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.
Highlights
Studying the maximum and its hitting time for a random process, denoted (M, ρ), is an interest in numerous models in economics, biology, or data science
The law of the maximum of some standard diffusion processes is used in statistics to understand the quality of an estimator
From the agreement formula, we will rigorously derive the above expressions for the densities of M and ρ for all Bessel bridges. Those expressions have been properly proven in [33] (Section 11) by Pitman and Yor for M and Bessel bridges of index ν < −1/2, but our results are a generalization, and our rigorous proof of the expression for the density for ρ seems to be new. To deduce these results we will employ three tools already provided by others: the agreement formula, the Abel summation, and a series expansion of the density of the first hitting time of 1
Summary
Studying the maximum and its hitting time for a random process, denoted (M, ρ), is an interest in numerous models in economics, biology, or data science. Marc Yor have computed the joint density of the place and time of the maximum, (M, ρ), for three constrained Brownian motions: the standard excursion, the Brownian meander, and the standard reflected Brownian bridge Those expressions have been properly proven in [33] (Section 11) by Pitman and Yor for M and Bessel bridges of index ν < −1/2, but our results are a generalization, and our rigorous proof of the expression for the density for ρ seems to be new To deduce these results we will employ three tools already provided by others: the agreement formula, the Abel summation, and a series expansion of the density of the first hitting time of 1.
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