投资与再保险策略的随机微分博弈

发布时间：2018-05-31 11:12

本文选题：随机最优控制 + HJB方程　；参考：《南京师范大学》2013年硕士论文

【摘要】：本文利用随机微分博弈的思想探讨了两家保险公司在连续时间下的决策问题。其中,决策变量分别是风险投资金额以及再保险的自留比例。在我们所考虑的模型中,两家公司分别承担着不同的保险风险,两种风险符合一般漂移布朗运动,它们之间可以存在相关性。同时,公司以各自现有的财富进行投资。假设他们都可以投资于相同的无风险资产,而可供投资的风险资产是各不相同的,但可以存在相关性。对于这两家保险公司财富额,我们构造出一个相同的报酬函数,一家公司将采取策略使得其报酬函数达到最大,而另外一家公司则时刻采取策略使得报酬函数达到最小。这其实就是一个零和随机微分博弈问题。我们假设保险风险不可以像金融资产那样借贷与卖空,而风险资产可以借贷一定上限的金额而不可以卖空。在考虑折现因子的情况下,我们得到零和博弈的纳什均衡解,给出了验证定理。在一个特殊的报酬函数下,我们详细讨论了纳什均衡存在的条件,以此给出了值函数的定义域,最终得到值函数以及最优策略。我们发现这时的值函数具有一个由凸到凹的函数形式,这与我们一般遇见的值函数形式有些不同。在文章的最后,我们给出了一个数值例子,用来展示纳什均衡解。
[Abstract]:In this paper, the decision problem of two insurance companies under continuous time is discussed by using the idea of stochastic differential game. The decision variables are the amount of venture capital and the retention ratio of reinsurance. In the model we consider, the two companies bear different insurance risks respectively. The two risks are consistent with the general drift Brownian motion, and there can be correlation between them. At the same time, companies with their existing wealth to invest. Suppose they can all invest in the same risk-free assets, and the risk assets available for investment are different, but can be relevant. For the wealth of these two insurance companies, we construct a common reward function. One company will adopt a strategy to maximize its return function, while the other company will always adopt a strategy to minimize the return function. This is actually a zero sum stochastic differential game problem. We assume that insurance risk can not be borrowed and short sold like financial assets, and risk assets can borrow a certain amount instead of short selling. Considering the discount factor, we obtain the Nash equilibrium solution of the zero-sum game, and give the proof theorem. Under a special reward function, we discuss the condition of the existence of Nash equilibrium in detail, and give the domain of the value function, and finally get the value function and the optimal strategy. We find that the value function has a function form from convex to concave, which is somewhat different from the value function we meet. At the end of the paper, we give a numerical example to show the Nash equilibrium solution.
【学位授予单位】：南京师范大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F224.32;F840.69

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