Assume that we have a random variable $T$ that takes values in $[0, \infty)$, and we know that for any continuous integrable function we have that for all $k \in \mathbb{N}$ the following holds $$ \mathbb{E}[\left(f(T)\right)^k] = \left(\int_0^\infty f(t)g_k(t) dt\right)^k$$ for some bounded function $g_k$ that depends on $k$.

Is it true that, assuming that $T$ is independent from a Wiener process $W$, the following holds for each $k \in \mathbb{N}$ $$\mathbb{E}[(W(T))^k] = \mathbb{E}\left[\left(\int_0^\infty W(t) g_k(t) dt\right)^k\right].$$

What I would like to find out is that if we know all the moments of $f(T)$, via a nice simple formula, then if we can use this formula when we replace $f$ by a stochastic process with continuous paths. I hope that it makes sense.


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