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Stationary solutions of the stochastic differential equation with Lévy noise

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  • Behme, Anita
  • Lindner, Alexander
  • Maller, Ross

Abstract

For a given bivariate Lévy process (Ut,Lt)t>=0, necessary and sufficient conditions for the existence of a strictly stationary solution of the stochastic differential equation are obtained. Neither strict positivity of the stochastic exponential of U nor independence of V0 and (U,L) is assumed and non-causal solutions may appear. The form of the stationary solution is determined and shown to be unique in distribution, provided it exists. For non-causal solutions, a sufficient condition for U and L to remain semimartingales with respect to the corresponding expanded filtration is given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:1:p:91-108
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    References listed on IDEAS

    as
    1. Jaschke, Stefan, 2003. "A note on the inhomogeneous linear stochastic differential equation," Insurance: Mathematics and Economics, Elsevier, vol. 32(3), pages 461-464, July.
    2. 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.
    3. Gjessing, Håkon K. & Paulsen, Jostein, 1997. "Present value distributions with applications to ruin theory and stochastic equations," Stochastic Processes and their Applications, Elsevier, vol. 71(1), pages 123-144, October.
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    Cited by:

    1. Behme, Anita & Lindner, Alexander, 2012. "Multivariate generalized Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1487-1518.
    2. 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.
    3. Kevei, Péter, 2018. "Ergodic properties of generalized Ornstein–Uhlenbeck processes," Stochastic Processes and their Applications, Elsevier, vol. 128(1), pages 156-181.
    4. 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.

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