Optional Times and Associated Concepts

Let (Ω, F; F(t), t∈I) be a filtered measurable space. If I does not have a last element, extend I to I⊃{+∞}, were +∞ 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.