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If we have a set of experimental data: $$X=\{x_1,x_2,\ldots,x_N\}$$ is it possible to write down an equation of the kind: $$dx(t)=b(x(t))\,dt+\sigma(x(t))\,dB(t)$$ describing the process from which the data arise, in which $B(t)$ is a Brownian process, $b$ is a function of only $x(t)$ and $\sigma$ the standard deviation? In which cases is it impossible?


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Does each $x$ value have a corresponding $t$ value? –  Michael Hardy Oct 3 '12 at 15:26
@ Michael Hardy: Yes, you get every $x_k$ at time $t_k$ –  Riccardo.Alestra Oct 3 '12 at 16:14
....and are those values of $t$ a part of the observable data? –  Michael Hardy Oct 3 '12 at 16:38
@ Michael Hardy: yes they are... –  Riccardo.Alestra Oct 3 '12 at 16:43
Since you have only finitely many data points and an infinite-dimensional space of possible functions $b$, and the data, even if improbably, could emerge from a process with almost any values of $b$ and $\sigma$, I think what you need is statistical estimation rather than trying to find some unique solution. And at this point I think you would fare better at stats.stackexchange.com, and there you should mention that you observe pairs $(t_i,x_i),\ i=1,\ldots,N$, and that you want to estimate $b$ and $\sigma$. –  Michael Hardy Oct 4 '12 at 19:16

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