Exact spectral asymptotics of fractional processes

TL;DR

This paper derives exact spectral asymptotics for fractional processes, including fractional Brownian motion and related processes, revealing new effects in their eigenstructure through analytic methods.

Contribution

It provides the first explicit asymptotic analysis of eigenvalues and eigenfunctions for fractional processes beyond Brownian motion.

Findings

Exact asymptotics for fractional Brownian motion eigenvalues and eigenfunctions.

New spectral effects observed in fractional Ornstein-Uhlenbeck and integrated fractional Brownian motion.

Method based on Laplace transform analyticity for spectral asymptotics.

Abstract

Eigenproblems frequently arise in theory and applications of stochastic processes, but only a few have explicit solutions. Those which do, are usually solved by reduction to the generalized Sturm--Liouville theory for differential operators. This includes the Brownian motion and a whole class of processes, which derive from it by means of linear transformations. The more general eigenproblem for the {\em fractional} Brownian motion (f.B.m.) is not solvable in closed form, but the exact asymptotics of its eigenvalues and eigenfunctions can be obtained, using a method based on analytic properties of the Laplace transform. In this paper we consider two processes closely related to the f.B.m.: the fractional Ornstein--Uhlenbeck process and the integrated fractional Brownian motion. While both derive from the f.B.m. by simple linear transformations, the corresponding eigenproblems turn out to be much more complex and their asymptotic structure exhibits new effects.