On the Martingale Property of Certain Local Martingales

Details On the Martingale Property of Certain Local Martingales On the Martingale Property of Certain Local Martingales

2009-05-22

arxiv.org/abs/0905.3701v3

Mathematics

Probability

Appendix on local time of diffusions added; 27 pages, 1 figure; to appear in PTRF

Scientific paper

The stochastic exponential $Z_t=\exp\{M_t-M_0-(1/2) _t\}$ 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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