# When can a Gaussian Process solve an SDE?

Considering an SDE of the form:

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

... (where $$W_t$$ is a Weiner 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)?

• InfProbSciX, do you know of functions $\mu(X_t,t)$, $\sigma(t)$ where $X_t$ is not multivariate normal? – Sextus Empiricus Dec 4 '18 at 16:16
• @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? – InfProbSciX Dec 4 '18 at 16:26
• 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 4 '18 at 16:38
• 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. – InfProbSciX Dec 4 '18 at 16:47