Small ball probabilities and large deviations for grey Brownian motion

TL;DR

This paper investigates the small ball probabilities and large deviations of generalized grey Brownian motion, revealing polynomial decay rates and providing asymptotic expansions, with implications for understanding its probabilistic behavior.

Contribution

It extends small ball probability analysis to generalized grey Brownian motion, including asymptotic expansions and large deviations estimates for specific norms.

Findings

Uniform norm of generalized grey Brownian motion has an analytic density.

Small ball probabilities decay polynomially with degree two.

Large deviations estimates are established for uniform and Hölder norms.

Abstract

We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the H\"older norm, we also prove a large deviations estimate.