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On the first positive and negative excursion exceeding a given length

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  • Sirovich, Roberta
  • Testa, Luisa

Abstract

For a one-dimensional diffusion process X, we derive the Laplace transform and the moments of the first time at which the age of an excursion above (or below) the level x is longer than u. The result is then illustrated for diffusion processes that are found relevant in applications. In the context of pricing Parisian options, the Brownian motion and the Geometric Brownian motion are considered and the Laplace transform can be made explicit and explicit expression for the moments can be derived. In the context of neuronal modeling, the Ornstein–Uhlenbeck process and the Cox–Ingersoll–Ross process are considered and the Laplace transform and the moments must be approximated by numerical inversion.

Suggested Citation

  • Sirovich, Roberta & Testa, Luisa, 2019. "On the first positive and negative excursion exceeding a given length," Statistics & Probability Letters, Elsevier, vol. 150(C), pages 137-145.
    • Handle: RePEc:eee:stapro:v:150:y:2019:i:c:p:137-145
    DOI: 10.1016/j.spl.2019.03.008
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    References listed on IDEAS

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    1. Angelos Dassios & Jia Wei Lim, 2017. "An Analytical Solution For The Two-Sided Parisian Stopping Time, Its Asymptotics, And The Pricing Of Parisian Options," Mathematical Finance, Wiley Blackwell, vol. 27(2), pages 604-620, April.
    2. Angelos Dassios & Shanle Wu, 2010. "Perturbed Brownian motion and its application to Parisian option pricing," Finance and Stochastics, Springer, vol. 14(3), pages 473-494, September.
    3. Dassios, Angelos & Lim, Jia Wei, 2017. "An analytical solution for the two-sided Parisian stopping time, its asymptotics and the pricing of Parisian options," LSE Research Online Documents on Economics 60154, London School of Economics and Political Science, LSE Library.
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