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:

$0=X_0\le X_{t_1}\le \cdots .$

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

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

References

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