Moments of generalized fractional polynomial processes
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912928260685824 |
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| author | Assefa, Johannes Keller-Ressel, Martin |
| author_facet | Assefa, Johannes Keller-Ressel, Martin |
| contents | 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 $α$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_13854 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moments of generalized fractional polynomial processes Assefa, Johannes Keller-Ressel, Martin Probability 60G22 (Primary), 60J99, 60K50 (Secondary) 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 $α$. |
| title | Moments of generalized fractional polynomial processes |
| topic | Probability 60G22 (Primary), 60J99, 60K50 (Secondary) |
| url | https://arxiv.org/abs/2501.13854 |