Nonlocality, Nonlinearity, and Time Inconsistency in Stochastic Differential Games

TL;DR

This paper extends the mathematical theory of stochastic differential games by establishing existence, uniqueness, and stability results for nonlocal fully nonlinear parabolic systems, incorporating behavioral factors like time inconsistency relevant to financial economics.

Contribution

It generalizes the theory of stochastic differential games to include nonlocality and time inconsistency, providing new well-posedness results and a multidimensional Feynman--Kac formula.

Findings

Proved existence and uniqueness of solutions for nonlocal nonlinear systems.

Established stability analysis for solutions.

Developed a multidimensional Feynman--Kac formula under nonlocality.

Abstract

This paper proves 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. The derived 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 economics including robust stochastic controls and games 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).