View source for Hartman–Watson distribution

The '''Hartman-Watson distribution''' is an [[absolutely continuous probability distribution]] which arises in the study of [[Brownian motion|Brownian]] [[Functional (mathematics)|functional]]s. It is named after [[Philip Hartman]] and [[Geoffrey S. Watson]], who encountered the distribution while studying the relationship between [[Brownian motion]] on the n-[[sphere]] and the [[von Mises distribution]].<ref>{{cite journal|first1=Philip |last1=Hartman |first2=Geoffrey S. |last2=Watson|title=Normal" Distribution Functions on Spheres and the Modified Bessel Functions|volume=2 |journal=The Annals of Probability|number=4|publisher=Institute of Mathematical Statistics |pages=593 -- 607 |date=1974 |doi=10.1214/aop/1176996606}}</ref> Important contributions to the distribution, such as an explicit form of the density in integral representation and a connection to [[Brownian motion|Brownian]] exponential functionals, came from [[Marc Yor]].<ref name="Yor">{{cite journal|first=Marc |last=Yor |title=Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson |journal=Z. Wahrscheinlichkeitstheorie verw Gebiete |volume=53|pages=71-95|date=1980|doi=10.1007/BF00531612}}</ref>

In [[financial mathematics]], the distribution is used to compute the prices of [[Asian option|Asian options]] with the [[Black-Scholes model]].
== Hartman-Watson Distribution ==
=== Definition ===
The '''Hartman-Watson distributions''' are the probability distributions <math>(\mu_r)_{r>0}</math>, which satisfy the following relationship between the [[Laplace transform]] and the [[Bessel's differential equation#Modified Bessel functions|modified Bessel function]] of first kind:
:<math>\int_0^\infty e^{-u^2t/2}\mu_r(\mathrm{d}t)=\frac{I_{|u|}(r)}{I_0(r)}\quad</math> for <math>u\in \R,\; r>0</math>,
where <math>I_\nu(r)</math> denoted the modified Bessel function defined as
:<math>I_\nu(t) := \sum_{n=0}^\infty \frac{(\frac{t}{2})^{2n+\nu}}{\Gamma(n+\nu+1)n!}.</math><ref>{{cite journal|first=Marc |last=Yor |title=Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson |journal=Z. Wahrscheinlichkeitstheorie verw Gebiete |volume=53|pages=72|date=1980|doi=10.1007/BF00531612}}</ref>

=== Explicit representation ===
The ''unnormalized density'' of the Hartman-Watson distribution is
:<math>\vartheta(r,t):=\frac{r}{(2\pi^3 t)^{1/2}}e^{\pi^2/2t}\int_0^{\infty}e^{-x^2/2t-r\cosh(x)}\sinh(x)\sin\left(\frac{\pi x}{t}\right)\mathrm{d}x</math>
for <math>r>0,\;t>0</math>.

It satisfies the equation
:<math>\int_0^\infty e^{-u^2t/2}\vartheta(r,t)\mathrm{d}t=I_{|u|}(r) \quad \text{für}\;\;r>0.</math><ref>{{cite journal|first=Marc |last=Yor |title=Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson |journal=Z. Wahrscheinlichkeitstheorie verw Gebiete |volume=53|pages=84-85|date=1980|doi=10.1007/BF00531612}}</ref>

The density of the Hartman-Watson distribution is defined on <math>\mathbb{R}_+</math> and given by
:<math>f_r(t)=\frac{\vartheta(r,t)}{I_0(t)}\quad \text{für}\;\;r>0,\;t\geq 0</math>
or explicitly
:<math>f_r(t)=\frac{r}{(2\pi^3 t)^{1/2}}\frac{\exp\left(\pi^2/2t\right)\int_0^{\infty}\exp\left(-x^2/2t-r\cosh(x)\right)\sinh(x)\sin\left(\frac{\pi x}{t}\right)\mathrm{d}x}{
\sum\limits_{n=0}^\infty 2^{-2n}t^{2n}/(n!)^2}\quad</math> for <math>r>0,\;t\geq 0</math>.

== Connection to Brownian exponential functionals ==
The following result by Yor (<ref>{{cite journal |first=Marc |last=Yor|title=On Some Exponential Functionals of Brownian Motion|journal=Advances in Applied Probability |volume=24 |number=3 |date=1992 |pages=509–531 |doi=10.2307/1427477}}</ref>) establishes a connection between the unnormalized Hartman-Watson density <math>\vartheta(r,t)</math> and [[Brownian motion|Brownian]] exponential functionals.

Let <math>(B_t^{(\mu)})_{t\geq 0}:=(B_t+\mu t)_{t\geq 0}</math> be a one-dimensional [[Brownian motion]] starting in <math>0</math> with drift <math>\mu\in\R</math>. Let <math>A^{(\mu)}:=(A^{\mu}_t)_{t\geq 0}</math> be the following Brownian functional
:<math>A^{(\mu)}_t=\int_0^t\exp\left(2B_s^{(\mu)}\right)\mathrm{d}s\quad</math> for <math>\;t\geq 0</math>
Then the distribution of <math>(A^{(\mu)}_t,B^{(\mu)}_t)</math> for <math>t>0</math> is given by
:<math>P\left(A^{(\mu)}_t\in \mathrm{d}u,B^{(\mu)}_t\in \mathrm{d}x\right)=e^{\mu x-\mu^2t/2}\exp\left(-\frac{1+e^{2x}}{2u}\right)\vartheta(e^{x}/u,t)\frac{1}{u}\mathrm{d}u \mathrm{d}x</math>
where <math>u>0</math> und <math>x\in\R</math>.<ref>{{cite journal|first1=Hiroyuki |last1=Matsumoto |first2=Marc |last2=Yor|title=Exponential functionals of Brownian motion, I: Probability laws at fixed time|volume=2 |journal=Probability Surveys|publisher=Institute of Mathematical Statistics and Bernoulli Society |pages=312 - 347 |date=2005 |doi=10.1214/154957805100000159|arxiv=math/0511517 }}</ref>

<math>P\left(X\in \mathrm{d}x,Y\in \mathrm{d}y\right)</math> is an alternative notation for a probability measure <math>\lambda(dx,dy)</math>.

== References ==
{{reflist}}

{{ProbDistributions|continuous-semi-infinite}}
[[Category:Continuous distributions]]