# Kolmogorov’s continuity theorem

Let $X={\{{X}_{t}\}}_{t\ge 0}$ be a process satisfying the following condition$:$ For all $T>0$ there exist positive constants $\alpha $, $\beta $, $D$ such that

$$E[{|{X}_{t}-{X}_{s}|}^{\alpha}]\le D{|t-s|}^{1+\beta}\mathrm{\hspace{0.17em}\hspace{0.17em}\hspace{0.25em}0}\le s,t\le T.$$ |

Then there exists a continuous^{} modification of $X$.

Title | Kolmogorov’s continuity theorem |
---|---|

Canonical name | KolmogorovsContinuityTheorem |

Date of creation | 2013-03-22 15:43:43 |

Last modified on | 2013-03-22 15:43:43 |

Owner | georgiosl (7242) |

Last modified by | georgiosl (7242) |

Numerical id | 9 |

Author | georgiosl (7242) |

Entry type | Theorem |

Classification | msc 60G07 |

Related topic | DistributionsOfAStochasticProcess |