On the martingale property of certain local martingales

@article{Mijatovi2009OnTM,
  title={On the martingale property of certain local martingales},
  author={Aleksandar Mijatovi{\'c} and Mikhail Urusov},
  journal={Probability Theory and Related Fields},
  year={2009},
  volume={152},
  pages={1-30}
}
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… 
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