  The stochastic exponential $Z_t=\exp\{M_t-M_0-(1/2) <M,M>_t\}$ 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.
