Let $\mathbf{\Omega}=(\Omega,\mathcal{F},(\mathcal{F}\_t)\_{t \in [0,\infty)},\mathbb{P})$ be a filtered probability space satisfying the Usual Conditions.


Let $P \colon [0,\infty) \times \mathbb{R} \times \mathcal{B}(\mathbb{R}) \to [0,1]$ be a family of Markov transition probabilities on $\mathbb{R}$ (that is: $P(t,x,\cdot)$ is a probability measure, $P(t,x,A)$ is measurable in $(t,x)$, and the Chapman-Kolmogorov relations hold).


Let $(X\_t)\_{t \geq 0}$ be a real-valued progressively measurable stochastic process over the filtered probability space $\mathbf{\Omega}$. Suppose that $X$ has the *strong Markov property* with transition probabilities $P$; that is, for every $(\mathcal{F}\_t)$-stopping time $\tau \colon \Omega \to [0,\infty]$, every $t \geq 0$ and every $A \in \mathcal{B}(\mathbb{R})$,
$$ \mathbb{P}(X\_{\tau+t} \in A | \mathcal{F}\_\tau) \ = \ P(t,X\_\tau,A) \hspace{4mm} \mathbb{P}\textrm{-a.e. on }\{\tau < \infty\}. $$



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> > Is it necessarily the case that for every $(\mathcal{F}\_t)$-stopping time $\tau \colon \Omega \to [0,\infty]$, for every $\mathcal{F}\_\tau$-measurable function $s \colon \Omega \to [0,\infty]$ and every $A \in \mathcal{B}(\mathbb{R})$,
> >  $$ \mathbb{P}(X\_{\tau+s} \in A | \mathcal{F}\_\tau) \ = \ P(s,X\_\tau,A) \hspace{4mm} \mathbb{P}\textrm{-a.e. on }\{\tau,s < \infty\}\,? $$
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(I once saw a paper that used the "strong Markov property" at a certain point, but at this point it actually seemed to be using the above "really strong Markov" property. In the context, it was actually fairly easy to prove the necessary claim just by restricting to rational times, since everything was continuous and the sets in question were open/closed; but still, it would be interesting to know if the logic holds in general.)



Does the strong Markov property imply the "really strong Markov" property?
(I'll only do the case in which $S:\Omega\to[0,\infty)$. A truncation argument reduces the general case to this special case.)


Consider a bounded $\mathcal F\otimes
\mathcal B[0,\infty)$ measurable function $F$. Then for $\mathcal F\_\tau$ measurable $S:\Omega\to[0,\infty)$,
$$
\Bbb E[F(\cdot,S) \mid\mathcal F\_\tau](\omega) = \Bbb E[F(\cdot,u) \mid\mathcal F\_\tau](\omega)\Big|\_{u=S(\omega)},
$$
for $\Bbb P$-a.e. $\omega$ in $\{\tau<\infty\}$. This is an evident consequence of the "usual" SMP as displayed in the question provided $F$ is of the form $F(\omega,u)=G(\omega)H(u)$. The assertion then follows for general $F$ by the functional form of the monotone class theorem. To finish, take $F(\omega,u):=1\_A(X\_{\tau(\omega)+u}(\omega))$, which has the required measurability because $X$ is progressive, by hypothesis.


