Switching Diffusion

Abstract

This chapter provides an introduction to switching diffusions. First the existence and uniqueness of the solution of associated stochastic differential equations, weak continuity, Feller, and strong Feller properties are established. Also given here are the definition of regularity and criteria ensuring such regularity. Moreover, smooth dependence on initial data is presented.

The rest of the chapter is arranged as follows. After this short introductory section, Section 2.2 presents the general setup for switching processes. Section 2.3 is concerned with regularity. Section 2.4 deals with weak continuity of the pair of process (X(t); α(t)). Section 2.5 proceeds with Feller properties. Section 2.6 goes one step further to obtain strong Feller properties. Section 2.7 presents smooth dependence properties of solutions of the switching diffusions. Section 2.8 gives remarks on how nonhomogeneous cases in which both the drift and diffusion coefficients depend explicitly on time t can be handled. Finally, Section 2.9 provides additional notes and remarks.