Conditions for Gaussianity of SDE

Question

Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq s<T$, the follwoing SDE with data has a strong solution $X_t$: $$ X_t^{x,s} = x + \int_0^t \mu(s,X_s)ds + \sum_{k=1}^m \int_s^t \sigma_k(s,X_s)dW_s^k, $$ where $(W^1,\dots,W^m)$ is an $m$-dimensional Brownian motion. Under what conditions on $\mu$ and the $\sigma_k$ is $X_t^{x,s}$ Gaussian?

Obviously this is true when $\mu$ and the $\sigma_k$ are constants; but how far can we relax our assumptions on these functions?

Answer 1

From the definition of the Itô stochastic integral, it is clear that the process $(X_t)$ will be Gaussian if (i) $\mu(s,\cdot)$ is affine -- that is, $\mu(s,x)=a(s)+b(s)x$ for some regular enough functions $a$ and $b$ and all appropriate $s$ and $x$ and (ii) $\sigma_k(s,x)$ does not depend on $x$. An example of such a process is the Ornstein--Uhlenbeck one.

Other than that, there are probably a few isolated exotic examples. Indeed, more generally, there seem to be few examples of non-affine transformations of Gaussian distributions that are Gaussian as well.