On the martingale property of certain local martingales

The stochastic exponential ${Z_t= {\rm exp}\{M_t-M_0-(1/2)\langle M,M\rangle_t\}}$ of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where ${M_t=\int_0^t b(Y_u)\,dW_u}$ and Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.