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Stretched non-local Pearson diffusions

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  • Beghin, Luisa
  • Leonenko, Nikolai
  • Papić, Ivan
  • Vaz, Jayme

Abstract

We define a novel class of time-changed Pearson diffusions, termed stretched non-local Pearson diffusions, where the stochastic time-change model has the Kilbas-Saigo function as its Laplace transform. Moreover, we introduce a stretched variant of the Caputo fractional derivative and prove that its eigenfunction is, in fact, the Kilbas-Saigo function. Furthermore, we solve fractional Cauchy problems involving the generator of the Pearson diffusion and the Fokker-Planck operator, providing both analytic and stochastic solutions, which connect the newly defined process and fractional operator with the Kilbas-Saigo function. We also prove that stretched non-local Pearson diffusions share the same limiting distributions as their standard counterparts. Finally, we investigate fractional hyperbolic Cauchy problems for Pearson diffusions, which resemble time-fractional telegraph equations, and provide both analytical and stochastic solutions. As a byproduct of our analysis, we derive a novel representation and an asymptotic formula for the Kilbas-Saigo function with complex arguments, which, to the best of our knowledge, are not currently available in the existing literature.

Suggested Citation

  • Beghin, Luisa & Leonenko, Nikolai & Papić, Ivan & Vaz, Jayme, 2026. "Stretched non-local Pearson diffusions," Stochastic Processes and their Applications, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:spapps:v:195:y:2026:i:c:s0304414925002984
    DOI: 10.1016/j.spa.2025.104854
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    References listed on IDEAS

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    1. Julie Lyng Forman & Michael Sørensen, 2008. "The Pearson Diffusions: A Class of Statistically Tractable Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 438-465, September.
    2. Lotfi Boudabsa & Thomas Simon, 2021. "Some Properties of the Kilbas-Saigo Function," Mathematics, MDPI, vol. 9(3), pages 1-24, January.
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    1. Luisa Beghin & Nikolai Leonenko & Jayme Vaz, 2026. "Relaxation Equations with Stretched Non-local Operators: Renewals and Time-Changed Processes," Journal of Theoretical Probability, Springer, vol. 39(2), pages 1-38, June.

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