# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2020028

## Nonzero-sum differential game of backward doubly stochastic systems with delay and applications

 1 Institute for Financial Studies and School of Mathematics, Shandong University, Jinan 250100, China 2 School of Mathematics and Quantitative Economics, and Shandong Key Laboratory of Blockchain Finance, Shandong University of Finance and Economics, Jinan 250014, China

* Corresponding author: Yufeng Shi

Received  August 2019 Revised  March 2020 Published  June 2020

Fund Project: This work was supported by National Key R & D Program of China (2018YFA0703900), National Natural Science Foundation of China (11871309, 11671229, 11371226, 11301298), Natural Science Foundation of Shandong Province of China (ZR2019MA013), the Special Funds of Taishan Scholar Project (Grant No. tsqn20161041), and Fostering Project of Dominant Discipline and Talent Team of Shandong Province Higher Education Institutions

This paper is concerned with a kind of nonzero-sum differential game of backward doubly stochastic system with delay, in which the state dynamics follows a delayed backward doubly stochastic differential equation (SDE). To deal with the above game problem, it is natural to involve the adjoint equation, which is a kind of anticipated forward doubly SDE. We give the existence and uniqueness of solutions to delayed backward doubly SDE and anticipated forward doubly SDE. We establish a necessary condition in the form of maximum principle with Pontryagin's type for open-loop Nash equilibrium point of this type of game, and then give a verification theorem which is a sufficient condition for Nash equilibrium point. The theoretical results are applied to study a nonzero-sum differential game of linear-quadratic backward doubly stochastic system with delay.

Citation: Qingfeng Zhu, Yufeng Shi. Nonzero-sum differential game of backward doubly stochastic systems with delay and applications. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020028
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