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Consider the following diffusion process $$ dX_t = \mu\,dt+\sigma(t,X_t)\,dW_t $$ where $X,W$ are 1-dimensional and. Is it true that given a history $(X_s,s\leq t)$ for each $s< t$ one can find $\sigma(s,X_s)$? I guess, yes since it can be found through the qudratic variation for which the only history is needed. On the other hand I have doubts that $\mu$ can be found since by the equivalent change of measure we can do to $\mu$ whatever we want on the finite horizon. I wonder why there is a difference.

Another example: consider a Poisson process with an intensity $\lambda>0$. Again, by the equivalent change of measure we can change $\lambda$. What is the difference between parameters $\mu,\lambda$ which cannot be observed and $\sigma$ which can be found based on the history?

To make question more particular, let us assume that $X_t$ is given as above with $\mu\in \mathbb R$ unknown and $N_t$ is the Poisson process with some unknown intensity $\lambda>0$. Correspondent natural filtrations we denote respectively by $\mathcal F^X_t$ and $\mathcal F^N_t$. How to prove that there are is $x\in\mathbb R,t<\infty$ such that $\{\mu<x\}\in \mathcal F^X_t$ and $\{\lambda<x\}\in\mathcal F^N_t$.

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Just making sure I understand your question, since $[X]_t = \int_0^t \sigma^2(s, X_s) \mathrm{d} s$ we could recover $\sigma$ from a single trajectory, but determining $\mu$ requires an ensemble ? – Sasha Sep 27 '11 at 15:20
@Sasha: Yes. I am talking about only single trajectories on the finite horizon. – Ilya Sep 27 '11 at 15:49

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