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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.

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  • $\begingroup$ You are intentionally writing $df(X_{t})$ rather than $dX_{t}$? $\endgroup$
    – Tobsn
    Commented Jul 20, 2022 at 5:00
  • $\begingroup$ Yes, so this integral is really 2 pieces from ito's lemma, one with dW_t and another with dt $\endgroup$
    – Asasuser
    Commented Jul 20, 2022 at 13:48
  • $\begingroup$ 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? $\endgroup$
    – Asasuser
    Commented Jul 20, 2022 at 14:03

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