Markov状态转换下的跳扩散风险理论的新模型与新算法

Jump-diffusion risk models with Markovian switching are the new emerging models which have become of great importance to the finance and insurance industry. The main purpose of this project is to establish some new kinds of jump-diffusion risk models with Markovian regime switching, and investigate the numerical solutions of stochastic differential equations (SDEs) with jump and Markovian switching which used to describe the corresponding new risk models. The main focuses in this project include:.(1) Researches on the ruin probability and Gerber-Shiu discounted penalty function in jump-diffusion risk model with Markovian switching for whether or not possessing linear dividend barrier. Investigating their properties respectively and giving feasible simulating analysis for some special cases..(2) Discussing the optimal portfolio selection problem for an investor who must invest the surplus into a Markovian-modulated market, which consists of risky and riskless financial assets in the conditions of the jump-diffusion model of the financial market with consideration for a random replacement of the operating regimes of the market. In this subproject, we plan to investigate the corresponding Hamilton-Jacobi-Bellman (HJB) equations under some optimization criterions (for example, minimizing the ruin probability or maximizing the expected utility, etc.). By solving or numerical solving the correspond HJB equations, we wish to derive the optimal strategies for these kinds of portfolio selection problem..(3) Researches on the numerical solutions of jump-diffusion SDEs with Markovian switching. In this subproject, some more efficient schemes such as jump-adapted scheme and predictor-corrector Euler-Maruyama method will be given. The convergence rates and stabilities of the corresponding schemes should be further studied. In addition, we plan to compare the observed stability regions with the numerical methods developed in this project in order to simulating the risk models with Markovian switching we proposed.

Markov状态转换下的跳扩散风险模型是目前金融与保险中十分关注的新兴模型,具有重要的理论和应用意义。本项目主要将研究Markov状态转换下的几类新的跳扩散风险模型，并对刻画此类模型的带Markov状态转换的跳扩散方程的数值解理论进行研究。主要内容包括：(1)带线性红利界和不带红利界的Markov转换下的跳扩散风险模型的破产概率、罚函数等刻画、性质研究及相应的模拟分析。(2)研究将盈余资产投资于股票市场和债券市场的带Markov转换下的跳扩散模型的最优投资组合问题，给出合理的优化准则（如最小化破产概率、最大化期望效用等）下最优值函数的HJB方程，通过分析和数值求解研究相应的最优投资策略问题。(3)带Markov状态转换的跳扩散方程的数值解理论的研究，给出几种计算效率高的算法，例如跳适应算法、预估校正法等，研究算法的收敛速度及稳定性，以及相应算法的稳定域，更好地为风险模型的数值模拟提供基础。

带Markov状态转换的跳扩散风险模型的理论性质研究以及算法研究，无论在理论中还是实际应用中有着重要的意义。本研究项目的主要结果包括两部分的内容：（1）带Markov状态转换的跳扩散方程的稳定性及数值解理论；（2）带Markov状态转换的跳扩散风险模型的理论研究；. 在第一部分的研究中，我们研究了带Markov状态转换的跳扩散方程及时滞跳扩散方程解的各种稳定性，如 稳定性、指数稳定性、鲁棒稳定性等，推广了前人的结果。在数值解方面，考虑到状态转换的因素等，给出几种计算效率高的算法，例如预估校正法、补偿随机 法等，研究算法的收敛速度及稳定性，以及相应算法的稳定域等。这部分的研究取得了很好的进展。. 在第二部分的研究中，我们研究了带Markov状态转换的几类风险模型的生存概率、Gerber-Shiu罚函数等的性质，并进行了模拟分析，特别对两状态转换的情形，得到了较好的结果。项目还对Markov状态转换下随机波动率模型的MCQMC算法，以及Markov状态转换下GARCH族模型的Bayes估计问题进行了研究，得到了一些有意义的结果。. 本项目在研期间共发表SCI论文4篇（带标注的2篇），培养并毕业博士生2名、与项目研究密切相关的毕业硕士生3名。