Let $X=(X_t)_{t\geq 0}$ be a stochastic process on $\mathbb{R}$ with independent increments. Then, assuming that all the relevant quantities exist, one can prove by simple algebraic manipulations that $$ \frac{d}{dt}Var(X_t)=\lim_{\Delta t\to 0} \frac{E((X_{t+\Delta t}-X_t)^2)}{\Delta t}. $$
At the end of this short PDF (https://sites.me.ucsb.edu/~moehlis/moehlis_papers/appendix.pdf) the authors make a comment that seems to suggest that the same conclusion also holds in the case of a time-homogeneous Markov process without the hypothesis of independent increments. Is that actually true? And if yes, how can it be proved?
To give some context, this question arose from a step in the derivation of the Fokker-Planck equation (see linked PDF). Indeed, it could also be stated as $$ D^{(2)}=\frac{1}{2}\frac{d}{dt}Var(X_t), $$ where $D^{(2)}$ is the second Kramers-Moyal coefficient.