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Considering an SDE of the form

$$dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t ,$$

where $W_t$ is a Wiener process, is there a set of necessary and sufficient conditions on the structure of the functions $\mu$ and $\sigma$ such that $X_t$ is multivariate normal (found perhaps by solving the Fokker-Planck equation)?

STJ
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adityar
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    InfProbSciX, do you know of functions $\mu(X_t,t)$, $\sigma(t)$ where $X_t$ is *not* multivariate normal? – Sextus Empiricus Dec 04 '18 at 16:16
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    @MartijnWeterings I'm not too sure about this but, if $\mu(X_t) = (X_t + a)^{-1}$, for $a$ close to $0$, I probably wouldn't expect $X_t$ to be multivariate normal. I'll do a few simulations now to check. Would it be surprising to find an $X_t$ that's _not_ mvn? – adityar Dec 04 '18 at 16:26
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    I do not know much about this stuff, and I was actually wondering *how* such SDE that depends on a Weiner process, would be creating a $X_t$ that is not multivariate normal. I myself imagine that it (not being multivariate normal) can be caused by terms like $X_t dW_t$ (like in a geometric Brownian motion) or anything else that would cause some interaction of the different $dW_t$. – Sextus Empiricus Dec 04 '18 at 16:38
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    That's an interesting comment and I think you're right. $\mu = 0, \sigma(X_t) = 1/X_t$ generates some weird looking (multivariate) distributions that aren't normal. I think that a lot of GPs can't actually be represented as SDEs and vice versa so I'm really interested in where they intersect. The OU process is an obvious example, but it's too restrictive, and on the other hand both GPs and state space models are incredibly powerful. – adityar Dec 04 '18 at 16:47
  • Looking back at this - there's a relationship between some stationary gaussian processes and state space models that has been shown by Simo Särkkä and Arno Solin. An example is shown in this slide deck titled [State Space Representation of Gaussian Processes](http://gpss.cc/gpss13/assets/Sheffield-GPSS2013-Sarkka.pdf). – adityar Jan 08 '21 at 20:28

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