Existence of increasing modification

Question

I have the following, seemingly simple question:

Consider a stochastic process $(X_t)$ satisfying $X_s\le X_t$ a.s. for all $s\le t.$ 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 prove it.

Thanks in advance!

Answer 1

Let $(\tilde X_t)$ be a separable modification of $(X_t)$; for example, see Theorem 38.1 (page 558) in Billingsley's Probability and Measure (2nd edition). If $D$ is the separant, and we define $$N=\bigcup_{s<t;\, s,t\in D}\left\{\omega: \tilde X_s(\omega)>\tilde X_t(\omega) \right\},$$ then $\mathbb{P}(N)=0$, and for $\omega\notin N$ we have $t\mapsto \tilde X_t(\omega)$ is a non-decreasing function.