Factorization, basic notions and applications

Abstract

The idea to define an equivalence relation on the basic spaceW in terms of sample paths of some random processX _t (w) coinciding up to timet is the basis of the important concept of saturated filtration. Here we exploit the same idea to represent spaceW in the formW=W ^τ×W′, whereW ^τ is the space of all equivalence classes with respect to the above relation corresponding to a stopping timeτ andW′ is an image ofW under some shifting operator. This representation allows us to work with spaceW ^τ×W′ rather thanW and to investigate more precisely the properties of some random objects given onW.