Levy martingale characterization
Theorem (Levy’s martingale characterisation).
Let $W\mathit{}\mathrm{(}t\mathrm{)}\mathrm{,}t\mathrm{\ge}\mathrm{0}$, be a stochastic process^{} and let ${\mathrm{F}}_{t}\mathrm{=}\sigma \mathrm{(}{W}_{s}\mathrm{,}s\mathrm{\le}t\mathrm{)}$ be the filtration^{} generated by it. Then $W\mathit{}\mathrm{(}t\mathrm{)}$ is a Wiener process^{} if and only if the following conditions hold:

1.
$W(0)=0$ almost surely;

2.
The sample paths $t\mapsto W(t)$ are continuous almost surely;

3.
$W(t)$ is a martingale^{} with respect to the filtration ${\mathcal{F}}_{t}$;

4.
${W(t)}^{2}t$ is a martingale with respect to ${\mathcal{F}}_{t}$.
Title  Levy martingale characterization 

Canonical name  LevyMartingaleCharacterization 
Date of creation  20130322 15:12:48 
Last modified on  20130322 15:12:48 
Owner  skubeedooo (5401) 
Last modified by  skubeedooo (5401) 
Numerical id  5 
Author  skubeedooo (5401) 
Entry type  Theorem 
Classification  msc 60J65 