Time-inconsistent recursive zero-sum stochastic differential games

Qingmeng Wei; Zhiyong Yu

doi:10.3934/mcrf.2018045

Article Contents

2018, Volume 8, Issue 3&4: 1051-1079. Doi: 10.3934/mcrf.2018045

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Time-inconsistent recursive zero-sum stochastic differential games

Qingmeng Wei^1, and
Zhiyong Yu^2, ,

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2.
School of Mathematics, Shandong University, Jinan 250100, China

* Corresponding author: Zhiyong Yu
* Corresponding author: Zhiyong Yu

Dedicated to Professor Jiongmin Yong’s 60 Birthday

Received: August 2017

Revised: March 2018

Published: September 2018

This work is supported in part by the National Natural Science Foundation of China (11471192, 11401091, 11571203), the Nature Science Foundation of Shandong Province (JQ201401), the Fundamental Research Funds of Shandong University (2017JC016), and the Fundamental Research Funds for the Central Universities (2412017FZ008)

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Abstract

In this paper, a kind of time-inconsistent recursive zero-sum stochastic differential game problems are studied by a hierarchical backward sequence of time-consistent subgames. The notion of feedback control-strategy law is adopted to constitute a closed-loop formulation. Instead of the time-inconsistent saddle points, a new concept named equilibrium saddle points is introduced and investigated, which is time-consistent and can be regarded as a local approximate saddle point in a proper sense. Moreover, a couple of equilibrium Hamilton-Jacobi-Bellman-Isaacs equations are obtained to characterize the equilibrium values and construct the equilibrium saddle points.

Keywords:

Time-inconsistency,
stochastic differential game,
equilibrium saddle point,
equilibrium Hamilton-Jacobi-Bellman-Isaacs equation,
recursive criterion functional.

Mathematics Subject Classification: Primary: 49N70; Secondary: 60H10.

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