  In this paper, the open-loop and closed-loop local and remote stochastic
nonzero-sum game (LRSNG) problem is investigated. Different from previous
works, the stochastic nonzero-sum game problem under consideration is
essentially a special class of two-person nonzero-sum game problem, in which
the information sets accessed by the two players are inconsistent. More
specifically, both the local player and the remote player are involved in the
system dynamics, and the information sets obtained by the two players are
different, and each player is designed to minimize its own cost function. For
the considered LRSNG problem, both the open-loop and closed-loop Nash
equilibrium are derived. The contributions of this paper are given as follows.
Firstly, the open-loop optimal Nash equilibrium is derived, which is determined
in terms of the solution to the forward and backward stochastic difference
equations (FBSDEs). Furthermore, by using the orthogonal decomposition method
and the completing square method, the feedback representation of the optimal
Nash equilibrium is derived for the first time. Finally, the effectiveness of
our results is verified by a numerical example.
