Local martingale

A local martingale is an adapted right- continuous stochastic process on a filtered probability space, so that an increasing sequence of stopping times exists almost surely, such that for all a martingale.

So there is a localization of the Martingalbegriffs. Local martingales play a role in the theory of stochastic integration, precisely corresponds to the class of potential integrators to semimartingales, sums of local martingales and adapted processes of finite variation.

The concept of local martingale is a far-reaching generalization of the Martingalbegriff, there are examples of uniformly integrable local martingales which are not martingales.