BMAP风险模型的优化问题及Sinc数值计算

Batch arrival markov process (BMAP) risk model is an extension of the markov modulated risk model and is harder than the markov modulated model. BMAP not only has important theoretical significance, but also has important and wide application prospects. The complex characteristic determines its derivation and calculation are very difficult. Sinc numerical algorithm has a unique advantage on solving BMAP risk model. At present, there are no scholars to study the optimization problem in the BMAP risk model, and the study of the Sinc numerical algorithm in risk models is also just beginning. This project studies some optimal problems in the BMAP risk model and the Sinc numerical solution of the HJB equation. We will study it in the following aspects. ①The stochastic optimal control questions in the BMAP risk models. Based on the classical risk model, some financial behavior can be regard as occurring in some random discrete points, these actions including: dividends, investment and observation, etc. We want to use the BMAP to describe these random time points. The theory of Markov Decision Process is applied to analyze the questions. On the other hand, the model that the parameters and the random variables of payments are modulated by the MAP will be studied. Such models can reflect the change of the market condition and the uncertainty of the payment. We will use the stochastic control theory to analyze the optimal dividend or investment problem. ②The numerical solution of HJB equation. For the risk model with jumps, it is very difficult to obtain the analytical solution. Based on the basic thought of the Sinc numerical method, combined with the characteristics of the HJB equation, Sinc numerical solution of the value functions and the optimal strategies will be studied.

成批到达马氏链（BMAP）风险模型是马氏调制风险模型的推广，在难度上远超马氏调制模型，不仅有重大的理论意义，而且有重要而又广阔的应用前景。其复杂特性决定了其推导和计算极为困难，Sinc数值算法在BMAP风险模型的求解上具有独到优势。当前，针对BMAP风险模型的优化问题尚无学者开展研究，而对风险模型中Sinc数值算法的探索也刚刚起步。本项目研究内容如下：①BMAP风险模型中的优化问题。针对现实保险市场中存在分红、投资、注资等金融行为的状况，构建能够刻画更多金融信息的BMAP风险模型，探讨处理BMAP风险模型优化问题的一般思路，解决BMAP风险模型下最优分红，最优投资和消费等问题值函数和最优策略的求解。②求HJB方程的Sinc数值解。针对带跳风险模型无法求HJB方程解析解的现状，以Sinc数值算法的基本思想为基础，结合HJB方程的特点，求优化问题值函数和最优策略的Sinc数值解。

本项目研究基于BMAP的风险模型的构建和优化问题求解。我们用BMAP过程同时描述经济环境的变化及保险金融市场所涉及到的随机行为,研究了BMAP风险模型中的最优投资和消费问题，MAP风险模型的最优分红问题，资金注入问题，得到了解BMAP优化问题的一般思路，具体优化问题的最优策略和值函数；同时研究了MAP风险模型和对偶风险模型中一些重要的量，如期望折现罚金函数及期望折现分红总量，并用Sinc 数值算法给出了数值解。本项目的研究成果将马氏决策过程理论的应用推广到BMAP过程，在理论上拓展了马氏决策理论和BMAP的研究领域。由BMAP过程来同时描述经济环境的随机变化及保险金融市场随机金融行为是非常符合实际情况的，因此具有很好的现实意义。同时本项目用Sinc数值算法求解了MAP调制风险模型的Gerber-Shiu函数和分红总量，扩宽了Sinc数值算法的应用范围。通过研究我们也发现Sinc数值算法求解HJB方程并不合适。