# Levy martingale characterization

###### Theorem (Levy’s martingale characterisation).

Let $W(t),t\geq 0$, be a stochastic process and let $\mathcal{F}_{t}=\sigma(W_{s},s\leq t)$ be the filtration generated by it. Then $W(t)$ is a Wiener process if and only if the following conditions hold:

1. 1.

$W(0)=0$ almost surely;

2. 2.

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

3. 3.

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

4. 4.

$|W(t)|^{2}-t$ is a martingale with respect to $\mathcal{F}_{t}$.

Title Levy martingale characterization LevyMartingaleCharacterization 2013-03-22 15:12:48 2013-03-22 15:12:48 skubeedooo (5401) skubeedooo (5401) 5 skubeedooo (5401) Theorem msc 60J65