This paper is concerned with a type of time-symmetric stochastic system, namely the so-called forward–backward doubly stochastic differential equations (FBDSDEs), in which the forward equations are delayed doubly stochastic differential equations (SDEs) and the backward equations are anticipated backward doubly SDEs. Under some monotonicity assumptions, the existence and uniqueness of measurable solutions to FBDSDEs are obtained. The future development of many processes depends on both their current state and historical state, and these processes can usually be represented by stochastic differential systems with time delay. Therefore, a class of nonzero sum differential game for doubly stochastic systems with time delay is studied in this paper. A necessary condition for the open-loop Nash equilibrium point of the Pontriagin-type maximum principle are established, and a sufficient condition for the Nash equilibrium point is obtained. Furthermore, the above results are applied to the study of nonzero sum differential games for linear quadratic backward doubly stochastic systems with delay. Based on the solution of FBDSDEs, an explicit expression of Nash equilibrium points for such game problems is established.

中文翻译：

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一类时间对称随机系统及相关博弈

本文涉及一种时间对称的随机系统，即所谓的前向-后向双随机微分方程（FBDSDE），其中前向方程是延迟的双随机随机微分方程（SDE），并且可以预期向后方程向后双重SDE。在某些单调性假设下，可获得FBDSDE的可测解的存在性和唯一性。许多过程的未来发展取决于它们的当前状态和历史状态，这些过程通常可以由具有时间延迟的随机差分系统表示。因此，本文研究了一类具有时滞的双随机系统的非零和微分对策。建立了彭特里亚金型极大原理的开环纳什平衡点的必要条件，并为纳什平衡点获得了充分的条件。此外，以上结果被用于研究具有延迟的线性二次后向双随机系统的非零和微分博弈。基于FBDSDE的解，建立了针对此类博弈问题的Nash平衡点的显式表达式。