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On the martingale property of certain local martingales
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  • Published: 31 July 2010

On the martingale property of certain local martingales

  • Aleksandar Mijatović1 &
  • Mikhail Urusov2 

Probability Theory and Related Fields volume 152, pages 1–30 (2012)Cite this article

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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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Author information

Authors and Affiliations

  1. Department of Statistics, University of Warwick, Coventry, CV4 9AL, UK

    Aleksandar Mijatović

  2. Institute of Mathematical Finance, Ulm University, 89081, Ulm, Germany

    Mikhail Urusov

Authors
  1. Aleksandar Mijatović
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  2. Mikhail Urusov
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Corresponding author

Correspondence to Mikhail Urusov.

Additional information

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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  • Received: 07 July 2009

  • Revised: 05 July 2010

  • Published: 31 July 2010

  • Issue Date: February 2012

  • DOI: https://doi.org/10.1007/s00440-010-0314-7

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Keywords

  • Local martingales versus true martingales
  • One-dimensional diffusions
  • Separating times
  • Financial bubbles

Mathematics Subject Classification (2000)

  • 60G44
  • 60G48
  • 60H10
  • 60J60
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