Persistence probabilities of mixed FBM and other mixed processes

TL;DR

This paper analyzes the persistence probabilities of sums of two self-similar Gaussian processes, showing that their asymptotic behavior is dominated by the process with the higher self-similarity index, including mixed fractional Brownian motion.

Contribution

It establishes the asymptotic decay rate of persistence probabilities for sums of mixed Gaussian processes, extending results to mixed fractional Brownian motion.

Findings

Persistence probability decays polynomially with exponent 1 - max(1/2, H).

Asymptotic behavior is governed by the process with the larger self-similarity index.

Results apply to mixed fractional Brownian motion, confirming conjectures about their persistence.

Abstract

We consider the sum of two self-similar centred Gaussian processes with different self-similarity indices. Under non-negativity assumptions of covariance functions and some further minor conditions, we show that the asymptotic behaviour of the persistence probability of the sum is the same as for the single process with the greater self-similarity index. In particular, this covers the mixed fractional Brownian motion introduced in Cheridito (2001) and shows that the corresponding persistence probability decays asymptotically polynomially with persistence exponent $1-\max(1/2,H),$ where $H$ is the Hurst parameter of the underlying fractional Brownian motion.