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Kac-Ornstein-Uhlenbeck Processes: Stationary Distributions and Exponential Functionals

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  • Nikita Ratanov

    (Chelyabinsk State University)

Abstract

We study Ornstein-Uhlenbeck processes whose parameters are modulated by an external two-state Markov process. The conditional means of such a process for a given modulation follow an analogue of the Langevin equation, which is controlled by a pair of telegraph processes. Stationary distributions of such processes are described. The relationship between stationary distributions and the distribution of the corresponding exponential functional is also discussed. Of interest is the limiting behaviour of these processes under conditions similar to the Kac scaling. Limiting processes turn out to be different classes of ordinary Ornstein-Uhlenbeck processes.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:metcap:v:24:y:2022:i:4:d:10.1007_s11009-022-09956-z
    DOI: 10.1007/s11009-022-09956-z
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    References listed on IDEAS

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    1. Nikita Ratanov, 2007. "A jump telegraph model for option pricing," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 575-583.
    2. Vasicek, Oldrich, 1977. "An equilibrium characterization of the term structure," Journal of Financial Economics, Elsevier, vol. 5(2), pages 177-188, November.
    3. 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.
    4. Weiss, George H, 2002. "Some applications of persistent random walks and the telegrapher's equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 311(3), pages 381-410.
    5. Vasicek, Oldrich Alfonso, 1977. "Abstract: An Equilibrium Characterization of the Term Structure," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(4), pages 627-627, November.
    6. Masoliver, Jaume & Weiss, George H., 1992. "First passage times for a generalized telegrapher's equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 183(4), pages 537-548.
    7. Nikita Ratanov, 2021. "Ornstein-Uhlenbeck Processes of Bounded Variation," Methodology and Computing in Applied Probability, Springer, vol. 23(3), pages 925-946, September.
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