A Markov process which is not a strong markov process? Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in Oksendal's book on stochastic analysis), but I've never seen a process which satisfies one but not the other.
Many thanks
-Simon
 A: Reflected Brownian motion on the slit domain $D\setminus [0,\infty)\times \{0\}$ where $D$ is the unit disc in the plane. It is Not strong Markov for hitting times on the slit, but it is Markovian.
A: A standard example is Exercise 6.17 in Michael Sharpe's book General theory of Markov processes.
The process stays at zero for an exponential amount of time, then moves to the right at a uniform speed. 
A: Consider the following continuous Markov process X, starting from position x


*

*if x = 0 then Xt = 0 for all times.

*if x ≠ 0 then X is a standard Brownian motion starting from x.


This is not strong Markov (look at times at which it hits zero).
A: Let $X(t) = f(W(t) + \pi)$, where $W(t)$ is a standard Wiener process and 
$$f(x) = \begin{cases} (x,0), &  x\leq 0 \\\ \\\ (\sin x,1-\cos x), & 0 < x < 2\pi \\\ 
\\\ (x-2\pi,0), & x\geq 2\pi
\end{cases} $$
is a map from $\mathbb R$ to $\mathbb R^2$.  $X(t)$ is an $\mathbb R^2$-valued Markov process on $\mathbb R_+$  which is not strongly Markovian. See "A Modern Approach to Probability Theory"
by Fristedt and Gray (1997, pp. 626–627).
If the time set  is discrete, the ordinary Markov property implies  the  strong Markov property.
A: 
Can anyone give an example of a Markov process which is not a strong Markov process?

The Markov property implies the strong Markov property but the other way around is not true.
'Strong' refers to more rules/conditions that define the property. As a consequence it will be a less restrictive situation. The strong condition applies to a wider set of cases.
The Markov property means that conditional on the present value, the future distribution of a stochastic model does not depend on the past values.
The strong Markov property means that conditional on the present value, and conditional on the present state being a stopping time condition, the future distribution of a stochastic model does not depend on the past values.
