proof of the dΓ©but theorem

Let (β„±)tβˆˆπ•‹ be a right-continuous filtration ( on the measurable spaceMathworldPlanetmathPlanetmath (Ξ©,β„±), It is assumed that 𝕋 is a closed subset of ℝ and that β„±t is universally complete for each tβˆˆπ•‹.

If AβŠ†π•‹Γ—Ξ© is a progressively measurable set, then we show that its dΓ©but


is a stopping time.

As A is progressively measurable, the set A∩((-∞,t)Γ—Ξ©) is ℬ⁒(𝕋)Γ—β„±t-measurable. By the measurable projection theorem it follows that

{D(A)<t}={Ο‰βˆˆΞ©:(s,Ο‰)∈A∩((-∞,t)Γ—Ξ©)Β for someΒ sβˆˆπ•‹}

is in β„±t. If there exists a sequence tnβˆˆπ•‹ with tn>t and tnβ†’t, then


On the other hand, if t is not a right limit point of 𝕋 then


In either case, {D⁒(A)≀t} is in β„±t, so D⁒(A) is a stopping time.

Title proof of the dΓ©but theorem
Canonical name ProofOfTheDebutTheorem
Date of creation 2013-03-22 18:39:15
Last modified on 2013-03-22 18:39:15
Owner gel (22282)
Last modified by gel (22282)
Numerical id 6
Author gel (22282)
Entry type Proof
Classification msc 60G40
Classification msc 60G05