A Type of Time-Symmetric Stochastic System and Related Games

Qingfeng Zhu,Hui Zhang,Jiaqiang Wen,Yufeng Shi

doi:10.3390/sym13010118

Abstract

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.

Highlights

Received: 17 December 2020Accepted: 6 January 2021Published: 12 January 2021Publisher’s Note: MDPI stays neu-In 1994, Pardoux and Peng [1] put forward the following backward doubly stochastic differential equations (BDSDEs): p(t) = ξ + Z T tF (s, p(s), q(s))ds + ← −G (s, p(s), q(s)) d B(s) − →
The future evolution of a lot of processes depends on their current state, and on their historical state, and these processes can usually be characterized by stochastic differential equations with time delay
To figure out the above nonzero sum differential game problem, it is natural to involve the adjoint equation, which is a kind of anticipated BDSDE

Summary

Introduction

In 1994, Pardoux and Peng [1] put forward the following backward doubly stochastic differential equations (BDSDEs): p(t) = ξ +. We will discuss this direction, that is, the following nonzero sum differential game driven by doubly stochastic systems with time delay. It is necessary to explore the following general FBDSDE with the forward equation being a delayed doubly SDE and the backward equation being the anticipated BDSDE: dy(t). Games for a doubly stochastic system with time delay, and use the solution of the above general FBDSDE to give an explicit expression of the unique equilibrium point.

Notations and Formulation of Problems

The General FBDSDE

Necessary Maximum Principle

Sufficient Maximum Principle

Applications in LQ Doubly Stochastic Games with Delay

The Proof of Theorem 1

Conclusions