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.
