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 such that

:

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 $t\; geq\; 0$, $mathbb\{P\}\; (X\_\{t\}\; =\; ilde\{X\}\_\{t\})\; =\; 1$.

Example

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

References

* Theorem 2.2.3