Kolmogorov continuity theorem: Difference between revisions

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==Statement of the theorem==

Let <math>X : [0, + \infty) \times \Omega \to \mathbb{R}^{n}</math> be a stochastic process, and suppose that for all times <math>T > 0</math>, there exist constants <math>\alpha, \beta, D K > 0</math> such that

:<math>\mathbb{E} \left[ | X_{t} - X_{s} |^{\alpha} \right] \leq DK | t - s |^{1 + \beta}</math>

for all <math>0 \leq s, t \leq T</math>. Then there exists a continuous version of <math>X</math>, i.e. a process <math>\tilde{X} : [0, + \infty) \times \Omega \to \mathbb{R}^{n}</math> such that