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.
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.
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