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Some time change representations of stable integrals, via predictable transformations of local martingales

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  • Kallenberg, Olav

Abstract

From the predictable reduction of a marked point process to Poisson, we derive a similar reduction theorem for purely discontinuous martingales to processes with independent increments. Both results are then used to examine the existence of stochastic integrals with respect to stable Lévy processes, and to prove a variety of time change representations for such integrals. The Knight phenomenon, where possibly dependent but orthogonal processes become independent after individual time changes, emerges as a general principle.

Suggested Citation

  • Kallenberg, Olav, 1992. "Some time change representations of stable integrals, via predictable transformations of local martingales," Stochastic Processes and their Applications, Elsevier, vol. 40(2), pages 199-223, March.
    • Handle: RePEc:eee:spapps:v:40:y:1992:i:2:p:199-223
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    Cited by:

    1. Xuekang Zhang & Huisheng Shu & Haoran Yi, 2023. "Parameter Estimation for Ornstein–Uhlenbeck Driven by Ornstein–Uhlenbeck Processes with Small Lévy Noises," Journal of Theoretical Probability, Springer, vol. 36(1), pages 78-98, March.
    2. Shen, Leyi & Xia, Xiaoyu & Yan, Litan, 2022. "Least squares estimation for the linear self-repelling diffusion driven by α-stable motions," Statistics & Probability Letters, Elsevier, vol. 181(C).
    3. Matyas Barczy & Leif Doering & Zenghu Li & Gyula Pap, 2013. "Stationarity and ergodicity for an affine two factor model," Papers 1302.2534, arXiv.org, revised Sep 2013.
    4. Wiktorsson, Magnus, 2002. "Simulation of stochastic integrals with respect to Lévy processes of type G," Stochastic Processes and their Applications, Elsevier, vol. 101(1), pages 113-125, September.
    5. Mijatović, Aleksandar & Uribe Bravo, Gerónimo, 2022. "Limit theorems for local times and applications to SDEs with jumps," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 39-56.
    6. Zhang, Xuekang & Yi, Haoran & Shu, Huisheng, 2019. "Nonparametric estimation of the trend for stochastic differential equations driven by small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 8-16.
    7. Long, Hongwei, 2009. "Least squares estimator for discretely observed Ornstein-Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 79(19), pages 2076-2085, October.
    8. Barndorff-Nielsen, Ole E. & Benth, Fred Espen & Pedersen, Jan & Veraart, Almut E.D., 2014. "On stochastic integration for volatility modulated Lévy-driven Volterra processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 812-847.
    9. Zanzotto, P. A., 1997. "On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion," Stochastic Processes and their Applications, Elsevier, vol. 68(2), pages 209-228, June.
    10. Hu, Yaozhong & Long, Hongwei, 2009. "Least squares estimator for Ornstein-Uhlenbeck processes driven by [alpha]-stable motions," Stochastic Processes and their Applications, Elsevier, vol. 119(8), pages 2465-2480, August.
    11. Shu, Huisheng & Jiang, Ziwei & Zhang, Xuekang, 2023. "Parameter estimation for integrated Ornstein–Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 199(C).

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