# Strong Markov property and time-homogeneity

Let $$(\Omega, \mathcal{A}, P, (X_n)_{n \in \mathbb{N}})$$ be a discrete-time stochastic process on a state space $$S$$ (which we assume to be finite and discrete for simplicity). Denote by $$\mathcal{F}_n := \sigma(X_0, X_1, \dots, X_n)$$ the filtration generated by $$X_n$$.

For $$\sigma$$-algebras $$\mathcal{F}$$, $$\mathcal{G}$$ and $$\mathcal{H}$$ denote by $$\mathcal{F} \perp\!\!\!\perp_{\mathcal{G}} \mathcal{H}$$ conditional independence of $$\mathcal{F}$$ and $$\mathcal{H}$$ given $$\mathcal{G}$$.

$$(X_n)$$ is called a Markov chain if it satisfies the Markov property: $$X_{n+1} \perp\!\!\!\perp_{X_n} \mathcal{F}_n$$ for all $$n$$, that is $$P(X_{n+1} = j \mid \mathcal{F}_n) = P(X_{n+1} = j \mid X_n)$$ almost surely. A Markov chain $$(X_n)$$ is called time-homogeneous if this conditional probability is independent of $$n$$.

For a process $$(X_n)_{n \in \mathbb{N}}$$ (a priori not a Markov chain) consider the following properties:

1. $$X_{N+1} \perp\!\!\!\perp_{X_N} \mathcal{F}_N$$ a.s. on $$\{ N < \infty \}$$ for all $$\mathcal{F}_n$$-stopping times $$N$$
2. $$X_{N+1} \perp\!\!\!\perp_{N, X_N} \mathcal{F}_N$$ a.s. on $$\{ N < \infty \}$$ for all $$\mathcal{F}_n$$-stopping times $$N$$

Both properties imply that $$(X_n)$$ is a Markov chain. Which of these two properties is the right definition of "strong Markov property"? Does any of these properties imply that $$(X_n)$$ is time-homogeneous?

Remark: In most of the literature one considers a time-homogeneous Markov chain $$(X_n)$$ equipped with a set of probability measures $$P_i$$ for $$i \in S$$ and defines for such processes the strong Markov property differently, namely as $$P_i(X_{N+1} = j \mid \mathcal{F}_N) = P_{X_N}(X_1 = j)$$ $$P_i$$-almost surely. Above, I'm interested in the cases when we have only one probability measure and without forcing time-homogeneity a priori.