Abstract
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 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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We are grateful to Peter Bank, Nicholas Bingham, Mark Davis, Yuri Kabanov, Ioannis Karatzas, Walter Schachermayer, and two anonymous referees for valuable suggestions. This paper was written while the second author was a postdoc in the Deutsche Bank Quantitative Products Laboratory, Berlin.
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Mijatović, A., Urusov, M. On the martingale property of certain local martingales. Probab. Theory Relat. Fields 152, 1–30 (2012). https://doi.org/10.1007/s00440-010-0314-7
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DOI: https://doi.org/10.1007/s00440-010-0314-7
Keywords
- Local martingales versus true martingales
- One-dimensional diffusions
- Separating times
- Financial bubbles