Increasing process

{{one source |date=March 2024}}

An increasing process is a stochastic process...

:$(X\_t)\_\{t\; \backslash in\; M\}$

...where the random variables $X\_t$ which make up the process are increasing almost surely and adapted:

:$0=X\_0\; \backslash leq\; X\_\{t\_1\}\; \backslash leq\; \backslash cdots\; .$

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

Consider a stochastic process $(\backslash Chi\_t)$ satisfying $X\_t\; \backslash leq\; X\_s$ a.s. for all $t\; \backslash leq\; s$  My question is: Does there exist a modification $\backslash breve\{X\}$ of ,$X$ which almost surely has increasing sample paths $t\; \backslash mapsto\; \backslash breve\{X\}\_t(\backslash omega)$?{{Cite web |title=Increasing stochastic process |url=https://mathoverflow.net/questions/168757/increasing-stochastic-process |access-date=2023-05-06 |website=MathOverflow |language=en}}