Kolmogorov continuity theorem: Difference between revisions

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Revision as of 22:17, 26 January 2021

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==Statement==

Let <math>(S,d)</math> be some complete metric space, and let <math>X :\colon [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>

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

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