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On stationary solutions of delay differential equations driven by a Lévy process

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  • Gushchin, Alexander A.
  • Küchler, Uwe

Abstract

The stochastic delay differential equationis considered, where Z(t) is a process with independent stationary increments and a is a finite signed measure. We obtain necessary and sufficient conditions for the existence of a stationary solution to this equation in terms of a and the Lévy measure of Z.

Suggested Citation

  • Gushchin, Alexander A. & Küchler, Uwe, 2000. "On stationary solutions of delay differential equations driven by a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 88(2), pages 195-211, August.
    • Handle: RePEc:eee:spapps:v:88:y:2000:i:2:p:195-211
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    Cited by:

    1. Fasen, Vicky, 2006. "Extremes of subexponential Lévy driven moving average processes," Stochastic Processes and their Applications, Elsevier, vol. 116(7), pages 1066-1087, July.
    2. Nielsen, Mikkel Slot, 2020. "On non-stationary solutions to MSDDEs: Representations and the cointegration space," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3154-3173.
    3. Li, Zhi & Long, Qinyi & Xu, Liping & Wen, Xueqi, 2022. "h-stability for stochastic Volterra–Levin equations," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    4. Uwe Küchler & Vyacheslav Vasiliev, 2005. "Sequential Identification of Linear Dynamic Systems with Memory," Statistical Inference for Stochastic Processes, Springer, vol. 8(1), pages 1-24, January.
    5. John Appleby & Markus Riedle & Catherine Swords, 2013. "Bubbles and crashes in a Black–Scholes model with delay," Finance and Stochastics, Springer, vol. 17(1), pages 1-30, January.
    6. Uwe Küchler & Michael Sørensen, 2010. "A simple estimator for discrete-time samples from affine stochastic delay differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 13(2), pages 125-132, June.
    7. Hutt, Axel & Atay, Fatihcan M., 2007. "Spontaneous and evoked activity in extended neural populations with gamma-distributed spatial interactions and transmission delay," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 547-560.
    8. Li, Zhi & Zhang, Wei, 2017. "Stability in distribution of stochastic Volterra–Levin equations," Statistics & Probability Letters, Elsevier, vol. 122(C), pages 20-27.
    9. Küchler, Uwe & Gapeev, Pavel V., 2003. "On Large Deviations in Testing Ornstein-Uhlenbeck Type Models with Delay," SFB 373 Discussion Papers 2003,45, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    10. Basse-O’Connor, Andreas & Nielsen, Mikkel Slot & Pedersen, Jan & Rohde, Victor, 2019. "Multivariate stochastic delay differential equations and CAR representations of CARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 129(10), pages 4119-4143.
    11. Reiß, M. & Riedle, M. & van Gaans, O., 2006. "Delay differential equations driven by Lévy processes: Stationarity and Feller properties," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1409-1432, October.

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