Survival exponents for fractional Brownian motion with multivariate time

TL;DR

This paper derives the asymptotic probability decay of multivariate fractional Brownian motion remaining below a fixed level within a spherical domain, revealing a dependence on the Hurst index and dimension.

Contribution

It establishes the log-asymptotics of the non-exceedance probability for multivariate fractional Brownian motion in spherical domains, extending understanding of its boundary behavior.

Findings

Probability decays as (H-d)logT for large T

Asymptotic behavior depends on Hurst index and dimension

Provides a precise asymptotic formula for boundary crossing probabilities

Abstract

Fractional Brownian motion, H-FBM , of index with d-dimensional time is considered in a spherical domain that contains 0 at its boundary. The main result : the log-asymptotics of probability that H-FBM does not exceed a fixed positive level is (H-d)logT(1+o(1)), where T>>1 is radius of the domain.