Let $(X_t)_{t\geq 0}$ be a stochastic process adapted to $(\mathscr{F}_t)_{t\geq 0}$.
The (weak) Markov property says $X_{t+s}|\mathscr{F}_t\sim X_{t+s}|X_t$.
The strong Markov property says that if $T$ is a stopping time, then $X_{T+s}|\mathscr{F}_T\sim X_{T+s}|X_T$.
Let's call property ($\star$) the one that says that if $S$ is a stopping time, then $X_{S+t}|\mathscr{F}_t\sim X_{S+t}|X_t$.
The difference with the SMP is that in the case of property ($\star$) I am conditioning with respect to a non-random sigma algebra and the stopping time appears "only before the bar of conditional expectation". So apparently it is a different way of generalising the weak Markov property. Broadly speaking, I would like to know whether this property ($\star$) has a name and under what conditions it is implied by either the WMP or the SMP.
My progress so far is the following:
- In general property $(\star)$ is not implied by the SMP. It's not difficult to find counterexamples in the discrete case, and the answer by Julian Newman gives a counterexample in the continuous case too.
- At page 58, Theorem 3, in the book Markov Processes, Brownian Motion, and Time Symmetry by Chung and Walsh, it is shown that $X_{S+t}|\mathscr{F}_t\sim X_{S+t}|X_t,S$ for Feller processes and if $S$ is $\mathscr{F}_t$ measurable. (What they do in the proof is fine but the way they state the final result in their equation $(5)$ is, in my opinion, wrong, since the RHS should depend also on $S$ and not only $X_t$. See also the discussion here.)
QUESTIONS:
- Is it true that property $(\star)$ holds for right-continuous measurable processes which satisfy the WMP in the case when $S$ is a stopping time with respect to $(\mathscr{F}^Y_r)_{r\geq 0}$, where $(Y_r)_{r\geq 0}=(X_{t+r})_{r\geq 0}$? In this case I think I can prove it, because we have $S=f(X_{t+r_1},X_{t+r_2},...)$ and by the DCT $\mathbb{E}(g(X_{t+S})\,|\,\mathscr{F}_t)=\lim_n\mathbb{E}(g(X_{t+S_n})\,|\,\mathscr{F}_t)$ for all bounded $g$, where $S_n=2^{-n}\lceil2^n S\rceil$ is a discretisation of $S$, and we can use the WMP on each $X_{t+r_k}$ to conclude that $\mathbb{E}(g(X_{t+S_n})\,|\,\mathscr{F}_t)=\mathbb{E}(g(X_{t+S_n})\,|\,X_t)$.
- More generally, can something can be said about the case when $S$ is not $\mathscr{F}_t$ measurable nor is a stopping time with respect to $(\mathscr{F}^Y_r)_{r\geq 0}$, but is just a generic stopping time with respect to $(\mathscr{F}^X_t)_{t\geq 0}$? In this case, I expect something like $X_{S+t}|\mathscr{F}_t\sim X_{S+t}|X_t,Z$ to be true, where $Z$ is some random variable that should capture the information about $S$ available at time $t$ (but which I don't know how to specify).
Your property (⋆) seems very strange; I think the point of "stronger versions" of the Markov property is that these are properties that automatically follow from the basic Markov property in discrete time but become more of an issue in continuous time, due to the uncountability of the timeline. But in the case of your property (⋆), I see no reason to expect a typical Markov process even in discrete time to have this property.
Anyway, as an example of a Markov process with all the "strength" of Markov that you could possibly ever hope for, but which fails your property (⋆): Just take a Wiener process $(W_t)_{t \geq 0}$, with $(\mathscr{F}_t)_{t \geq 0}$ its natural filtration. For the stopping time $S$, take $$ S = \min\{s \geq 2 : W_s=W_1\}. $$ For each $t \in (1,2]$, on the positive-probability event $$ E_t:=\{W_1>0 \ \text{ and }\ W_t<0\} $$ we have $\mathbb{P}(W_{S+t}>0|W_t) < \tfrac{1}{2}$ but [since $W_S=W_1$ and $W_1$ is $\mathscr{F}_t$-measurable and every $s \geq 2$ has $W_{s+t}-W_s$ being independent of $\mathscr{F}_t$ and distributed $\text{Normal}(0,t)$] $$ \mathbb{P}(W_{S+t}>0|\mathscr{F}_t) = \int_{-W_S}^\infty \mathrm{pdf}[\text{Normal}(0,t)](x) \; dx > \tfrac{1}{2}. $$