Stochastic maximum principle for hybrid optimal control problems under partial observation

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, cannot be directly available. Such kind of partially observed control problem has wide applications in finance, management, engineering, and so on. The main contribution of this paper includes three aspects. Firstly, we adopt a novel non-linear filtering method and obtain a general filtering equation such that the partially observed problem is converted into a completely observed one. Our method relies on some delicate stochastic analysis technique and is essentially different from the traditional filtering methods for hybrid diffusions. Secondly, 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. The derived maximum principle takes a simple form and is convenient to implement. Finally, in order to illustrate the theoretical results, we solve a linear quadratic example by using the maximum principle and get an observable optimal control.