模型不确定性及违约风险下的最优投资问题

数学物理学报 ›› 2016, Vol. 36 ›› Issue (2): 362-379.

模型不确定性及违约风险下的最优投资问题

郑箫箫¹, 孙中洋¹, 张鑫²

1. 南开大学数学科学学院天津 300071;
2. 东南大学数学系南京 210096

收稿日期:2015-09-19 修回日期:2016-01-11 出版日期:2016-04-25 发布日期:2016-04-25
作者简介:郑箫箫,xxzh1022@163.com;孙中洋,zysun_nk@126.com;张鑫,nku.x.zhang@gmail.com
基金资助:
国家自然科学基金(11371020)资助

Optimal Portfolio Problems for an Insurance Company Under Default Risk and Model Uncertainty

Zheng Xiaoxiao¹, Sun Zhongyang¹, Zhang Xin²

1 School of Mathematical Sciences, Nankai University, Tianjin 300071;
2 Department of Mathematics, Southeast University, Nanjing 210096

Received:2015-09-19 Revised:2016-01-11 Online:2016-04-25 Published:2016-04-25
Supported by:
Supported by the NSFC (11371020)

摘要/Abstract

摘要：

该文研究了一个同时具有模型不确定性和违约风险的随机最优投资组合问题.假设在金融市场中包含三种资产:银行账户(无风险资产),股票资产及可违约债券.考虑一个保险公司把保费盈余投资在这三种资产上来最大化其效用函数.把模型的不确定性因素考虑进去,此时问题转化为一个在金融市场与保险公司之间的零和微分博弈问题.首先考虑了跳扩散风险模型而后又考虑了扩散逼近模型.在这两个模型中通过动态规划准则导出了Hamilton-Jacobi-Bellman-Isaacs (HJBI)方程,从而求出了最优投资策略,并给出了验证定理.

关键词: 随机微分博弈, HJBI方程, 可违约债券, 模型不确定性, CARA效用最大化

Abstract:

In this paper, we investigate a stochastic portfolio optimization problem with model uncertainty and default risk. We assume that an insurer can invest his money into financial market where a savings account, a stock and a corporate bond are available, and aim to maximize the CARA utility of the terminal wealth. Furthermore, to take the model uncertainty into consideration, we formulate the optimization problem as a zero-sum stochastic differential game problem between market and the insurer. By using dynamic programming principle, we derive the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, and then find the optimal policy under the "worst-case" scenario for both jump-diffusion model and its diffusion approximation.

Key words: Stochastic differential game, HJBI equation, Defaultable bond, Model uncertainty, CARA utility maximization

中图分类号:

O211.6

郑箫箫, 孙中洋, 张鑫. 模型不确定性及违约风险下的最优投资问题[J]. 数学物理学报, 2016, 36(2): 362-379.

Zheng Xiaoxiao, Sun Zhongyang, Zhang Xin. Optimal Portfolio Problems for an Insurance Company Under Default Risk and Model Uncertainty[J]. Acta mathematica scientia,Series A, 2016, 36(2): 362-379.

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