Forgive me for the poorly researched question. I'm currently working on a computer science project involving training a neural stochastic differential equation, and I've run into a problem while dealing with the loss.

Suppose I have an Ito process which is a solution to $$ dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t $$ Is there an estimate for $H(X_t)$ for fixed time $t$, in terms of $\mu$ and $\sigma$? I know about the Fokker-Planck equation but I have yet to figure out how I might use it to estimate $H(X_t)$.

Naively I could sample a bunch of $X_t$'s and bin them to obtain a kernel density estimate, from which I could approximate $H(X_t)$, but this is computationally inefficient in the context I'm working, and I wouldn't be able to propagate gradients through it.

Even a useful upper bound on $H(X_t)$ in terms of the drift and diffusion would be helpful. Any ideas or prior research someone can point me to? Thanks!