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If $X$ is a stochastic process, a.s. continuos and such that $\forall t \geq 0, X_t \in L^1_\omega$, is its mean function $t \rightarrow E[X_t]$ continuos?

I can show it if $X \in L^1_\omega L^{\infty}_t$ which means that $E[\sup_{t \geq0} |X_u|] < \infty$ by the dominated convergence theorem.

Is it true in general?

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  • $\begingroup$ Have a look at the counterexample presented here; the process has continuous sample paths but $t \mapsto E(X_t)$ fails to be continuous (.... which follows from a computation which is very similiar to the one which you find in the linked question). $\endgroup$
    – saz
    May 20 '19 at 17:50

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