Consistent control of stochastic systems with delay

Abstract

Thisthesis focuses on solving two research topics in distinct contexts using stochas tic control methods. The first topic develops a theory addressing a broad class of time-inconsistent stochastic control problems characterized by stochastic differential de layed equations (SDDEs), indicating the absence of a Bellman optimality principle. The approach involves framing these problems within a game theoretic framework and seeking subgame perfect Nash equilibrium strategies. For a general controlled process with delay and a reasonably broad objective functional, we extend the standard Bellman equation into a system of nonlinear equations. This extension facilitates the determination of both the equilibrium strategy and the equilibrium value function. Importantly, to exemplify the theory’s applicability, we delve into specific example such mean-variance portfolio with state dependent risk aversion problem with delay. By extending the theoretical founda tions, this analysis provides insights into addressing and resolving time inconsistencies in a practical example. In the second topic studies an equilibrium investment-reinsurance /new business and investment strategy for mean-variance insurers with state dependent risk aversion, the insurers are allowed to purchase proportional reinsurance, acquire new business and invest in a financial market, where both the surplus and the price process of risky stocks of the insurers are assumed to follow geometric Levy process. Under the influence of performance-related capital inflow/outflow, the wealth process of the investor