Stochastic Differential Equations and Related Topics

研究方向为正反向随机微分方程（FBSDEs）和随机偏微分方程（SPDEs）及其在随机控制和随机金融/保险理论中的应用。本研究的主要贡献包括：提出了正-倒向鞅问题（FBMP）的新概念，以及研究FBSDEs弱解的适定性的一般框架；以及对完全非线性spde的随机特性的研究，这是随机粘度解理论的潜在基本组成部分。本文还将研究一类具有跳跃和可能的超线性增长系数的FBSDEs的适定性，以及它在一类具有一般保险模型的最优投资/再保险问题中的应用。PI还建议继续他对随机控制和随机金融/保险问题的研究。两个特殊的问题：一个涉及“万能可变寿命”保险的动态定价，另一个涉及由正常鞅（或满足结构方程的鞅）驱动的系统，将受到强烈的关注。后者也被认为是所提出的最优再保险问题的理论延伸。几乎所有提出的项目都有很强的应用背景，特别是在随机控制和随机金融/保险方面。这些问题中的许多问题反映了保险产品和养老金计划风险证券化的新趋势，其结果有望对精算数学、金融数学产生更广泛的影响，并可能因良好的合同设计而使保险界受益。关于FBSDEs弱解的拟议项目，特别是FBMP的新概念，将填补迄今为止理论的空白。提出的关于全非线性spde的随机特性的项目将为随机粘度解的概念带来新的见解，并有望大大推进一般理论。

The principal investigator proposes to study several long-standing problems involving forward-backward stochastic differential equations (FBSDEs) and stochastic partial differential equations (SPDEs), as well as their applications in stochastic control and stochastic finance/insurance theory. The main contributions of the research include a new notion of forward-backward martingale problem (FBMP), along with a general framework for studying the well-posedness of the weak solutions of FBSDEs; and a study of stochastic characteristics for fully nonlinear SPDEs, a potential fundamental building block of the theory of stochastic viscosity solutions. The well-posedness of a class of FBSDEs with jumps and possibly super-linear growth coefficients, as well as its application to a class of optimal investment/reinsurance problems with general insurance models will also be investigated. The PI also proposes to continue his research on stochastic control and stochastic finance/insurance problems. Two particular problems: one involving the dynamic pricing of the "Universal Variable Life" insurance, and the other involving systems driven by normal martingales (or martingales satisfying structure equations) will receive strong attention. The latter is also considered as a theoretical extension of the proposed optimal reinsurance problem.Almost all the proposed projects have strong background in applications, especially in stochastic control and stochastic finance/insurance. Many of these problems reflect the new trend of securitisation of risks in insurance products and pension plans, and the results are expected to have broader impact on actuarial mathematics, financial mathematics, and could benefit the insurance community for good contract designs. The proposed project on the weak solutions for FBSDEs, especially the new notion of FBMP, will fill the gap in the theory, which has been left open so far. The proposed project on stochastic characteristics of fully nonlinear SPDEs will bring new insight to the notion of stochastic viscosity solutions, and is expected to substantially advance the general theory.