Geometrical optics of large deviations of fractional Brownian motion

TL;DR

This paper extends the geometrical optics approach to analyze large deviations in fractional Brownian motion, providing new solutions for tail distributions of various functionals and demonstrating its effectiveness through exact and novel results.

Contribution

It develops a formalism for large deviations of fractional Brownian motion, including optimal paths, and applies it to solve previously unsolved tail distribution problems.

Findings

Validated the formalism on known distributions of fBM.

Derived new large-deviation tail distributions for fBM functionals.

Demonstrated the approach's effectiveness for non-Markovian processes.

Abstract

It has been shown recently that the optimal fluctuation method -- essentially geometrical optics -- provides a valuable insight into large deviations of Brownian motion. Here we extend the geometrical optics formalism to two-sided, $-\infty<t<\infty$, fractional Brownian motion (fBM) on the line, which is "pushed" to a large deviation regime by imposed constraints. We test the formalism on three examples where exact solutions are available: the two- and three-point probability distributions of the fBm and the distribution of the area under the fBm on a specified time interval. Then we apply the formalism to several previously unsolved problems by evaluating large-deviation tails of the following distributions: (i) of the first-passage time, (ii) of the maximum of, and (iii) of the area under, fractional Brownian bridge and fractional Brownian excursion, and (iv) of the first-passage area distribution of the fBm. An intrinsic part of a geometrical optics calculation is determination of the optimal path -- the most likely realization of the process which dominates the probability distribution of the conditioned process. Due to the non-Markovian nature of the fBm, the optimal paths of a fBm, subject to constraints on a finite interval $0<t\leq T$, involve both the past $-\infty<t<0$ and the future $T<t<\infty$.