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Moments of generalized fractional polynomial processes

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  • Assefa, Johannes
  • Keller-Ressel, Martin

Abstract

We derive a moment formula for generalized fractional polynomial processes, i.e., for polynomial-preserving Markov processes time-changed by an inverse Lévy-subordinator. If the time change is inverse α-stable, the time-derivative of the Kolmogorov backward equation is replaced by a Caputo fractional derivative of order α, and we demonstrate that moments of such processes are computable, in a closed form, using matrix Mittag-Leffler functions. The same holds true for cross-moments in equilibrium, generalizing results of Leonenko, Meerschaert and Sikorskii from the one-dimensional diffusive case of second-order moments to the multivariate, jump-diffusive case of moments of arbitrary order. We show that also in this more general setting, fractional polynomial processes exhibit long-range dependence, with correlations decaying as a power law with exponent α.

Suggested Citation

  • Assefa, Johannes & Keller-Ressel, Martin, 2026. "Moments of generalized fractional polynomial processes," Stochastic Processes and their Applications, Elsevier, vol. 195(C).
    • Handle: RePEc:eee:spapps:v:195:y:2026:i:c:s0304414926000335
    DOI: 10.1016/j.spa.2026.104901
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    References listed on IDEAS

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    1. Christa Cuchiero & Sara Svaluto-Ferro, 2021. "Infinite-dimensional polynomial processes," Finance and Stochastics, Springer, vol. 25(2), pages 383-426, April.
    2. Christa Cuchiero & Martin Keller-Ressel & Josef Teichmann, 2012. "Polynomial processes and their applications to mathematical finance," Finance and Stochastics, Springer, vol. 16(4), pages 711-740, October.
    3. Cuchiero, Christa & Di Persio, Luca & Guida, Francesco & Svaluto-Ferro, Sara, 2024. "Measure-valued affine and polynomial diffusions," Stochastic Processes and their Applications, Elsevier, vol. 175(C).
    4. Kei Kobayashi, 2011. "Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 24(3), pages 789-820, September.
    5. 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.
    6. Chen, Zhen-Qing, 2017. "Time fractional equations and probabilistic representation," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 168-174.
    7. Filipović, Damir & Larsson, Martin & Pulido, Sergio, 2020. "Markov cubature rules for polynomial processes," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1947-1971.
    8. Christa Cuchiero & Francesco Guida & Luca di Persio & Sara Svaluto-Ferro, 2021. "Measure-valued affine and polynomial diffusions," Papers 2112.15129, arXiv.org.
    9. Gourieroux, Christian & Jasiak, Joann, 2006. "Multivariate Jacobi process with application to smooth transitions," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 475-505.
    10. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    11. Damir Filipović & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Swiss Finance Institute Research Paper Series 17-60, Swiss Finance Institute.
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