Stochastic maximum principle for hybrid optimal control problems under partial observation

Abstract: 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.