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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.

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