Before revision:  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.
After revision:  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.