Controlled Diffusion Processes

Abstract

In the remainder of these notes we consider a class of $\mathbb {R}^d$–valued Markov processes $\{x(t),t\ge 0\}$ called (Markov) diffusion processes. These are processes that are characterized in a suitable sense by a function $b:[0,\infty )\times \mathbb {R}^d\rightarrow \mathbb {R}^d$ called the drift vector, and a $d\times d$ matrix D on $[0,\infty )\times \mathbb {R}^d$ called the diffusion matrix, which is assumed to be symmetric and nonnegative definite. In the extreme case in which $D\equiv 0$, the zero matrix, the process $x(\cdot )$ is the solution of an ordinary differential equation $\dot{x}(t)=b(t,x(t)), t\ge 0$. At the other extreme, if $b\equiv 0$ and $D\equiv I$ the identity matrix, then $x(\cdot )$ is a Markov process called Wiener process or Brownian motion. (See Example 5.3, above, or any introductory book on stochastic analysis or stochastic differential equations, for instance, Arnold (1974), Evans (2013), Mikosch (1998), Øksendal (2003),...)