Research on stochastic differential equations seems to be exclusively focused on the Brownian motion noise, where the solution is a nowhere differentiable function.
I am instead interested in stochastic equations of the form
$$dy/dt=f(y)+e(t),$$
where $e(t)$ is a smooth random function, e.g. a smooth Gaussian process.
Is there any interesting theory about such equations? I'm specifically interested in parameter inference, but any other references will be appreciated too.
I don't think such a theory exists.
My understanding of stochastic SDEs is that they're a reinterpretation of a stochastic integral; that is, a stochastic SDE can be rewritten as a stochastic integral, and it is in terms of said integrals we should think of stochastic SDEs.
A key property of stochastic integrals is that they are continuous-time martingales, and there your project ends because all continuous-time martingales are time-shifted Brownian motion. That is, for any continuous-time martingale $M$ there is a Brownian motion $B$ s.t. $M_t = B_{\langle M, M \rangle_t}$ for all $t$, with $\langle M, M \rangle_t$ being the quadratic variation of $M$. Since $B$ is nowhere differentiable $M$ is nowhere differentiable; this includes all stochastic integrals.
Brownian motion is special because it is the only Gaussian process that is also a martingale and has independent increments (two properties that turn out to be redundant; one implies the other). I'm pretty sure no one knows of a differentiable process that is also a martingale; it certainly isn't a Gaussian process if it exists (and I don't think it does).
In short, talking about SDEs involving smooth stochastic processes means we would need to abandon martingales and build a brand new theory for stochastic integration. No one is in any hurry to do that.
I doubt there's much literature - most stochastic processes we care about are not differentiable.
However if there is a particular case you're interested in you can simply solve things pathwise - no stochastic calculus involved.
