Increasing process explained

An increasing process is a stochastic process...

(X _{t)} _{t}

...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 … .

A continuous increasing process is such a process where the set

M is continuous.

Consider a stochastic process

(\Chi _{t)} satisfying X _{t} \leq X _{s}  a.s. for all t \leq 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)} ?^[1]

Notes and References

1. Web site: Increasing stochastic process . 2023-05-06 . MathOverflow . en.