proof of Martingale criterion (continuous time)


1. Let X be a martingaleMathworldPlanetmath. By the optional sampling theoremMathworldPlanetmath we have E(Xc|τ)=Xcτ=Xττc. Since conditional expectations are uniformly integrable the first direction follows.

2. Let (τk)k1 be a local sequence of stopping times (i.e. τk a.s. and Xτk martingale k). For each t+ we have XτktXt,k almost surely. The set

{Xτkt:k} {Xτ:τstopping time,τc}

is uniformly integrable (take c=t). It follows that Xtτk1Xt,k. Since the martingale property is stable under 1 convergence, X is a martingale. ∎

Title proof of Martingale criterion (continuous time)
Canonical name ProofOfMartingaleCriterioncontinuousTime
Date of creation 2013-03-22 18:54:28
Last modified on 2013-03-22 18:54:28
Owner karstenb (16623)
Last modified by karstenb (16623)
Numerical id 4
Author karstenb (16623)
Entry type Proof
Classification msc 60G07
Classification msc 60G48