Differential games of partial information forward-backward doubly stochastic differential equations and applications

TL;DR

This paper introduces a new class of differential game problems involving forward-backward doubly stochastic differential equations with partial information, providing theoretical conditions for equilibrium and saddle points, and illustrating with a linear-quadratic example.

Contribution

It develops necessary and sufficient conditions for equilibrium and saddle points in a novel class of stochastic differential games with partial information and doubly stochastic systems.

Findings

Derived explicit equilibrium expressions using stochastic filtering techniques.

Established theoretical conditions for equilibrium and saddle points in complex stochastic games.

Illustrated the application with a linear-quadratic nonzero-sum differential game.

Abstract

This paper is concerned with a new type of differential game problems of forwardbackward stochastic systems. There are three distinguishing features: Firstly, our game systems are forward-backward doubly stochastic differential equations, which is a class of more general game systems than other forward-backward stochastic game systems without doubly stochastic terms; Secondly, forward equations are directly related to backward equations at initial time, not terminal time; Thirdly, the admissible control is required to be adapted to a sub-information of the full information generated by the underlying Brownian motions. We give a necessary and a sufficient conditions for both an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. Finally, we work out an example of linear-quadratic nonzero-sum differential games to illustrate the theoretical applications. Applying some stochastic filtering techniques, we obtain the explicit expression of the equilibrium point.