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Volterra-type Ornstein–Uhlenbeck processes in space and time

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  • Pham, Viet Son
  • Chong, Carsten

Abstract

We propose a novel class of temporo-spatial Ornstein–Uhlenbeck processes as solutions to Lévy-driven Volterra equations with additive noise and multiplicative drift. After formulating conditions for the existence and uniqueness of solutions, we derive an explicit solution formula and discuss distributional properties such as stationarity, second-order structure and short versus long memory. Furthermore, we analyze in detail the path properties of the solution process. In particular, we introduce different notions of càdlàg paths in space and time and establish conditions for the existence of versions with these regularity properties. The theoretical results are accompanied by illustrative examples.

Suggested Citation

  • Pham, Viet Son & Chong, Carsten, 2018. "Volterra-type Ornstein–Uhlenbeck processes in space and time," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 3082-3117.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:9:p:3082-3117
    DOI: 10.1016/j.spa.2017.10.012
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    1. Peter J. Brockwell & Yasumasa Matsuda, 2017. "Continuous auto-regressive moving average random fields on ℝ-super-n," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 833-857, June.
    2. Stefan Tappe, 2012. "Some Refinements of Existence Results for SPDEs Driven by Wiener Processes and Poisson Random Measures," International Journal of Stochastic Analysis, Hindawi, vol. 2012, pages 1-24, December.
    3. Chong, Carsten, 2017. "Stochastic PDEs with heavy-tailed noise," Stochastic Processes and their Applications, Elsevier, vol. 127(7), pages 2262-2280.
    4. P. Brockwell, 2001. "Lévy-Driven Carma Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(1), pages 113-124, March.
    5. Michele Nguyen & Almut E. D. Veraart, 2017. "Spatio-temporal Ornstein–Uhlenbeck Processes: Theory, Simulation and Statistical Inference," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(1), pages 46-80, March.
    6. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    Cited by:

    1. Pham, Viet Son, 2020. "Lévy-driven causal CARMA random fields," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7547-7574.

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