Kolmogorov continuity theorem: Difference between revisions

Revision as of 14:33, 7 December 2015

Revision as of 19:52, 9 February 2016

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Let <math>(S,d)</math> be some metric space, 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} \left[ d(X_{t}X_t, X_{s}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 S</math> such that

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

* for every time <math>t \geq 0</math>, <math>\mathbb{P} (X_{t}X_t = \tilde{X}_{t}_t) = 1.</math>

Furthermore, the paths of <math>\tilde{X}</math> are almost surely [[Hoelder continuity|<math>\gamma</math>-Hölder continuous]] for every <math>0<\gamma<\tfrac\beta\alpha</math>.

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==Example==

In the case of [[Brownian motion]] on <math>\mathbb{R}^{n}</math>, the choice of constants <math>\alpha = 4</math>, <math>\beta = 1</math>, <math>K = n (n + 2)</math> will work in the Kolmogorov continuity theorem.

==See Alsoalso==

[[Kolmogorov extension theorem]]