Moments of generalized fractional polynomial processes

TL;DR

This work develops a framework for moments of generalized fractional polynomial processes obtained by time-changing polynomial Markov processes with inverse Lévy subordinators. When the time change is $α$-stable, the Kolmogorov backward equation becomes a Caputo fractional equation and moments admit closed-form expressions via matrix Mittag-Leffler functions, with cross-moments in equilibrium and a clear long-range dependence structure due to the heavy-tailed time change. The authors extend standard polynomial-process moment formulas to multivariate jump-diffusions under time change, providing explicit representations such as $\mathbb{E}_x[p(X_{L_t})]=H(x)^T E_α(t^α A)\vec{p}$ and $\mathbb{E}_μ[p(X_{t+s}) q(X_t)]=v^T \overrightarrow{\mathbf{m}}(\vec{q}, e^{sA}\vec{p})$, along with equilibrium correlations that decay as a power law in $t$ with exponent $α$. They also propose a conjecture on state-dependent subordination, illustrating a Volterra-system-based polynomial solution that hints at polynomial preservation under non-Markovian, state-dependent time changes, and outlining directions for constructing semi-Markov analogues of fractional polynomial dynamics.

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 $α$.

Theorems & Definitions (43)

• Definition 1
• Lemma 2
• Lemma 3
• proof
• Proposition 4
• Remark 5
• proof
• Example 6
• Lemma 7
• proof