Weiner Process Markovian property

Question

Wiener process definition:

1. $W(t) \sim \mathcal{N}(0, \sigma\sqrt{t})$
2. $W(t)$ is Markovian
3. $W(t+t')-W(t)$ and $W(t)$ are independent.

I don't know of a case where $2$ is true but $3$ is not & vice versa. From Markovian property definition, the future at $t+t'$ should depend on $t$. So the definition itself should mean $3$ is true. What am I missing here? Can you give examples for both cases?

Answer 1

3 does not have to hold in order for 2 to hold. What the Markov property says is that $W(t+t')-W(t)$, conditional on the value of $W(t)$, does not depend on $W(s)$ for any $s<t$. It can still depend on $W(t)$ itself, but in the specific case of the Wiener process it doesn't. Basically for the Wiener process, you move the same way regardless of where you start; it is spatially homogeneous. So knowing $W(t)=x$ just translates the distribution of $W(t+t')$ to be centered at $x$ instead of $0$.

For an example where 2 holds and 3 does not, consider basically any SDE with coefficients depending on position, for instance $dX_t=X_t dW_t$.

I am less confident about whether 3 implies 2. What does imply 2 is $W(t_3)-W(t_2)$ being independent of $W(t_2)-W(t_1)$ whenever $t_1<t_2<t_3$. Maybe you can massage 3 into this but I would have to play with it.