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 $X_t$, 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!

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!