proof of Martingale criterion

Let (τk)k1 be a localizing sequence of stopping times for X. Then:

Λ{τkn} ΛΩa.s.k,n

since {τkn}k1{τkn}n.

k=1{τkn} =Ωa.s., sinceτk,a.s.

Now assume EXn-<,nn0 (the case EXn+< being analogous).

1) We have EXn-<n.

We proceed by (backward) inductionMathworldPlanetmath. For n=n0 the statement holds.


(Xτk)- =(Xτkn-)nsubmartingale

We have:

{τkn}Xn-1-𝑑P ={τkn}Xτk(n-1)-𝑑P

Where the first to second line is the submartingale property and the last line follows by induction hypothesis.

Using Fatou we get:

Xn-1-𝑑P =limkXn-1-Λ{τkn}dP
lim infkXn-1-Λ{τkn}𝑑P

2) We have Xn1(n).

We have Xτkn+Xn+ a.s., k,n. With Fatou we get:

EXn+ lim infkEXτkn+
=EX0+lim infkEXτkn-
=EX0+lim infkE(j=0n-1Xj-Λ{τk=j}+Xn-Λ{τkn})

With 1) Xn1 follows.


X is a martingale, because XnτkXn a.s. k and:

|Xnτk| j=0n|Xj|1(1-bound)

Thus XnτkL1Xn,kn by bounded convergence theoremMathworldPlanetmath. Hence X must be martingale and we are done. ∎

Title proof of Martingale criterion
Canonical name ProofOfMartingaleCriterion
Date of creation 2013-03-22 18:34:51
Last modified on 2013-03-22 18:34:51
Owner karstenb (16623)
Last modified by karstenb (16623)
Numerical id 5
Author karstenb (16623)
Entry type Proof
Classification msc 60G07