Kolmogorov continuity theorem

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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 $X : \left[0, + infty\right) imes Omega o mathbb\left\{R\right\}^\left\{n\right\}$ 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 $ilde\left\{X\right\} : \left[0, + infty\right) imes Omega o mathbb\left\{R\right\}^\left\{n\right\}$ such that

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

Example

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

References

* Theorem 2.2.3

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