Increasing process

An increasing process is a stochastic process...

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

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

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

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

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

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