Kolmogorov continuity theorem: Difference between revisions

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==Statement of the theorem==

Let <math>(S,d)</math> be some complete metric space such that d is <math>\mathcal{B}(S)\otimes\mathcal{B}(S)</math> measurable, and let <math>X : [0, + \infty) \times \Omega \to S</math> be a stochastic process. Suppose that for all times <math>T > 0</math>, there exist positive constants <math>\alpha, \beta, K</math> such that

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