A stochastic linear-quadratic differential game with time-inconsistency

Qinglong Zhou; Gaofeng Zong; Qinglong Zhou; Gaofeng Zong

doi:10.3934/era.2022131

Electronic Research Archive

2022, Volume 30, Issue 7: 2550-2567. doi: 10.3934/era.2022131

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A stochastic linear-quadratic differential game with time-inconsistency

Qinglong Zhou ^{1
,
,},
Gaofeng Zong ²

1.
Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, China
2.
School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan 250014, China

Received: 14 September 2021 Revised: 14 December 2021 Accepted: 07 February 2022 Published: 11 May 2022

We consider a general stochastic linear-quadratic differential game with time-inconsistency. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derive a sufficient condition for equilibrium strategies via a system of forward-backward stochastic differential equation. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given.
- time-inconsistency,
- stochastic linear-quadratic differential game,
- equilibrium strategy,
- BSDE,
- forward-backward stochastic differential equation
Citation: Qinglong Zhou, Gaofeng Zong. A stochastic linear-quadratic differential game with time-inconsistency[J]. Electronic Research Archive, 2022, 30(7): 2550-2567. doi: 10.3934/era.2022131

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Abstract

We consider a general stochastic linear-quadratic differential game with time-inconsistency. The time-inconsistency arises from the presence of quadratic terms of the expected state as well as state-dependent term in the objective functionals. We define an equilibrium strategy, which is different from the classical one, and derive a sufficient condition for equilibrium strategies via a system of forward-backward stochastic differential equation. When the state is one-dimensional and the coefficients are all deterministic, we find an explicit equilibrium strategy. The uniqueness of such equilibrium strategy is also given.

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