Markov Processes

Abstract

Let {x(•), ℝ(•)} be a stochastic process with state space (X, H) on a filtered probability space (Ω, ℱ, P; ℱ(t), t ∈ I. The process is called a Markov process if when s < t and A ∈ H, then

$$ P\left\{ {x(t) \in A\left| {F(s)} \right.} \right\} = P\left\{ {x(t) \in A\left| {x(s} \right.)} \right\}a.s. $$

(1.1)