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Given a probability space, we say that $(X_t)_{t \geq 0}$ is Markov w.r.t its own filtration $(\mathcal F_t)$ if for all $s<t$,

$$ P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot | X_s)$$

Are you aware of weaker but similar properties such as

$$P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot | X_0, X_s) $$

or more interestingly,

$$ P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot |X_0, X_s, \int_0^s X_u du )$$

If so, what are the names of such properties and where could I read about them? If it is not present in the literature, can you give an example of a non-trivial process satisfying

$$ P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot |X_0, X_s, \int_0^s X_u du )$$

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