Kolmogorov continuity theorem

Kolmogorov continuity theorem

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.

tatement of the theorem

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

:mathbb{E} left [ | X_{t} - X_{s} |^{alpha} ight] leq D | t - s |^{1 + eta}

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

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


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


* Theorem 2.2.3

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