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.
This research was supported by a grant from the Ecole Normale Supérieure, Paris which gave me the opportunity to visit the Department of Statistics at the University of California, Berkeley from April to July 2018. The research was done under the supervision of Jim Pitman. I am grateful to him for proposing this work, for always being patient and helpful, and for introducing me to different aspects of mathematics. I also thank Satya Majumdar and Michael Kearney for their advice and for drawing my attention to . Finally, I thank an anonymous reviewer for helping me to improve this paper.
"Time and place of the maximum for one-dimensional diffusion bridges and meanders." Probab. Surveys 18 1 - 43, 2021. https://doi.org/10.1214/18-PS312