# Process with Markov property but not strong Markov property

I'm trying to find a simple example of a stochastic process with the Markov property, but not the strong Markov property, to give me an intuitive understanding of the distinction between them.

All the processes I can think of off the top of my head seem to have either both or neither of these properties.

Thanks.

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An example is $X_t=\max\{t-T,0\}$, where $T$ is exponentially distributed.
For every fixed nonnegative $t$, conditionally on $\mathscr F_t^X$, $(X_{t+s})_{s\ge0}$ is distributed like $(X_s)_{s\ge0}$ on $[X_t=0]$ and like $(X_t+s)_{s\ge0}$ on $[X_t>0]$. But $(X_{T+s})_{s\ge0}$ is not distributed like $(X_s)_{s\ge0}$ on $\Omega=[X_T=0]$.
Okay, imagine a process whose volatility is time-dependent: $dF_t = f(t) dW_t$. This is Markov, since the distribution of future states only depends on the present state, but it is not strong Markov, because the distribution of $F_{\tau + t} - F_{\tau}$ is not independent of $\tau$.