I'm reading Section 5.4 Markov property from these notes, i.e.,
$\newcommand{\Ex}{\mathbb E}\newcommand{\diff}{\,\mathrm d}$Let $B$ be a standard Brownian motion and $(\mathcal F_t, t \ge 0)$ its natural filtration. Let $f, g:\mathbb R_{\ge 0} \times \mathbb R \to \mathbb R$ be jointly continuous in $(t, x)$ and Lipschitz in $x$. Let $X$ be the strong solution of $$ \begin{cases} \diff X_t &= f(t, X_t) \diff t + g(t, X_t) \diff B_t, \\ X_{0} &= x_0. \end{cases} \quad (\star) $$
Let $h:\mathbb R \to \mathbb R$ be bounded continuous. It can be shown that if $T \ge t \ge 0$ then $$ \Ex [h(X_T) | \mathcal F_t] = \varphi (X_t) = \Ex [h(X_T) | X_t], $$ where $\varphi (x) := \Ex [h(X^{t, x}_T)]$ with $X^{t, x}$ the strong solution of $$ \begin{cases} \diff X_s &= f(s, X_s) \diff s + g(s, X_s) \diff B_s, \quad \forall s \in [t, T],\\ X_t &= x. \end{cases} $$
It's mentioned in this answer that if $X$ is time-homogeneous then $X$ is a Markov process.
Could you elaborate on the proof or the references that the solution $X$ of $(\star)$ is a Markov process?
Here is a reference for the result: Schilling, Partzsch, Boettcher; Th.18.13. p. 285.
One can consult also the newer edition: Schilling; Th.21.23. p. 403.