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 \dots .

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

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

References

↑ "Increasing stochastic process". MathOverflow. Retrieved 2023-05-06.

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[1] "Increasing stochastic process". MathOverflow. Retrieved 2023-05-06.

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