Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

doi:10.1007/s11464-020-0889-y

Front. Math. China

2020, Vol. 15

Issue (6) : 1307-1326 https://doi.org/10.1007/s11464-020-0889-y

RESEARCH ARTICLE

Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

Qingfeng ZHU^1,², Lijiao SU¹, Fuguo LIU³, Yufeng SHI²(

), Yong’ao SHEN¹, Shuyang WANG⁴

¹. School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, and Shandong Key Laboratory of Blockchain Finance, Jinan 250014, China
². Institute for Financial Studies and School of Mathematics, Shandong University, Jinan 250100, China
³. Department of Mathematics, Changji University, Changji 831100, China
⁴. School of Informatics, Xiamen University, Xiamen 361005, China

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Abstract

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.

Keywords Non-zero sum stochastic differential game mean-field backward doubly stochastic differential equation (BDSDE) Nash equilibrium point aximum principle

Corresponding Author(s): Yufeng SHI

Issue Date: 05 February 2021

Cite this article:

Qingfeng ZHU,Lijiao SU,Fuguo LIU, et al. Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games[J]. Front. Math. China, 2020, 15(6): 1307-1326.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-020-0889-y
https://academic.hep.com.cn/fmc/EN/Y2020/V15/I6/1307

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