次流动性市场中保险人的最优清算与最优投资策略研究

In the illiquidity market, a large transaction always leads to the price impact. An investor should consider the price impact and the volatility risk of the price of the asset when he is trading. The insurer, as a large investor, is faced with a risk process. Besides the price impact and the volatility risk of the price of the asset, an insurer should also take into account the ruin risk when he invests in an illiquidity market. Most of the existing literature focuses on an ordinary investor (without the risk process). In this project, by the stochastic dynamic programming method, we will study the optimal liquidation strategies and optimal investment strategies for an insurer in the illiquidity market. The following problems will be studied: the optimal liquidation strategy under the continuous-time first generation model and the second generation model by minimizing the ruin probability and maximizing the expected utility; the optimal liquidation strategy and optimal investment strategy under the assumption that trading can only be executed at the arrival times of a Poisson process; proposing the stochastic liquidity model characterized by Markov chain and studying the optimal problems under this model; the optimal investment strategy for the CARA and CRRA investor under the model with transit price impact. Because of the complexities of the problems and the models, we will develop some novel methods to study the value functions and optimal strategies. By considering the closed form solution and numerical solution, we will analyze the sensitivities of the value functions and the optimal strategies with respective to some model parameters.

在次流动市场中，大宗交易往往会产生价格效应。当投资者交易时，需要考虑价格效应和资产价格的波动风险。作为一个大的投资者，保险人具有一个风险过程。除了上述两种风险，保险人还需考虑破产风险。现有文献大都以一般投资者（没有风险过程）为研究对象。本项目将利用随机动态规划方法研究在次流动市场中保险人的最优清算及最优投资策略，主要内容包括: 分别以最小化破产概率与最大化期望效用为最优化准则, 在连续时间第一代模型与第二代模型下，考虑保险人的最优清算策略；假设交易时间限制在泊松过程到达时刻，考虑保险人的最优清算策略及最优投资策略；提出以马氏链来刻画的随机流动性模型，并考虑该模型下相应的最优化问题；考虑CARA或CRRA投资者在滑过效应模型下的最优投资策略。由于问题和模型的复杂性，需要采取一些新的方法来研究值函数及最优策略。通过考虑显示解或数值解，还将讨论值函数及最优策略与一些模型参数之间的关系。

本项目在具有实际经济意义、较为一般化的模型下，利用随机动态规划方法研究了在次流动市场中保险人的最优清算及最优投资策略。围绕含有时间不一致性偏好的最优养老金管理问题、含有随机系数的最优均值-方差投资组合与资产-负债管理问题、带有债务管理的最优分红及投资问题、基于马氏链调节扩散过程的时间不一致性最优控制问题以及马氏链驱动的倒向随机Volterra积分方程等方面进行了研究。同时对最优投资再保险及分红策略、交易费用影响下的保险公司最优分红、注资和再保险策略、破产终端值影响下保险公司最优分红、注资和再保险策略进行了研究。研究成果发表在国际著名的精算学和应用概率统计杂志及国内顶级杂志上，如 Insurance: Mathematics and Economics, Applied stochastic Models in Business and Industry, Journal of Industrial and Management Optimization，SIAM Journal on Control and Optimization, Journal of Computational and Applied Mathematics, Communications in Statistics - Theory and Methods，Mathematical Control and Related Fields, Stochastics: An International Journal of Probability and Stochastic Processes, Stochastic Analysis and Applications，Economic Modelling，Frontiers of Mathematics in China，IMA Journal of Management Mathematics，ASTIN Bulletin, 及《中国科学》等，共发表论文34篇，其中33篇论文被SCIE/SSCI检索。课题所涉及的研究内容均是目前国际上精算界研究的前沿和热点问题。一方面本课题的研究驱动了对几类应用随机过程和应用概率问题的研究，另一方面本课题有鲜明的现实背景，研究成果有望对保险公司的定价、投资、风险控制与管理具有一定的指导作用。