Let $T$ be a stopping time and $X$ a stochastic process such that $X_t \in L^1$ for any $t\ge 0$. Suppose that the stopping time has finite expectation. What can be said about $\mathbb E[|X_T|]$? Is it bounded by the product of the expectation of $T$ and some constant which does not depend upon $T$?
The application I have in mind is the case where $X$ is a Brownian motion. In one case, I would like to consider the subcase where $T$ is even bounded.
Have a nice day,
Kehrwert
If $X_t$ is merely integrable, then you can really say nothing. Take, for example, $X_t = f(t)$ (non-random!), then $\mathbb E[|X_t|] = \mathbb E[|f(T)|]$ can be even infinite.
For the Wiener process, of course, the situation is different: it is well known that $$ \mathbb{E} [|W_T|] \le \sqrt{\mathbb{E}[T]} $$ (since $W_t^2 - t$ is a martingale).