First entrance and exit times and the intrinsic topology in the state space

Abstract

Let X = (x _t , ζ ,ℳ _t P _x ) be a Markov process on a state space (E,ℬ). Forevery ω ∈ Ω _t we denote by F⁸ _t = F⁸_t (ω) the range of the sample function during the period [s,t] (i.e. the set of points x _u (ω) with u ∈ [s, t]). We call the function

$$\tau \left( \omega \right) = \sup \left\{ {t{:^8}_t \cap \Gamma = \emptyset } \right\} = \sup \left\{ {t:{F^8}_t \subseteq E\backslash \Gamma } \right\}$$

(4.1)

the first entrance time after time s into the set Г or the first exit time after time s from E\Г *. Let ℱ be any class of subsets of the space E. We call the upper bound of the first exit times after time s from all sets Г∈ ℱ the first exit time after time s from ℱ.

Translated by J. Fabius.