Time-Inconsistent Stochastic Linear-quadratic Differential Game

TL;DR

This paper studies a complex stochastic differential game with time-inconsistent objectives, proposing a new equilibrium concept, deriving conditions for equilibrium strategies, and providing explicit solutions in special cases.

Contribution

It introduces a novel equilibrium strategy for time-inconsistent stochastic linear-quadratic differential games and derives explicit solutions in deterministic, one-dimensional cases.

Findings

Established a sufficient condition for equilibrium strategies.

Derived explicit equilibrium strategies in one-dimensional deterministic cases.

Proved uniqueness of the equilibrium strategy in these cases.

Abstract

We consider a general time-inconsistent stochastic linear-quadratic differential game. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derived a sufficient conditions for equilibrium strategies via a system of forward-backward stochastic differential equations. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given.