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On the Martingale Property of Certain Local Martingales

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  • Aleksandar Mijatovic
  • Mikhail Urusov

Abstract

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.

Suggested Citation

  • Aleksandar Mijatovic & Mikhail Urusov, 2009. "On the Martingale Property of Certain Local Martingales," Papers 0905.3701, arXiv.org, revised Oct 2010.
    • Handle: RePEc:arx:papers:0905.3701
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    References listed on IDEAS

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    Cited by:

    1. S. M. Ould Aly, 2013. "Monotonicity Of Prices In Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(03), pages 1-23.
    2. Aleksandar Mijatović & Mikhail Urusov, 2012. "Deterministic criteria for the absence of arbitrage in one-dimensional diffusion models," Finance and Stochastics, Springer, vol. 16(2), pages 225-247, April.
    3. Aleksandar Mijatovi'c & Mikhail Urusov, 2011. "A note on a paper by Wong and Heyde," Papers 1105.3918, arXiv.org.
    4. Rheinländer, Thorsten & Schmutz, Michael, 2013. "Self-dual continuous processes," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1765-1779.
    5. Baldeaux, Jan & Ignatieva, Katja & Platen, Eckhard, 2018. "Detecting money market bubbles," Journal of Banking & Finance, Elsevier, vol. 87(C), pages 369-379.
    6. Thorsten Rheinlander & Michael Schmutz, 2012. "Self-dual continuous processes," Papers 1201.6516, arXiv.org.

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