  This paper is concerned with a Stackelberg stochastic differential game with
asymmetric noisy observation, with one follower and one leader. In our model,
the follower cannot observe the state process directly, but could observe a
noisy observation process, while the leader can completely observe the state
process. Open-loop Stackelberg equilibrium is considered. The follower first
solve an stochastic optimal control problem with partial observation, the
maximum principle and verification theorem are obtained. Then the leader turns
to solve an optimal control problem for a conditional mean-field
forward-backward stochastic differential equation, and both maximum principle
and verification theorem are proved. An linear-quadratic Stackelberg stochastic
differential game with asymmetric noisy observation is discussed to illustrate
the theoretical results in this paper. With the aid of some Riccati equations,
the open-loop Stackelberg equilibrium admits its state estimate feedback
representation.
