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 <math>X : [0, + \infty) \times \Omega \to \mathbb{R}^{n}</math> be a stochastic process, and suppose that for all times <math>T > 0</math>, there exist constants <math>\alpha, \beta, D > 0</math> such that
- <math>\mathbb{E} \left[ | X_{t} - X_{s} |^{\alpha} \right] \leq D | t - s |^{1 + \beta}</math>
for all <math>0 \leq s, t \leq T</math>. Then there exists a continuous version of <math>X</math>, i.e. a process <math>\tilde{X} : [0, + \infty) \times \Omega \to \mathbb{R}^{n}</math> such that
- <math>\tilde{X}</math> is sample continuous;
- for every time <math>t \geq 0</math>, <math>\mathbb{P} (X_{t} = \tilde{X}_{t}) = 1</math>.
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>D = n (n + 2)</math> 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.
