provide key insights to practitioners and academic researchers. A significant number of papers are expected to be published in leading journals of related fields over the course of this research grant. Undergraduate students, graduate students, and postdoctoral fellows will be intensively involved and trained in this proposed research program.