Another Look at the Hartman-Watson Distributions

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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.


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

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Correspondence to Maciej Wiśniewolski.

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Jakubowski, J., Wiśniewolski, M. Another Look at the Hartman-Watson Distributions. Potential Anal 53, 1269–1297 (2020).

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  • 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