Considering the SDE $dX_t=b(t,X_t)dt+\sigma (t,X_t)dW_t$ ($W$ is Brownian motion)
If there exists weak solution $(X,W),(\Omega ,\mathscr{F} ,P),\{\mathscr{F}_t\}$, is $X$ Markov process?
I know that if this SDE is well posed (i.e. for all $x\in \mathbb{R}^d$, there exists unique weak solution whose initial value is $x$), $X$ holds Strong Markov property.
My question: When $X$ holds Markov property?