Abstract
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.
Highlights
The recent paper of Lyasoff [17] has shed a new light on the long studied subject of finding the distribution of an important additive functional of geometric Brownian motion, that is At = t 0e2Bu d u, where (Bt, t ≥ 0)is a real typicalBrownian motion, and on Hartman-Watson distributions
Our goal is to study the probabilistic nature of HW distributions, to find their connections with modified Bessel functions I and K and to provide a new numerically tractable description of HW distributions
We discover that g(t, ·) satisfies the following parabolic partial differential equation (PDE): g(t, x) ∂t
Summary
The recent paper of Lyasoff [17] has shed a new light on the long studied subject of finding the distribution of an important additive functional of geometric Brownian motion, that is. Yor [28] established the distribution of At by applying the so-called Lamperti relation, which connects geometric Brownian motion, the additive functional A and an appropriate (squared) Bessel process This resulted in finding the density of the joint distribution of the vector (e2Bt , At ). We prove that the function u defined by x u(t, x) = I0(x − y) is another solution of the parabolic PDE satisfied by u above, and for this solution we are able to produce some identities, analogous to these obtained earlier for u We show yet another application of the time–space convolution identity. Using it and the theory of Volterra integral equations we obtain a new description of the modified Bessel functions In for n ∈ N (Theorem 5.10). It turns out that all In depend on I0, I0 and on some well defined generating sequences
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