马尔科夫体制转换金融保险模型中的随机控制问题研究

本项目将在青年基金项目关于马尔科夫体制转换金融保险模型研究的基础上，拟利用随机过程分析理论及随机控制理论深入拓展研究马尔科夫体制转换金融保险模型下的相关随机控制问题。我们首先将通过动态规划原理、随机极大值原理、等价鞅测度变换理论、随机控制对偶理论解决马尔科夫体制转换金融保险模型中的期望终端财富和消费效用最大化。其次,我们还将利用倒向随机微分程理论研究带跳和无跳马尔科夫体制转换金融市场下的递归效用随机最优控制问题。再者，我们将从以下三个方面研究马尔科夫体制转换模型下的最优投资组合问题：1.均值-方差投资组合问题；2.有违约证券资产的马尔科夫体制转换跳扩散金融市场模型下的投资组合问题；3.以Roy提出的安全第一准则方法。该项目研究内容是金融和保险中随机控制问题的最新课题，是倒向随机微分方程、随机控制以及数理金融领域的交叉研究，本项目的研究将促进随机最优控制，数理金融与精算等其他理论的发展。

This project is based upon NFSC research project: Ruin and stochastic control problems in the Markov regime-swtiching finance and insurance models. We aim at a deeper investigation of Markov regime-switching financial and risk models through the accumulation of stochastic process, stochastic analysis and stochastic control theory. As it to the problems of expected utility maximization from terminal wealth and/or consumption, we apply techniques such as dynamic programming, stochastic maximum principle, measure transform and duality theory. Secondly, on the basis of Backward Stochastic Differential Equation theory, we are concerned with stochastic recursive utility portfolio optimization problem in the context of both Markov reigme-switching diffusion models and jump-diffusion models. Further, we study the optimal portfolio selection problem from three aspects：1. Mean-variance portfolio selection problem; 2.Portfolio selection problems in Markov regime-switching jump-diffusion market model with defaultable securities; 3. Roy’s Safety-First principle. The project which focus on many an innovative topic is committed to integrate the research of stochastic analysis, backward stochastic differential equations, stochastic control and mathematical finance, simultaneously, to promote the development of stochastic optimal control and finance-insurance field.