Kolmogorov continuity theorem

Let $(S,d)$ be some complete separable metric space, and let $X\backslash colon\; [0,\; +\; \backslash infty)\; \backslash times\; \backslash Omega\; \backslash to\; S$ be a stochastic process. Suppose that for all times $T\; >\; 0$, there exist positive constants $\backslash alpha,\; \backslash beta,\; K$ such that

:$\backslash mathbb\{E\}\; [d(X\_t,\; X\_s)^\backslash alpha]\; \backslash leq\; K\; |\; t\; -\; s\; |^\{1\; +\; \backslash beta\}$

for all $0\; \backslash leq\; s,\; t\; \backslash leq\; T$. Then there exists a modification $\backslash tilde\{X\}$ of $X$ that is a continuous process, i.e. a process $\backslash tilde\{X\}\backslash colon\; [0,\; +\; \backslash infty)\; \backslash times\; \backslash Omega\; \backslash to\; S$ such that

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

Furthermore, the paths of $\backslash tilde\{X\}$ are locally $\backslash gamma$-Hölder-continuous for every .

In the case of Brownian motion on $\backslash mathbb\{R\}^n$, the choice of constants $\backslash alpha\; =\; 4$, $\backslash beta\; =\; 1$, $K\; =\; n\; (n\; +\; 2)$ will work in the Kolmogorov continuity theorem. Moreover, for any positive integer $m$, the constants $\backslash alpha\; =\; 2m$, $\backslash beta\; =\; m-1$ will work, for some positive value of $K$ that depends on $n$ and $m$.

• Kolmogorov extension theorem

• {{cite book | author= Daniel W. Stroock, S. R. Srinivasa Varadhan | authorlink=Daniel W. Stroock, S. R. Srinivasa Varadhan | title=Multidimensional Diffusion Processes | publisher=Springer, Berlin | year=1997 | isbn=978-3-662-22201-0}} p. 51

Category:Theorems about stochastic processes