Increasing process

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Increasing process" – news · newspapers · books · scholar · JSTOR (March 2024)

An increasing process is a stochastic process...

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

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

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

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

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

This probability-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e