A general time-inconsistent stochastic optimal control problem with delay

In this paper, we develop a theory addressing a broad class of time-inconsistent stochastic control problems characterized by stochastic differential delayed 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, where we formulate the optimal investment mean-variance problem within a game theoretic framework and then applying a stochastic control theory with delay. Next by solving the extended HJB equations and constructing an exponential martingale process the explicit- form solution of the optimal investment strategy and the corresponding equilibrium value function are derived. This analysis not only extends the theoretical foundations but also provides insights into addressing and resolving time inconsistency in a practical example.