Degenerate Diffusion Processes

In this chapter, we show that the degenerate points may separate the state space into different regions for multiple classes, and we discuss the optimization of multi-class degenerate diffusion processes. We also show that under some conditions, the performance function of finite-horizon optimization problems, or the potential function of the long-run average optimization problems, is semi-smooth at degenerate points and smooth at non-degenerate points. Thus, degenerate points coincide with semi-smooth points. Furthermore, there are some special features at the degenerate points: the local time at these points are zero, and the process can only move toward one direction. Therefore, the effect of semi-smoothness of a function can be ignored at these degenerate points in the Ito-Tanaka formula. With these special features in consideration, various optimization problems such as long-run average, finite-horizon, optimal stopping, and singular control, become simpler.