Talk:Kolmogorov continuity theorem

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The condition

\forall t \geq 0,\ \mathbb{P} (X_{t} = \tilde{X}_{t}) = 1

means that X is a modification of the sample continuous process \tilde{X}. (This condition is even weaker than assuming that X is a modification of a process that has surely continuous paths.) But what we want here is that X has a continuous version, that is, there exists a (sample) continuous process \tilde{X} such that

\mathbb{P} (X_t = \tilde{X}_t \ \forall t \geq 0 ) = 1.

Probabble (talk) 15:03, 3 May 2008 (UTC)