# stochastic integral when initial distribution is stationary

let $$X_t$$ be an Ito diffusion with drift $$\mu(x)$$ and noise $$\sigma(x)$$ such that there exists a stationary distribution $$\rho_0(x)$$ for the process.

suppose that I want to calculate $$\mathbb{E}\left[\int_0^h f(X_t)df(X_t)\right]$$ where here the expectation is taken both with respect to realizations of the process and with respect to the initial condition. suppose $$f$$ is $$C^{\infty}(\mathbb{R})$$.

1- In the particular case when the initial condition is itself sampled from $$\rho_0$$, is there a straightforward solution to the above?

2- Is it possible to rewrite this expectation in terms of the transition probability $$p_t(y|x)$$ that solves the forward Kolmogorov equation?

I apologize if this is obvious, I am a beginner in stochastic calculus.

• You are intentionally writing $df(X_{t})$ rather than $dX_{t}$? Commented Jul 20, 2022 at 5:00
• Yes, so this integral is really 2 pieces from ito's lemma, one with dW_t and another with dt Commented Jul 20, 2022 at 13:48
• But even if it was $dX_t$ instead of df(X_t), would the fact that the initial condition is sampled from the stationary distribution provide a simpler expression for the expectation? Commented Jul 20, 2022 at 14:03