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.

中文翻译：

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具有时滞的后向双随机系统的非零和微分博弈及其应用

本文研究了一种具有时滞的后向双随机系统的非零和微分对策，其中状态动力学遵循一个时滞的后向双随机微分方程（SDE）。为了解决上述博弈问题，自然需要包含伴随方程，这是一种预期的前向双SDE。我们给出了延迟向后双SDE和预期向前双SDE的解决方案的存在性和唯一性。我们以庞特里亚金类型为这种博弈的开环纳什均衡点建立了以最大原理形式的必要条件，然后给出了一个证明定理，这是纳什均衡点的充分条件。