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

Kai Du; Jianhui Huang; Zhen Wu

doi:10.3934/mcrf.2018028

Article Contents

2018, Volume 8, Issue 3&4: 653-678. Doi: 10.3934/mcrf.2018028

This issue Previous Article Weak laws of large numbers for sublinear expectation Next Article Inverse S-shaped probability weighting and its impact on investment

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

1.
School of Mathematics, Shandong University, Jinan 250100, China
2.
Zhongtai Securities Institute for Financial Study, Shandong University, Jinan 250100, China
3.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

* Corresponding author: Zhen Wu
* Corresponding author: Zhen Wu

Received: March 2017

Revised: December 2017

Published: September 2018

The work of K. Du is partially supported by the PolyU-SDU Joint Research Center (JRC) on Financial Mathematics. K. Du acknowledges the National Natural Sciences Foundations of China (11601285). J. Huang acknowledges the financial support by RGC Grant PolyU 153005/14P, 153275/16P, 5006/13P. 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

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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.

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Mathematics Subject Classification: Primary: 93E20, 91A23; Secondary: 93E03, 91A10, 93E14.

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