Increasing process

An increasing process is a stochastic process...

$(X_{t})_{t\in M}$

...where the random variables $X_{t}$ which make up the process are increasing almost surely and adapted:

$0=X_{0}\leq X_{t_{1}}\leq \cdots .$

A continuous increasing process is such a process where the set $M$ is continuous.

Consider a stochastic process $(\mathrm {X} _{t})$ satisfying $X_{t}\leq X_{s}$ a.s. for all $t\leq s$  My question is: Does there exist a modification ${\breve {X}}$ of ,$X$ which almost surely has increasing sample paths $t\mapsto {\breve {X}}_{t}(\omega )$?