# Existence of the solution for SDE with Gaussian Process

I'm interested in the existence of the solution for a non-Ito SDE. Sloppy notation but assume a SDE given by

$$\dot{x}=f(x),\quad f(x)\sim GP(0,k(x,x')),$$

where $$f$$ is a Gaussian Process with kernel $$k$$. It's quite easy to show that for some kernels (Lipschitz, etc.) all realizations of $$f$$ lead to ODEs with unique solutions. However, is there a way to transfer this information to the SDE? E.g. as all realizations lead to well-behaved ODEs -> there exists a solution for the SDE? Or is there any other way how to deal with this problem?

Please note that the question is about the existence and uniqueness of the solution of the SDE (like the picard-lindelöf theorem for ODEs). Not the solution itself.