title: Increasing process
text: # 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:
  
      $0=X_{0}\leq X_{t_{1}}\leq \cdots .$
  
  A continuous increasing process is such a process where the set $M$ is continuous.
  
  Consider a stochastic process $(\mathrm {X} _{t})$ satisfying $X_{t}\leq X_{s}$ 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 $t\mapsto {\breve {X}}_{t}(\omega )$?
  
  