Nonlocality, nonlinearity, and time inconsistency in stochastic differential games

Abstract

This paper studies the well-posedness of a class of nonlocal fully nonlinear parabolic systems, which nest the equilibrium Hamilton–Jacobi–Bellman (HJB) systems that characterize the time-consistent Nash equilibrium point of a stochastic differential game (SDG) with time-inconsistent (TIC) preferences. The nonlocality of the parabolic systems stems from the flow feature (controlled by an external temporal parameter) of the systems. This paper proves the existence and uniqueness results as well as the stability analysis for the solutions to such systems. We first obtain the results for the linear cases for an arbitrary time horizon and then extend them to the quasilinear and fully nonlinear cases under some suitable conditions. Two examples of TIC SDG are provided to illustrate financial applications with global solvability. Moreover, with the well-posedness results, we establish a general multidimensional Feynman–Kac formula in the presence of nonlocality (time inconsistency).