Discontinuous Markov Processes

A Markov process $x(t,w),t\geq0,\omega\in\Omega$, on a measurable space $(\mathscr E,\mathfrak B)$ is called a discontinuous process, if for every $\omega\in\Omega$ and $t\geq0$ there exists an $\varepsilon>0$ such that $x(t,\omega)=x(t+h,\omega)$ for all $h\in(0,\varepsilon]$. In this paper infinitesimal operators of all discontinuous processes are calculated. The results of these calculations imply the step-function processes described.