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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.

FreshDresch
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Ron
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1 Answers1

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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 $\mathcal F_t=\sigma(\theta(s);0\leqslant s\leqslant t)$.

Did
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  • Thank you, but still I do not understand fully. I agree with you that I did not provide full information about $f$ and $\theta(t)$. But in general, Does any stochastic process is Markov with respect to its natural filtration? (this was the main question I intended to ask) – Ron Jul 24 '14 at 15:15
  • "Does any stochastic process is Markov with respect to its natural filtration?" Of course not (otherwise the thing would not have a name, you see... :-)). – Did Jul 24 '14 at 15:25