2025/01/23 by Assefa, Johannes, Keller-Ressel, Martin
#60G22 (Primary) #60J99 #60K50 (Secondary) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2501.13854
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.