Kolmogorov continuity theorem: Difference between revisions

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==Statement of the theorem==

Let <math>(S,d)</math> be some metric space, and let <math>X : [0, + \infty) \times \Omega \to \mathbb{R}^{n}S</math> be a stochastic process, and. supposeSuppose that for all times <math>T > 0</math>, there exist positive constants <math>\alpha, \beta, K</math> such that

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

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

* <math>\tilde{X}</math> is [[sample continuous process|sample continuous]];