Transformation of Measure Induced by Adapted Shifts

Abstract

In this chapter we consider the following setup: (Ω,F, (F _t , t ∈ [0, 1]), Θ) or, in short (Ω, F,.F.,Θ) is a filtered probability space, i.e., (F _t , t ∈ [0, 1]) is a right continuous, increasing family of sub-sigma fields of. F. We suppose that F ₀ contains all the Θ-negligible sets. In the rest of this chapter such a quadruple will be called a filtered probability space without further precision. W = (W _t, t ∈ [0, 1]) is a one dimensional Wiener process living on (Ω, F, F., Θ), i.e., ω ↦ W _t (ω) is F _t -measurable for all t ∈ [0, 1], for every 0 ≤ t < t+h ≤ 1, (W _t+h − W _t ) is a normal (0, h)-random variable which is independent of. F _t and moreover the paths (W _t , t ∈ [0, 1]) are almost surely continuous. Suppose that α = (α _t , t ∈ [0, 1]) is a measurable and adapted process.