  This paper is concerned with a partially observed hybrid optimal control
problem, where continuous dynamics and discrete events coexist and in
particular, the continuous dynamics can be observed while the discrete events,
described by a Markov chain, is not directly available. Such kind of problem is
first considered in the literature and has wide applications in finance,
management, engineering, and so on. There are three major contributions made in
this paper: First, we develop a novel non-linear filtering method to convert
the partially observed problem into a completely observed one. Our method
relies on some delicate stochastic analysis technique related to hybrid
diffusions and is essentially different from the traditional filtering
approaches. Second, we establish a new maximum principle based on the
completely observed problem, whose two-dimensional state process consists of
the continuous dynamics and the optimal filter. An important advantage of the
maximum principle is that it takes a simple form and is convenient to
implement. Finally, in order to illustrate the theoretical results, we solve a
linear quadratic (LQ) example using the derived maximum principle to get some
observable optimal controls.
