There might be different definitions of what a generalized Langevin equation is, but let us consider the following expression:

$$ \dot{x}_i = \frac{dx_i}{dt} = f_i(\mathbf{x}) + \sum\limits_{m=1}^{n} g_i^m(\mathbf{x}) \eta_m(t) $$

where $\mathbf{x}=\{x_i|1\le i\le k\}$ is the set of unknowns, the $f_{i}$ and $g_{i}$ are arbitrary functions and the $\eta_m$ are random functions of time, often referred to as "noise terms".

Now generally higher order equations can generally be transformed into first-order equation, so take an arbitrary first-order stochastic differential equation. Can we always write it in the form above?

In the first place, it seems that assuming a finite number of noise terms $n$ might be too restrictive. Take for instance noise that takes the form $\exp(\eta(t) + \mathbf{x})$. If we expand this as a series, each of the terms can be written in the form $g^m(\mathbf{x}) \xi_m(t)$ for some stochastic variable $\xi_m(t) \sim \eta(t)^m$, but the series would in principle be infinite.

Hence, it is not clear that in general a noise term that depends on $\mathbf{x}$ can be separated into a deterministic term that depends on $\mathbf{x}$ and a stochastic term that only depends on $t$.

Nevertheless, one could still imagine that there might be transformations that can put any arbitrary SDE into the above form. Are there any proofs on whether such transformations exist for general classes of stochastic differential equations?

EDIT: for simplicity, let us assume that the stochastic system under consideration is Markovian.