Difference density map

In mathematics, the Kolmogorov continuity theorem is a theorem that guarantees that a stochastic process that satisfies certain constraints 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 $X:[0,+\mathrm{\infty })\times \mathrm{\Omega }\to {\mathbb{ℝ}}^n$ be a stochastic process, and suppose that for all times $T>0$, there exist positive constants $\alpha ,\beta ,K$ such that

$\mathbb{𝔼}\left[\right|X_t-X_s{|}^{\alpha }]\le K|t-s{|}^{1+\beta }$

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

• $\tilde{X}$ is sample continuous;
• for every time $t\ge 0$, $\mathbb{\mathbb{P}}(X_t={\tilde{X}}_t)=1.$

Example

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

References

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