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Usually, we start with an SDE for $X(t)$ and then try to solve it and find the corresponding probability density function $p(x,t)$. But what if we start with a pdf - can we find a corresponding SDE? How can this be done? Would we use the Fokker-Plank equation and try to "guess" the mean and diffusion coefficient? Or is there an approach that doesn't require guessing? Would such SDE be unique?

Confounded
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  • If you have the Fokker-Plank equation you don't need to 'guess' the cofficients, you can read them off. – Tobsn Feb 09 '21 at 07:25
  • @Tobsn Thank you for your comment. Yes, clearly if the FP equation is already available than we can just read them off. However, what I meant by "using the FP" is to take the derivatives of the pdf (which is assumed to be the only "thing" available to us) and try to "construct" the FP. – Confounded Feb 09 '21 at 08:55
  • From the density kernel you can determine the generator of the diffusion from which you can read of FPeq and the SDE, respectively. – Tobsn Feb 09 '21 at 14:23
  • @Tobsn And how does one "determine the generator of the diffusion"? And, for that matter, what is meant by "the generator of the diffusion"? Thank you – Confounded Feb 09 '21 at 14:29
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    $\mathcal{A}f(x)=\lim_{t\to 0}\frac{1}{t}\left(\int f(y)p_{t}(x,dy)-f(x)\right)$. What the generator $\mathcal{A}$ is/does you'll find in the literature. – Tobsn Feb 09 '21 at 14:34
  • @Tobsn And what is the function $f(x)$ in your formula? – Confounded Feb 09 '21 at 14:55

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