I'm trying to understand more about Ito Diffusions and the Strong Markov Property. In Oksendal's book (cited at the end), theorem 7.2.4 shows that if we have an Ito diffusion of the form $dX_t=b(X_t)dt+\sigma(X_t)dW_t$, where $b$ and $\sigma$ are regular enough so that a unique strong solution exists, then $\{X_t\}_{t\geq 0}$ has the Strong Markov Propery.
I've been searching for generalizations of this result for an time inhomogeneous Ito diffusion of the form $dX_t=b(X_t,t)dt+\sigma(X_t,t)dW_t$ but I haven't been able to find anything.
Can someone tell me if the Strong Markov Property holds for the time inhomogeneous case? If the answer is "no" could you provide an example to understand what fails? Also, if the answer is "no", is considering the process $Y=[t,X_t]$ by extending the state space from $E$ to $E\times \mathbb{R}_+$ a good way around it?
Øksendal, Bernt, Stochastic differential equations. An introduction with applications., Universitext. Berlin: Springer. xxiii, 360 p. (2003). ZBL1025.60026.
Yes, it holds for inhomogeneous SDEs also. You can find a proof of the following following theorem in Xuerong Mao's book: "Stochastic Differential Equations and Applications", page 88. Consider the SDE $$dx_t = f(x_t,t)dt + g(x_t,t)dW_t \quad \text{ on } t_0 \leq t \leq T \tag*{(1)}$$ with initial value $x_{t_0} = x_0.$ Assume that there exists two positive constants $K$ and $\bar{K}$ such that
(i) (Lipschitz condition) for all $x,y \in \mathbb{R}^d$ and $t \in [t_0,T]$ $$|f(x,t)-f(y,t)|^2 \bigvee |g(x,t)-g(y,t)|^2 \leq \bar{K}|x-y|^2; $$ (ii) (Linear growth condition) for all $(x,t) \in \mathbb{R}^d \times [t_0,T]$ $$|f(x,t)|^2 \bigvee |g(x,t)|^2 \leq K(1+|x|^2).$$
Then there exists a unique solution $\{ x_t \}$ to Equation (1). Furthermore $\{ x_t \}$ is a strong Markov process.