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On a martingale associated to generalized Ornstein-Uhlenbeck processes and an application to finance

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  • Patie, Pierre

Abstract

In this paper we study the two-dimensional joint distribution of the first passage time of a constant level by spectrally negative generalized Ornstein-Uhlenbeck processes and their primitive stopped at this first passage time. By using martingales techniques, we show an explicit expression of the Laplace transform of the distribution in terms of new special functions. Finally, we give an application in finance which consists of computing the Laplace transform of the price of an European call option on the maximum on the yield in the generalized Vasicek model. The stable case is studied in more detail.

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  • Patie, Pierre, 2005. "On a martingale associated to generalized Ornstein-Uhlenbeck processes and an application to finance," Stochastic Processes and their Applications, Elsevier, vol. 115(4), pages 593-607, April.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:4:p:593-607
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    References listed on IDEAS

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    1. Olivier Scaillet & Boris Leblanc, 1998. "Path dependent options on yields in the affine term structure model," Finance and Stochastics, Springer, vol. 2(4), pages 349-367.
    2. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    2. Jacobsen, Martin & Jensen, Anders Tolver, 2007. "Exit times for a class of piecewise exponential Markov processes with two-sided jumps," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1330-1356, September.
    3. Bouasker, O. & Letifi, N. & Prigent, J.-L., 2016. "Optimal funding and hiring/firing policies with mean reverting demand," Economic Modelling, Elsevier, vol. 58(C), pages 569-579.
    4. Foucart, Clément & Vidmar, Matija, 2024. "Continuous-state branching processes with collisions: First passage times and duality," Stochastic Processes and their Applications, Elsevier, vol. 167(C).
    5. Duhalde, Xan & Foucart, Clément & Ma, Chunhua, 2014. "On the hitting times of continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4182-4201.
    6. Czarna, Irmina & Pérez, José-Luis & Rolski, Tomasz & Yamazaki, Kazutoshi, 2019. "Fluctuation theory for level-dependent Lévy risk processes," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5406-5449.
    7. Bankovsky, Damien & Sly, Allan, 2009. "Exact conditions for no ruin for the generalised Ornstein-Uhlenbeck process," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2544-2562, August.
    8. Ma, Rugang, 2015. "Lamperti transformation for continuous-state branching processes with competition and applications," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 11-17.

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