In the recent years, time-inconsistent stochastic control problems have received remarkable attentions in stochastic control, mathematical finance and economics. Time-inconsistency for a dynamic control problem means that the so-called Bellman's principle of optimality does not hold. In other words, a restriction of an optimal control for a specific initial pair on a later time interval might not be optimal for that corresponding initial pair. In order to deal with a time-inconsistent problem in a sophisticated way, the main approach in the literature is to regard the dynamic problem as a non-cooperative game, where decisions at every instant of time are selected by different players (which represent the incarnations of the controller). Nash equilibria are therefore considered instead of optimal controls.
In this talk, we investigate a time-inconsistent stochastic recursive control problem where the cost functional is defined by the solution to a backward stochastic Volterra integral equation (BSVIE, for short), which is a generalization of a backward stochastic differential equation (BSDE, for short). We show that the corresponding adjoint equations become extended backward stochastic Volterra integral equations (EBSVIEs, for short), and provide a necessary and sufficient condition for an open-loop Nash equilibrium control via variational methods.