ABSTRACT
Let us set F 0n = σ(X0, X1, . . . , Xn). We know that a function 9 on is F 0n - measurable if and only if9 = φ(X0, X1, . . . , Xn) for aE ⊗(n+1)-measurable function φ. In particular for an event A to belong to F 0n it means, intuitively, that one knows whether A has happened or not as soon as the values taken by X0, X1, . . . , Xn are known. However, in many cases, what is known at time n is not just the values of X0, X1, . . . , Xn, since also other information may be available, the values taken by other processes, for instance. This explains the following definition, Definition 2.1 We call filtered probability space a term
(,F , (Fn)n≥0,P)
where (,F ,P) is a probability space and (Fn)n≥0 is an increasing family of subσ -algebras ofF , called a filtration, on . Fn is called the σ -algebra of the events prior to time n. Definition 2.2 An adapted stochastic process on a measurable space (E,E ) is a term X = (,F , (Fn)n≥0, (Xn)n≥0,P) where (,F , (Fn)n≥0,P) is a filtered probability space and, for every n, Xn is a r.v. with values in (E,E ) and Fn-measurable (this last property is also expressed by saying that (Xn)n≥0 is adapted to the filtration (Fn)n≥0). •2.2 We give now two examples of those processes that will be our main concern.
