On the martingale property of stochastic exponentials

We present a necessary and sufficient condition for a stochastic exponential to be a true martingale. It is proved that the criteria for the true martingale property are related to whether a related process explodes. An alternative and interesting interpretation of this result is that the stochastic exponential is a true martingale if and only if under a 'candidate measure' the integrand process is square integrable over time. Applications of our theorem to problems arising in mathematical finance are also given.