An increasing process is a stochastic process...
( X t ) t ∈ M {\displaystyle (X_{t})_{t\in M}}
...where the random variables X t {\displaystyle X_{t}} which make up the process are increasing almost surely and adapted:
0 = X 0 ≤ X t 1 ≤ ⋯ . {\displaystyle 0=X_{0}\leq X_{t_{1}}\leq \cdots .}
A continuous increasing process is such a process where the set M {\displaystyle M} is continuous. Consider a stochastic process ( X t ) {\displaystyle (\mathrm {X} _{t})} satisfying X t ≤ X s {\displaystyle X_{t}\leq X_{s}} a.s. for all t ≤ s {\displaystyle t\leq s} My question is: Does there exist a modification X ˘ {\displaystyle {\breve {X}}} of , X {\displaystyle X} which almost surely has increasing sample paths t ↦ X ˘ t ( ω ) {\displaystyle t\mapsto {\breve {X}}_{t}(\omega )} ?
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