Increasing process

An increasing process is a stochastic process...

[math]\displaystyle{ (X_t)_{t \in M} }[/math]

...where the random variables [math]\displaystyle{ X_t }[/math] which make up the process are increasing almost surely and adapted:

[math]\displaystyle{ 0=X_0 \leq X_{t_1} \leq \cdots . }[/math]

A continuous increasing process is such a process where the set [math]\displaystyle{ M }[/math] is continuous.

Consider a stochastic process [math]\displaystyle{ (\Chi_t) }[/math] satisfying [math]\displaystyle{ X_t \leq X_s }[/math] a.s. for all [math]\displaystyle{ t \leq s }[/math]  My question is: Does there exist a modification [math]\displaystyle{ \breve{X} }[/math] of ,[math]\displaystyle{ X }[/math] which almost surely has increasing sample paths [math]\displaystyle{ t \mapsto \breve{X}_t(\omega) }[/math]?