Talk:Kolmogorov continuity theorem

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)