# Smooth stochastic differential equations

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.

• "Brownian motion is special because it is the only Gaussian process that is also a martingale" Worse, I believe it is the only Gaussian process that is even a semimartingale. By Bichteler Dellacherie, this makes it the only Gaussian process we can define a stochastic integral against. – user223391 Feb 20 '18 at 18:13
• At least fBm is not a semimartingale and I don't really know any continuous Gaussian processes other than that, lol – user223391 Feb 20 '18 at 18:32

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.

Since a Gaussian process can be interpreted as a solution of the stochastic heat equation (see e.g. here), my impression is that it should be possible to reframe the equation you're interested in as one driven by white noise. It would probably be second order, though.