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 ${ℱ}_t$;

4. ${|W(t)|}^2-t$ is a martingale with respect to ${ℱ}_t$.