Semigroups of Conditioned Shifts and Approximation of Markov Processes

Abstract

Let $\mathscr{L}$ be the space of processes, progressively measurable with respect to an increasing family of $\sigma$-algebras $\{\mathscr{F}_t\}$ and having finite mean. Then $\mathscr{J}(s)f(t) = E(f(t + s) \mid \mathscr{F}_t), f \in \mathscr{L}$, defines a semigroup of linear operators on $\mathscr{L}$. Using $\mathscr{J}(s)$ and known semigroup approximation theorems, techniques are developed for proving convergence in distribution of a sequence of (possibly non-Markov) processes to a Markov process. Results are also given which are useful in proving weak convergence. In particular for a sequence of Markov process $\{X_n(t)\}$ it is shown that if the usual semigroups $(T_n(t)f(x) = E(f(X_n(t)) \mid X(0) = x))$ converge uniformly in $x$ for $f$ continuous with compact support, then the processes converge weakly.