The research on the strong Markov property

Let $X(t, omega) = {x_t(omega): t geq 0} be a Markov process defined on a probability space $(Omega,mathcal{F}, P)$ and valued in a measurable space (E; mathcal{E} ). In this paper, we give the definitions of $sigms$-algebras prior to $alpha$ and post-$alpha$ and discuss their properties. At the same time, we prove that the strong Markov property holds for an arbitrary Markov process, that is, we prove that the Markov property is equivalent to the strong Markov property.