Optional Times and Associated Concepts

Abstract

Let (Ω, ℱ; ℱ(t), t∈Iel) be a filtered measurable space. If I does not have a last element, extend I to I∪{+ ∞}, where + ∪ is a new element ordered after every element of I, and define ℱ(+ ∞) as any sub σ algebra of ℱ containing ⋎_t<+∞ℱ(t). The choice of ℱ(+∞) within these limits is usually irrelevant. If I has a last element, that element will be denoted by +∞ in this section. The most common choices of I are the set ℤ⁺ (discrete parameter case) and the set ℝ⁺ (continuous parameter case). The index set I is thought of as representing a set of time points, and ℱ(t) then represents the class of events up to time t. The problems of defining what is meant by a random time T corresponding to the arrival time of an event whose arrival is determined by preceding events and of defining the class ℱ(T) of preceding events are solved by the following definitions.