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Asymptotic results for time-changed Lévy processes sampled at hitting times

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  • Rosenbaum, Mathieu
  • Tankov, Peter
Abstract
We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to [epsilon]. For a wide class of Lévy processes, we introduce a renormalization depending on [epsilon], under which the Lévy process converges in law to an [alpha]-stable process as[epsilon] goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.

Suggested Citation

  • Rosenbaum, Mathieu & Tankov, Peter, 2011. "Asymptotic results for time-changed Lévy processes sampled at hitting times," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1607-1632, July.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:7:p:1607-1632
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    References listed on IDEAS

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    Cited by:

    1. José E. Figueroa-López & Sveinn Ólafsson, 2016. "Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility," Finance and Stochastics, Springer, vol. 20(1), pages 219-265, January.
    2. José Figueroa-López & Sveinn Ólafsson, 2016. "Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility," Finance and Stochastics, Springer, vol. 20(1), pages 219-265, January.
    3. Mathieu Rosenbaum & Peter Tankov, 2011. "Asymptotically optimal discretization of hedging strategies with jumps," Papers 1108.5940, arXiv.org, revised Apr 2014.
    4. José E. Figueroa-López & Sveinn Ólafsson, 2016. "Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps," Finance and Stochastics, Springer, vol. 20(4), pages 973-1020, October.
    5. Jos'e E. Figueroa-L'opez & Ruoting Gong & Christian Houdr'e, 2012. "High-order short-time expansions for ATM option prices of exponential L\'evy models," Papers 1208.5520, arXiv.org, revised Apr 2014.
    6. Jos'e E. Figueroa-L'opez & Peter Tankov, 2012. "Small-time asymptotics of stopped L\'evy bridges and simulation schemes with controlled bias," Papers 1203.2355, arXiv.org, revised Jul 2014.
    7. Jos'e E. Figueroa-L'opez & Sveinn 'Olafsson, 2014. "Short-time expansions for close-to-the-money options under a L\'evy jump model with stochastic volatility," Papers 1404.0601, arXiv.org, revised Oct 2014.
    8. Jos'e E. Figueroa-L'opez & Sveinn 'Olafsson, 2015. "Short-time asymptotics for the implied volatility skew under a stochastic volatility model with L\'evy jumps," Papers 1502.02595, arXiv.org, revised Dec 2015.

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