Kolmogorov continuity theorem

Remarks

Idea of the proof. On the dyadic times $k 2^{-m}$, Markov's inequality and the hypothesis give $P(\abs{X_{(k+1)2^{-m}} - X_{k 2^{-m}}} > 2^{-\gamma m}) \le C 2^{-m(1 + \beta - \alpha \gamma)}$. Summing over the $2^m$ times in $[0, 1]$ and over $m$ gives a convergent series when $\gamma < \beta / \alpha$. By the Borel–Cantelli lemma, the increments on dyadic times are eventually small, so $X$ is uniformly Hölder continuous on the dyadic times, and one defines $\tilde X$ as its continuous extension.

Example: for a Wiener process $W$ and an integer $m \ge 2$, $W_t - W_s$ is normal with variance $\abs{t - s}$, so $E[\abs{W_t - W_s}^{2m}] = c_m \abs{t - s}^m$ for a constant $c_m$. The theorem applies with $\alpha = 2m$ and $\beta = m - 1$, and gives Hölder continuity of every order $\gamma < (m - 1)/(2m)$. Letting $m \to \infty$, Brownian paths are locally Hölder continuous of every order $\gamma < 1/2$.

The exponent $1 + \beta$ must exceed $1$. For a Poisson process $N$, $E[\abs{N_t - N_s}] = \abs{t - s}$, so the bound holds with $\alpha = 1$ and exponent $1$, but the paths of $N$ have jumps and $N$ has no continuous modification.

For processes indexed by $t \in [0, 1]^d$, the same conclusion holds with $\abs{t - s}^{d + \beta}$ in place of $\abs{t - s}^{1 + \beta}$.

Going deeper

For a student

The theorem bridges the finite-dimensional distributions of a process, which say nothing about path properties, and the regularity of its paths. It is used to construct Brownian motion and the Brownian sheet, to prove continuity of solutions of stochastic differential equations in their initial data, and to obtain Hölder exponents. Its proof is a chaining argument, a technique that also yields the Kolmogorov–Chentsov tightness criteria in spaces of continuous functions.