An extension of the mixed Novikov–Kazamaki condition

Given a continuous local martingale [Formula: see text], the associated stochastic exponential [Formula: see text] is a local martingale, but not necessarily a true martingale. To know whether [Formula: see text] is a true martingale is important for many applications, e.g., if Girsanov’s theorem is applied to perform a change of measure. We give several generalizations of Kazamaki’s results and finally construct a counterexample which does not satisfy the mixed Novikov–Kazamaki condition, but satisfies our conditions.