Strong Markov Processes

Let E be a metric space, and suppose that B is the Borel field generated by the open sets of E. A stochastic process is defined on E if a function x(t,ω )(0 t < ∞ ,ω ∈ Ω )$ and a system of probability measures ${ P}_x (x ∈ E)$ are given such that all ${ P}_x $ are defined on the Borel field generated by sets of type (1), and ${ P}_x \{ x(0,w) = x\} ≡ 1$. A random variable $τ (ω )$ is said to be independent of the future if for every s the $ω $-set $\{ τ (ω ) s\} $ belongs to the Borel field generated by sets of type (1) with $t s$. A measurable stochastic process on $E$ is called a strong Markov homogeneous process if for any $Γ _1 , ⋯ ,Γ _n ∈ B,0 < t_1 < t_2 < ⋯ < t_n $, and any $τ $ independent of the future (5) is true for almost all $ω $ such that $τ (ω ) < ∞ $. Every strong Markov homogeneous process is a Markov homogeneous process, but the converse is not true as shown by examples. The process is said to be a Feller process, if the space C of all bounded continuous functions on E is invariant under transformations Tₜ (t 0) defined by (7). It is proved that if a homogeneous Markov process is of the Feller type and if x(t,ω ) for all ω is a right-continuous function of t, then the process is a strong Markov process.