Differential games of partial information forward-backward doubly SDE and applications^∗

¹ Department of Building and Real Estate, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P.R. China
bscmhui@polyu.edu.hk
² School of Mathematics and Statistics, Shandong University, Weihai, Weihai 264209, P.R. China
xiaohua@sdu.edu.cn

Received: 29 August 2011
Revised: 8 February 2013

Abstract

This paper addresses a new differential game problem with forward-backward doubly stochastic differential equations. There are two distinguishing features. One is that our game systems are initial coupled, rather than terminal coupled. The other is that the admissible control is required to be adapted to a subset of the information generated by the underlying Brownian motions. We establish a necessary condition and a sufficient condition for an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. To illustrate some possible applications, an example of linear-quadratic nonzero-sum differential games is worked out. Applying stochastic filtering techniques, we obtain an explicit expression of the equilibrium point.