A question about Markov process

Question

Let a stochastic process $(x(t),\theta(t))$ be given by $$ \dot{x}(t)=f(x(t),\theta(t)) $$

for a well defined continuous function $f(\cdot,\cdot)$. Let $\mathcal{F}_t$ denote the natural filtration of $(x(t),\theta(t))$ on the interval $[0,t]$. Then, is it implicit that the stochastic process $(x(t),\theta(t))$ is Markov with respect to $\mathcal{F}_t$? Or do we have prove it explicitly?

May be the question looks too stupid. But any help will be of great use.

Answer 1

Then, is it implicit that the stochastic process $(x(t),\theta(t))$ is Markov with respect to $\mathcal{F}_t$? Or do we have prove it explicitly?

If ever you must prove it, with no further hypothesis, then some trouble might arise... Assume for example that $x(0)=0$ and $\dot x(t)=\theta(t)$, then $x(t)=\int\limits_0^t\theta(s)\mathrm ds$ is not always Markov with respect to the filtration $(\mathcal F_t)$. Note that $\mathcal F_t=\sigma(\theta(s);0\leqslant s\leqslant t)$.