Nonzero Sum Differential Game of Mean-Field BSDEs with Jumps under Partial Information

Xiaolan Chen,Qingfeng Zhu

doi:10.1155/2014/561382

Abstract

This paper is concerned with a kind of nonzero sum differential game of mean-field backward stochastic differential equations with jump (MF-BSDEJ), in which the coefficient contains not only the state process but also its marginal distribution. Moreover, the cost functional is also of mean-field type. It is required that the control is adapted to a subfiltration of the filtration generated by the underlying Brownian motion and Poisson random measure. We establish a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a partial information linear-quadratic (LQ) game.

Highlights

Game theory had been an active area of research and a useful tool in many applications, in biology and economics
We establish a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and give a verification theorem which is a sufficient condition for Nash equilibrium point
They established a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and gave a verification theorem which is a sufficient condition for Nash equilibrium point

Summary

Introduction

Game theory had been an active area of research and a useful tool in many applications, in biology and economics. Wang and Yu [19] developed some results for optimal control of BSDEs and established a maximum principle for partial information differential games of backward stochastic differential equations (BSDEs). They established a necessary condition in the form of maximum principle with Pontryagin’s type for open-loop Nash equilibrium point of this type of partial information game and gave a verification theorem which is a sufficient condition for Nash equilibrium point. Øksendal and Sulem [22] established a general maximum principle for forward-backward stochastic differential games for Ito-Levy processes with partial information and applied the theory to optimal portfolio and consumption problems under model uncertainty, in markets modeled by Ito-Levy processes.

Statement of the Problems

A Partial Information Necessary Maximum Principle

A Partial Information Sufficient Maximum Principle

Application in a Partial Information LQ Case

Conclusion