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Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes

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  • Lindner, Alexander
  • Maller, Ross

Abstract

The generalised Ornstein-Uhlenbeck process constructed from a bivariate Lévy process ([xi]t,[eta]t)t[greater-or-equal, slanted]0 is defined aswhere V0 is an independent starting random variable. The stationarity of the process is closely related to the convergence or divergence of the Lévy integral . We make precise this relation in the general case, showing that the conditions are not in general equivalent, though they are for example if [xi] and [eta] are independent. Characterisations are expressed in terms of the Lévy measure of ([xi],[eta]). Conditions for the moments of the strictly stationary distribution to be finite are given, and the autocovariance function and the heavy-tailed behaviour of the stationary solution are also studied.

Suggested Citation

  • Lindner, Alexander & Maller, Ross, 2005. "Lévy integrals and the stationarity of generalised Ornstein-Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 115(10), pages 1701-1722, October.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:10:p:1701-1722
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    Cited by:

    1. Behme, Anita & Lindner, Alexander & Reker, Jana & Rivero, Victor, 2021. "Continuity properties and the support of killed exponential functionals," Stochastic Processes and their Applications, Elsevier, vol. 140(C), pages 115-146.
    2. Ernstsen, Rune Ramsdal & Boomsma, Trine Krogh, 2018. "Valuation of power plants," European Journal of Operational Research, Elsevier, vol. 266(3), pages 1153-1174.
    3. Nikita Ratanov, 2022. "Kac-Ornstein-Uhlenbeck Processes: Stationary Distributions and Exponential Functionals," Methodology and Computing in Applied Probability, Springer, vol. 24(4), pages 2703-2721, December.
    4. Anita Behme & Alexander Lindner, 2015. "On Exponential Functionals of Lévy Processes," Journal of Theoretical Probability, Springer, vol. 28(2), pages 681-720, June.
    5. Bertoin, Jean, 2019. "Ergodic aspects of some Ornstein–Uhlenbeck type processes related to Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1443-1454.
    6. Behme, Anita & Di Tella, Paolo & Sideris, Apostolos, 2024. "On moments of integrals with respect to Markov additive processes and of Markov modulated generalized Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    7. Behme, Anita & Lindner, Alexander, 2012. "Multivariate generalized Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1487-1518.
    8. Endo, Kotaro & Matsui, Muneya, 2008. "The stationarity of multidimensional generalized Ornstein-Uhlenbeck processes," Statistics & Probability Letters, Elsevier, vol. 78(14), pages 2265-2272, October.
    9. Kostadinova, Radostina, 2007. "Optimal investment for insurers when the stock price follows an exponential Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 250-263, September.
    10. Behme, Anita & Chong, Carsten & Klüppelberg, Claudia, 2015. "Superposition of COGARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1426-1469.
    11. Kevei, Péter, 2018. "Ergodic properties of generalized Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 156-181.
    12. Behme, Anita & Lindner, Alexander & Maller, Ross, 2011. "Stationary solutions of the stochastic differential equation with Lévy noise," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 91-108, January.
    13. 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.
    14. Brandes, Dirk-Philip & Lindner, Alexander, 2014. "Non-causal strictly stationary solutions of random recurrence equations," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 113-118.
    15. Bankovsky, Damien, 2010. "Conditions for certain ruin for the generalised Ornstein-Uhlenbeck process and the structure of the upper and lower bounds," Stochastic Processes and their Applications, Elsevier, vol. 120(2), pages 255-280, February.
    16. Klüppelberg, Claudia & Kostadinova, Radostina, 2008. "Integrated insurance risk models with exponential Lévy investment," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 560-577, April.

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