Mathematics 275C – Stochastic Processes – T. Liggett I. The construction problem for continuous time Markov chains

I. The construction problem for continuous time Markov chains Reference: Part II of David Freedman, Markov Chains, Holden-Day, 1971. Suppose that S is a countable set, Ω = the set of all right continuous step functions ω : [0,∞) → S with finitely many jumps in any finite time interval, Xt(ω) = ω(t), and {P , x ∈ S} are probability measures on Ω that satisfy (i) P (X0 = x) = 1 for all x ∈ S, and (ii) the Markov property E(Y ◦ θt | Ft) = EtY a.s. P x for all x ∈ S and all bounded measurable Y on Ω. Let pt(x, y) = P (Xt = y) be the corresponding transition probabilities. Then (i) and (ii) imply