Nonlocality, Nonlinearity, and Time Inconsistency in Stochastic Differential Games

This paper presents the existence and uniqueness results (in the sense of maximally defined regularity) as well as the stability analysis for the solutions to a class of nonlocal fully-nonlinear parabolic systems, where the nonlocality stems from the flow feature (controlled by an external temporal parameter) of the systems. Our mathematical results generalize the theory of stochastic differential games to incorporate with behavioral factors such as time-inconsistent preferences, which facilitate developments of many studies in financial decision making including robust stochastic controls in finance and optimal investment and consumption under relative performance concerns. Moreover, with the well-posedness results, we establish a general multidimensional Feynman--Kac formula in the presence of nonlocality (time inconsistency)