Stochastic optimal control problems in risk management

随机控制理论产生于不确定条件下的决策问题；它研究如何根据预定的性能准则对随机动力系统进行优化。基于Bellman最优性原理，采用动态规划原理求解（连续时间）马尔可夫框架下的标准随机控制问题，得到Hamilton-Jacobi-Bellman方程。作为一种优雅而强大的技术，随机控制在经济、金融、保险和风险管理中有着广泛的应用。这些领域的最新发展给研究人员带来了许多新的挑战和机遇，以发展新的理论研究和数值计算方法。本研究的主要目的是利用随机控制理论作为主要技术来研究保险和经济学中的一些关键问题。具体来说，我们将考虑(1)年金化难题；(2)保险市场的逆向选择与优势选择；(3)可变年金的最优收费结构；(4)极端风险缓解；(5)一般保险均衡定价。预计本研究可以改善风险管理，并为从业者和学术研究人员提供关键见解。在这项研究资助的过程中，预计将在相关领域的主要期刊上发表大量论文。本科生、研究生和博士后将集中参与和培训这个拟议的研究项目。

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.