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Integrated stationary Ornstein–Uhlenbeck process, and double integral processes

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  • Abundo, Mario
  • Pirozzi, Enrica

Abstract

We find a representation of the integral of the stationary Ornstein–Uhlenbeck (ISOU) process in terms of Brownian motion Bt; moreover, we show that, under certain conditions on the functions f and g, the double integral process (DIP) D(t)=∫βtg(s)∫αsf(u)dBuds can be thought as the integral of a suitable Gauss–Markov process. Some theoretical and application details are given, among them we provide a simulation formula based on that representation by which sample paths, probability densities and first passage times of the ISOU process are obtained; the first-passage times of the DIP are also studied.

Suggested Citation

  • Abundo, Mario & Pirozzi, Enrica, 2018. "Integrated stationary Ornstein–Uhlenbeck process, and double integral processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 494(C), pages 265-275.
    • Handle: RePEc:eee:phsmap:v:494:y:2018:i:c:p:265-275
    DOI: 10.1016/j.physa.2017.12.043
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    References listed on IDEAS

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    1. Abundo, Mario, 2012. "An inverse first-passage problem for one-dimensional diffusions with random starting point," Statistics & Probability Letters, Elsevier, vol. 82(1), pages 7-14.
    2. Bazzani, Armando & Bassi, Gabriele & Turchetti, Giorgio, 2003. "Diffusion and memory effects for stochastic processes and fractional Langevin equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(3), pages 530-550.
    3. Ole E. Barndorff‐Nielsen & Neil Shephard, 2003. "Integrated OU Processes and Non‐Gaussian OU‐based Stochastic Volatility Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(2), pages 277-295, June.
    4. Aniello Buonocore & Luigia Caputo & Enrica Pirozzi & Luigi M. Ricciardi, 2011. "The First Passage Time Problem for Gauss-Diffusion Processes: Algorithmic Approaches and Applications to LIF Neuronal Model," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 29-57, March.
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    Cited by:

    1. Lefebvre, Mario, 2024. "Exact solution to a first-passage problem for an Ornstein–Uhlenbeck process with jumps and its integral," Statistics & Probability Letters, Elsevier, vol. 205(C).
    2. He, Yue & Kawai, Reiichiro, 2022. "Super- and subdiffusive positions in fractional Klein–Kramers equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 588(C).
    3. Giacomo Ascione & Yuliya Mishura & Enrica Pirozzi, 2021. "Fractional Ornstein-Uhlenbeck Process with Stochastic Forcing, and its Applications," Methodology and Computing in Applied Probability, Springer, vol. 23(1), pages 53-84, March.
    4. Mario Abundo & Enrica Pirozzi, 2019. "On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes," Mathematics, MDPI, vol. 7(10), pages 1-12, October.
    5. Giuseppe D’Onofrio & Claudio Macci & Enrica Pirozzi, 2018. "Asymptotic Results for First-Passage Times of Some Exponential Processes," Methodology and Computing in Applied Probability, Springer, vol. 20(4), pages 1453-1476, December.

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