一类正则-脉冲混合型随机微分对策的理论与应用

Model risk is one of the risks that must be incorporated in stochastic models of finance and insurance. Meanwhile stochastic optimal impulse control has a wide range of applications in the field of finance and insurance. In this project, we are going to apply the robust approach to investigate the stochastic optimal impulse control problems with model risk. And then we will formulate and develop a class of stochastic differential games involving regular-impulse controls. It is challenging to solve this class of regular-singular games. This is because the control strategy of each player is a combination of regular control and impulse control, the impulse control consists of a sequence of impulse times and a sequence of the impulses at the impulse times, and there is mutual coupling and influence between different control variables..In the existing literatures, Bellman’s dynamic programming and Pontryagin’s maximum principle are the two main approaches in solving these games. Both of them have their own limitations. The former cannot handle the stochastic differential games under insider information. And the adjoint variables of the latter are defined by the adjoint equation, which is a backward stochastic differential equation. In general it is hard to obtain its solution..In this project, we plan to establish new stochastic maximum principles by Malliavin calculus, ① under asymmetric partial informations; ②under asymmetric insider informations, for this class of stochastic differential games involving regular-impulse controls. We hope that the new adjoint variables are represented explicitly. So the new maximum principles will be able to break the limitations of the two classic approaches. .It is therefore that our research is of importance in theory and applications.

在金融保险中建立随机模型时，模型风险是必须要考量的一种风险。脉冲型随机最优控制在金融保险领域有着广泛的应用，本项目将采用鲁棒方法来处理模型风险，进而从脉冲型随机最优控制中，提取并推广一类正则-脉冲混合型的随机微分对策。这类对策中，博弈双方的控制变量是正则变量和脉冲变量的复合，脉冲变量既要控制脉冲时刻又要控制脉冲量，这些不同控制变量相互耦合相互制约，使得问题更具挑战性。目前Bellman动态规划原理和Pontryagin最大值原理是研究此类问题的最主要方法，但各有局限性，前者不适用于内幕信息下的随机微分对策，后者的对偶变量是由对偶方程定义，对偶方程是倒向随机微分方程，不易求出明确解。本项目拟应用Malliavin分析理论，分别在①不对称部分信息下②不对称内幕信息下，提出并证明新的最大值原理，力求使其中的对偶变量具有明确表达式，有望突破两种经典方法的局限性，因此具有重要的理论意义和应用前景。

脉冲型随机控制被广泛用于金融保险工程领域的建模，而建模本身会导致模型风险。本项目使用鲁棒方法处理模型风险，从脉冲型随机控制中提取并推广了一类正则-脉冲混合型随机微分对策，其系统的状态过程由Lévy扩散过程刻画，博弈双方的策略是正则随机控制和脉冲型随机控制的复合，博弈双方获取的信息不对称。目前动态规划原理和最大值原理是求解此类问题的最主要方法。本项目研究并解决了以下问题：（I）引入Malliavin分析、白噪声分析和向前积分等理论，利用变分法，针对正则-脉冲混合型随机微分对策，推导了鞍点（零和对策）和纳什均衡点（非零和对策）的最优性条件，并得到了此最优性条件与最大值原理的关系。新的最优性条件：(1)适用于不对称部分信息下和不对称内幕信息下的混合型随机微分对策，突破了动态规划原理的局限性；(2)具有明确表达式，由系统的系数和Malliavin导数表示，绕过了最大值原理中伴随方程这一倒向随机微分方程难以求解的困难。此项研究一定程度上实现了对现有经典方法的突破，丰富和发展了随机微分博弈的理论工具。（II）引入博弈论刻画模型风险，针对一类金融保险领域的热点问题——保险公司的最优投资-分红-融资问题，利用混合型随机微分对策建模，通过求解博弈问题的鞍点，对数学模型中普遍存在的模型风险进行刻画和度量，得到最坏情形下的控制策略，为金融保险领域的风险管理提供量化依据和理论基础，推动了正则-脉冲混合型随机微分对策在现实领域的应用。