Let $(\Omega,\mathcal F, (\mathcal F_t)_{t \in T}, P)$, $\, T \subseteq \mathbb R$, be a filtered probability space. A stochastic process $X=(X_t)_{t\geq 0}$ adapted to $\mathcal F_t$ is an $\mathcal F_t$-martingale if it is such that $E[|X_t|]<\infty$, $\forall t \geq 0$ and $E[X_t |\mathcal F_s]=X_s$, $\Omega$-a.s., for all $s < t$.
I am interested on an example, if it exists, of a process $Y=(Y_t)_{t\geq 0}$, possibly Markovian w.r.t to $\mathcal F_t$, such that $Z_s:=E[Y_t |\mathcal F_s]$ has the same law as $Y_s$ for all $s<t$, but which is not a martingale.
