Linear quadratic mean-field-game of backward stochastic differential systems

Kai Du,Zhen Wu,Jianhui Huang

doi:10.3934/mcrf.2018028

Abstract

This paper is concerned with a dynamic game of N weakly-coupled linear backward stochastic differential equation (BSDE) systems involving mean-field interactions. The backward mean-field game (MFG) is introduced to establish the backward decentralized strategies. To this end, we introduce the notations of Hamiltonian-type consistency condition (HCC) and Riccati-type consistency condition (RCC) in BSDE setup. Then, the backward MFG strategies are derived based on HCC and RCC respectively. Under mild conditions, these two MFG solutions are shown to be equivalent. Next, the approximate Nash equilibrium of derived MFG strategies are also proved. In addition, the scalar-valued case of backward MFG is solved explicitly. As an illustration, one example from quadratic hedging with relative performance is further studied.

Highlights

In recent years, the study of mean-field game (MFG) has attracted consistent and extensive research attentions because of its significant theoretical values and broad practical applications
This paper introduces a large-population system with individual states following a linear backward stochastic differential equations (BSDEs)
Inspired by above mentioned motivations, this paper studies the dynamic problem of large-population backward stochastic differential systems

Summary

Introduction

The study of mean-field game (MFG) has attracted consistent and extensive research attentions because of its significant theoretical values and broad practical applications. The methodology of MFG has provided an effective and tractable analysis framework to establish the approximate Nash equilibrium for weakly-coupled stochastic controlled system with mean-field. Backward mean-field game (BMFG), -Nash equilibrium, Hamiltoniantype consistency condition (HCC), Riccati-type consistency condition (RCC), backward stochastic differential equation (BSDE). K. Du acknowledges the National Natural Sciences Foundations of China (11601285). Z. Wu acknowledges the Natural Science Foundation of China (61573217), the National High-level personnel of special support program and the Chang Jiang Scholar Program of Chinese Education Ministry

Objectives

Conclusion