Partial Information Stochastic Differential Games for Backward Stochastic Systems Driven by Lévy Processes

Fu Zhang,Maoning Tang,Qingxin Meng

doi:10.1155/2020/8563790

Abstract

In this paper, we consider a partial information two-person zero-sum stochastic differential game problem, where the system is governed by a backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. A sufficient condition and a necessary one for the existence of the saddle point for the game are proved. As an application, a linear quadratic stochastic differential game problem is discussed.

Highlights

Consider a partial information two-person zero-sum stochastic differential game problem, where the system is governed by the following nonlinear backward stochastic differential equation (BSDE), for any t ∈ [0, T], T
Where {W(t), 0 ≤ t ≤ T} is a standard d-dimensional Brownian motion and H(t) 􏼈Hi(t)∞ i 1, 0 ≤ t ≤ T􏼉 are Teugels martingales associated with a Levy processes {L(t), 0 ≤ t ≤ T}
Motivated by [7], Meng and Tang [8] studied the general full information stochastic optimal control problem for the forward stochastic systems driven by Teugels martingales and an independent multidimensional Brownian motion and proved the corresponding stochastic maximum principle

Summary

Introduction

Consider a partial information two-person zero-sum stochastic differential game problem, where the system is governed by the following nonlinear backward stochastic differential equation (BSDE), for any t ∈ [0, T], T y(t) ξ + 􏽚 f s, y(s), q(s), z(s), u1(t), u2(t)􏼁ds t dT. For all admissible open-loop controls (u1(·), u2(·)) ∈ G1 × G2 We denote this partial stochastic differential game by Problem (P). Motivated by [7], Meng and Tang [8] studied the general full information stochastic optimal control problem for the forward stochastic systems driven by Teugels martingales and an independent multidimensional Brownian motion and proved the corresponding stochastic maximum principle. To the best of our knowledge, there is little discussion on the partial information stochastic differential games for the system driven by Teugels martingales and an independent Brownian motion, which motives us to write this paper. A twoperson zero-sum stochastic differential game of linear backward stochastic differential equations with a quadratic cost criteria under partial information is discussed and the optimal control is characterized explicitly by the adjoint processes.

Preliminaries and Assumptions

A Partial Information Sufficient Maximum Principle

Partial Information Necessary Maximum Principle

Example