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Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 2, Pages 340–358 (Mi tvp288)  

This article is cited in 20 scientific papers (total in 20 papers)

Martingales and first passage times for Ornstein–Uhlenbeck processes with a jump component

A. A. Novikov

University of Technology, Sydney

Abstract: Using martingale technique, we show that a distribution of the first-passage time over a level for the Ornstein–Uhlenbeck process with jumps is exponentially bounded. In the case of absence of positive jumps, the Laplace transform for this passage time is found. Further, the maximal inequalities are also given when the marginal distribution is stable.

Keywords: exponential martingales, first-passage times, Ornstein–Uhlenbeck process, Laplace transform, moment Wald's identity, maximal inequalities, stable distribution.


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Theory of Probability and its Applications, 2004, 48:2, 288–303

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Received: 23.01.2003

Citation: A. A. Novikov, “Martingales and first passage times for Ornstein–Uhlenbeck processes with a jump component”, Teor. Veroyatnost. i Primenen., 48:2 (2003), 340–358; Theory Probab. Appl., 48:2 (2004), 288–303

Citation in format AMSBIB
\by A.~A.~Novikov
\paper Martingales and first passage times for Ornstein--Uhlenbeck processes with a jump component
\jour Teor. Veroyatnost. i Primenen.
\yr 2003
\vol 48
\issue 2
\pages 340--358
\jour Theory Probab. Appl.
\yr 2004
\vol 48
\issue 2
\pages 288--303

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    This publication is cited in the following articles:
    1. Patie P., “On a martingale associated to generalized Ornstein–Uhlenbeck processes and an application to finance”, Stochastic Process. Appl., 115:4 (2005), 593–607  crossref  mathscinet  zmath  isi  scopus
    2. Novikov A., Melchers R.E., Shinjikashvili E., Kordzakhia N., “First passage time of filtered Poisson process with exponential shape function”, Probabilistic Engineering Mechanics, 20:1 (2005), 57–65  crossref  isi  elib  scopus
    3. Jacobsen M., Jensen A.T., “Exit times for a class of piecewise exponential Markov processes with two–sided jumps”, Stochastic Process. Appl., 117:9 (2007), 1330–1356  crossref  mathscinet  zmath  isi  elib  scopus
    4. Jakubowski T., “The estimates of the mean first exit time from a ball for the alpha-stable Ornstein–Uhlenbeck processes”, Stochastic Process. Appl., 117:10 (2007), 1540–1560  crossref  mathscinet  zmath  isi  scopus
    5. A. A. Novikov, “On Distributions of First Passage Times and Optimal Stopping of AR(1) Sequences”, Theory Probab. Appl., 53:3 (2009), 419–429  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. Avram F., Usabel M., “The Gerber-Shiu expected discounted penalty-reward function under an affine jump-diffusion model”, Astin Bull., 38:2 (2008), 461–481  crossref  mathscinet  zmath  isi  scopus
    7. Novikov A., Kordzakhia N., “Martingales and first passage times of AR(1) sequences”, Stochastics, 80:2-3 (2008), 197–210  crossref  mathscinet  zmath  isi  scopus
    8. Borovkov K., Novikov A., “On exit times of Levy-driven Ornstein–Uhlenbeck processes”, Statist. Probab. Lett., 78:12 (2008), 1517–1525  crossref  mathscinet  zmath  isi  scopus
    9. Xing X., Zhang W., Wang Y., “The stationary distributions of two classes of reflected Ornstein–Uhlenbeck processes”, J. Appl. Probab., 46:3 (2009), 709–720  crossref  mathscinet  zmath  isi  elib  scopus
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    11. Abbring J.H., “Mixed Hitting-Time Models”, Econometrica, 80:2 (2012), 783–819  crossref  mathscinet  zmath  isi  elib  scopus
    12. Bo L., “First Passage Times of Reflected Ornstein–Uhlenbeck Processes with Two-Sided Jumps”, Queueing Syst., 73:1 (2013), 105–118  crossref  mathscinet  zmath  isi  scopus
    13. Bo L., Ren G., Wang Y., Yang X., “First Passage Times of Reflected Generalized Ornstein–Uhlenbeck Processes”, Stoch. Dyn., 13:1 (2013), 1250014  crossref  mathscinet  zmath  isi  elib  scopus
    14. Graczyk P., Jakubowski T., Luks T., “Martin Representation and Relative Fatou Theorem for Fractional Laplacian with a Gradient Perturbation”, Positivity, 17:4 (2013), 1043–1070  crossref  mathscinet  zmath  isi  scopus
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    18. Ma R., “Lamperti Transformation For Continuous-State Branching Processes With Competition and Applications”, Stat. Probab. Lett., 107 (2015), 11–17  crossref  mathscinet  zmath  isi  scopus
    19. Zhou J., Wu L., Bai Ya., “Occupation Times of Levy-Driven Ornstein–Uhlenbeck Processes With Two-Sided Exponential Jumps and Applications”, Stat. Probab. Lett., 125 (2017), 80–90  crossref  mathscinet  zmath  isi  scopus
    20. Ernstsen R.R., Boomsma T.K., “Valuation of Power Plants”, Eur. J. Oper. Res., 266:3 (2018), 1153–1174  crossref  mathscinet  zmath  isi  scopus
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