Mixed fractional Brownian motion: a spectral take

TL;DR

This paper analyzes mixed fractional Brownian motion using spectral methods, deriving eigenvalue approximations and exact small ball probabilities, revealing scale stratification related to the Hurst parameter.

Contribution

It introduces a spectral approach to mixed fractional Brownian motion, providing second-order eigenvalue approximations and exact small ball probability calculations.

Findings

Eigenvalues approximated asymptotically up to second order

Exact $L_2$-small ball probabilities computed

Scale stratification linked to Hurst parameter

Abstract

This paper provides yet another look at the mixed fractional Brownian motion (fBm), this time, from the spectral perspective. We derive an approximation for the eigenvalues of its covariance operator, asymptotically accurate up to the second order. This in turn allows to compute the exact $L_2$-small ball probabilities, previously known only at logarithmic precision. The obtained expressions show an interesting stratification of scales, which occurs at certain values of the Hurst parameter of the fractional component. Some of them have been previously encountered in other problems involving such mixtures.