Another Look at the Hartman-Watson Distributions

  • Jacek Jakubowski
  • Maciej WiśniewolskiEmail author
Open Access


The article deals with the Hartman-Watson distributions and presents a new approach to them by investigating a special function u. The function u is strictly related to the distribution of the exponential functional of Brownian motion appearing in the mathematical finance framework. The study of the latter leads to new explicit representations for the function u. One of them is through a new parabolic PDE. Integral relations of convolution type between Hartman-Watson distributions and modified Bessel functions are presented. It turns out that u can be represented as an integral convolution of itself and the modified Bessel function K0. Finally, excursion theory and a subordinator connected to the hyperbolic cosine of Brownian motion are involved in order to obtain yet another representation for u. Possible applications of the resulting explicit formulas are discussed, among others Monte Carlo evaluations of u.


Hartman-Watson distributions Additive functional of Brownian motion Asian options PDE Excursions of Brownian motion Lev́y measure Modified Bessel functions 

Mathematics Subject Classification (2010)

60G40 60G17 91G80 



We would like to thank the anonymous referee for his/her careful reading of the manuscript and valuable remarks.


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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WarsawWarszawaPoland

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