A new proof for the conditions of Novikov and Kazamaki

TL;DR

This paper introduces a new proof technique for the conditions ensuring a continuous, nonnegative local martingale is a true martingale, extending classical criteria through Föllmer's measure.

Contribution

It provides a novel proof that generalizes Novikov's and Kazamaki's conditions using Föllmer's measure, broadening their applicability.

Findings

Generalization of Novikov's condition

Extension of Kazamaki's criterion

Martingale property for nonnegative local martingales

Abstract

This paper provides a novel proof for the sufficiency of certain well-known criteria that guarantee the martingale property of a continuous, nonnegative local martingale. More precisely, it is shown that generalizations of Novikov's condition and Kazamaki's criterion follow directly from the existence of F\"ollmer's measure. This approach allows to extend well-known criteria of martingality from strictly positive to only nonnegative, continuous local martingales.