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Kolmogorov continuity theorem
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Wikipedia
In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constrains on the moments of its increments will be continuous (or, more precisely, have a "continuous version"). It is credited to the Soviet mathematician Andrey Nikolaevich Kolmogorov.

Statement of the theorem

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

mathbb{E} left[| X_{t} - X_{s} |^{alpha} right] leq D | t - s |^{1 + beta}

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

  • tilde{X} is sample continuous;
  • for every time t geq 0, mathbb{P} (X_{t} = tilde{X}_{t}) = 1.

Example

In the case of Brownian motion on mathbb{R}^{n}, the choice of constants alpha = 4, beta = 1, D = n (n + 2) will work in the Kolmogorov continuity theorem.

References

  • Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin. ISBN 3-540-04758-1. Theorem 2.2.3

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Last updated on Saturday February 16, 2008 at 10:07:33 PST (GMT -0800)
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