Increasing process

An increasing process is a stochastic process...

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

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

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

A continuous increasing process is such a process where the set ${\displaystyle M}$ is continuous.

Consider a stochastic process ${\displaystyle (\mathrm {X} _{t})}$ satisfying ${\displaystyle X_{t}\leq X_{s}}$ a.s. for all ${\displaystyle t\leq s}$  My question is: Does there exist a modification ${\displaystyle {\breve {X}}}$ of ,${\displaystyle X}$ which almost surely has increasing sample paths ${\displaystyle t\mapsto {\breve {X}}_{t}(\omega )}$?^[1]