Time and place of the maximum for one-dimensional diffusion bridges and meanders

TL;DR

This paper reviews and extends the analysis of the maximum and its timing for various constrained one-dimensional diffusion processes, including Brownian motions, Bessel processes, and skew Brownian bridges, using series representations and discussing their limitations.

Contribution

It generalizes existing series-based methods for maximum distributions to new diffusion processes like skew Brownian bridges and generalized Bessel meanders.

Findings

Series representations for maximum distributions extended to new diffusion types

Discussion of the applicability limits of the series methods

Connections made between different constrained diffusion processes

Abstract

For three constrained Brownian motions, the excursion, the meander, and the reflected bridge, the densities of the maximum and of the time to reach it were expressed as double series by Majumdar, Randon-Furling, Kearney, and Yor (2008). Some of these series were regularized by Abel summation. Similar results for Bessel processes were obtained by Schehr and Le Doussal (2010) using the real space renormalization group method. Here this work is reviewed, and extended from the point of view of one-dimensional diffusion theory to some other diffusion processes including skew Brownian bridges and generalized Bessel meanders. We discuss the limits of the application of this method for other diffusion processes.