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On exponential local martingales associated with strong Markov continuous local martingales

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  • Blei, Stefan
  • Engelbert, Hans-Jürgen
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    Abstract

    We investigate integral functionals , t>=0, where m is a nonnegative measure on and LY is the local time of a Wiener process with drift, i.e., Yt=Wt+t, t>=0, with a standard Wiener process W. We give conditions for a.s. convergence and divergence of Tt, t>=0, and T[infinity]. In the second part of the present note we apply these results to exponential local martingales associated with strong Markov continuous local martingales. In terms of the speed measure of a strong Markov continuous local martingale, we state a necessary and sufficient condition for the exponential local martingale associated with a strong Markov continuous local martingale to be a martingale.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 119 (2009)
    Issue (Month): 9 (September)
    Pages: 2859-2880

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    Handle: RePEc:eee:spapps:v:119:y:2009:i:9:p:2859-2880

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    Related research

    Keywords: Continuous local martingales Continuous strong Markov processes Stochastic differential equations Brownian motion Brownian motion with drift Integral functionals 0-1-laws Continuous exponential local martingales Stochastic exponentials Martingale property of stochastic exponentials;

    References

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    1. Tina Hviid Rydberg, 1997. "A note on the existence of unique equivalent martingale measures in a Markovian setting," Finance and Stochastics, Springer, vol. 1(3), pages 251-257.
    2. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48.
    3. Albert N. Shiryaev & Jan Kallsen, 2002. "The cumulant process and Esscher's change of measure," Finance and Stochastics, Springer, vol. 6(4), pages 397-428.
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    Cited by:
    1. Ruf, Johannes, 2013. "A new proof for the conditions of Novikov and Kazamaki," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 404-421.
    2. Mayerhofer, Eberhard & Muhle-Karbe, Johannes & Smirnov, Alexander G., 2011. "A characterization of the martingale property of exponentially affine processes," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 568-582, March.

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