ON THE MARTINGALE PROPERTY OF CERTAIN LOCAL MARTINGALES

Abstract. The stochastic exponential $Z_t=\exp\{M_t-M_0-(1/2)\langle M,M \rangle_t\}$ 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 financial bubbles in a one-dimensional setting.

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