Criteria for what makes a local optional martingale a true martingale

What makes an optional stochastic exponential a true optional martingale in a probability space where the underlying filtration is not right continuous nor complete. In this paper, we are going to answer this fundamental question. True martingales play an important role in many applications, just consider the large field of the fair pricing of derivative contracts. Therefore, it is important to give a natural, simple and verifiable condition for having a true martingale property in the context of the theory of optional martingales. The way we have done this here is by extending the method of Beneš to local optional martingales, giving a list of conditions for when an optional stochastic exponential of a local optional martingale is a true optional martingale.