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.