On a Class of Set-Valued Markov Processes

Abstract

Let $Z$ be a finite or countable set, $\textbackslashXi$ the set of subsets of $Z, \textbackslash{\textbackslashxi_t\textbackslash}$ a Markov process with state space $\textbackslashXi$. A process $\textbackslash{\textbackslashxi_t\textasciicircum\textbackslashast\textbackslash}$ with the same state space is called associate to $\textbackslash{\textbackslashxi_t\textbackslash}$ if $\textbackslashmathbf{P}_\textbackslashxi\textbackslash{\textbackslashxi_t \textbackslashcap \textbackslasheta \textbackslashneq \textbackslashvarnothing\textbackslash} = \textbackslashmathbf{P}_\textbackslasheta\textasciicircum\textbackslashast\textbackslash{\textbackslashxi_t\textasciicircum\textbackslashast \textbackslashcap \textbackslashxi \textbackslashneq \textbackslashvarnothing\textbackslash}$ whenever $\textbackslashxi$ and $\textbackslasheta$ are subsets of $\textbackslashmathbf{Z}$, at least one of which is finite. Criteria are found for the existence of a process associate to a given one. Examples and applications are given.