I have the following, seemingly simple question:

Consider a stochastic process $(X_t)$ satisfying $X_t\le X_s$ a.s. for all $t\le s.$ My question is: Does there exist a modification $\tilde{X}$ of $X$, which almost surely has increasing sample paths $t\mapsto\tilde{X}_t(\omega)$?

I assume such a modification exists, but I did not manage to proof it.

Thanks in advance!