On the time to reach maximum for a variety of constrained Brownian motions

TL;DR

This paper derives the joint probability density of the maximum and the time it occurs for various constrained Brownian motions, providing explicit formulas, analyzing moments and asymptotics, and validating results with simulations.

Contribution

It introduces explicit formulas for the joint distribution of maximum and time for constrained Brownian motions using path integrals and agreement formulae.

Findings

Explicit formulas for P(M,t_m) for excursions, meanders, and reflected bridges.

Analysis of moments and asymptotic behavior of P(t_m).

Theoretical results agree well with numerical simulations.

Abstract

We derive P(M,t_m), the joint probability density of the maximum M and the time t_m at which this maximum is achieved for a class of constrained Brownian motions. In particular, we provide explicit results for excursions, meanders and reflected bridges associated with Brownian motion. By subsequently integrating over M, the marginal density P(t_m) is obtained in each case in the form of a doubly infinite series. For the excursion and meander, we analyse the moments and asymptotic limits of P(t_m) in some detail and show that the theoretical results are in excellent accord with numerical simulations. Our primary method of derivation is based on a path integral technique; however, an alternative approach is also outlined which is founded on certain "agreement formulae" that are encountered more generally in probabilistic studies of Brownian motion processes.