Time-Inconsistent Recursive Stochastic Optimal Control Problems

A time-inconsistent stochastic optimal control problem with a recursive cost functional is studied. Equilibrium strategy is introduced, which is time-consistent and locally approximately optimal. By means of multiperson hierarchical differential games associated with partitions of the time interval, a family of approximate equilibrium strategy is constructed, and by sending the mesh size of the time interval partition to zero, an equilibrium Hamilton--Jacobi--Bellman (HJB) equation is derived through which the equilibrium value function can be identified and the equilibrium strategy can be obtained. Moreover, a well-posedness result of the equilibrium HJB equation is established under certain conditions, and a verification theorem is proved. Finally, an illustrative example is presented, and some comparisons of different definitions of equilibrium strategy are put in order.