Hartman–Watson distribution

The Hartman-Watson distribution is an absolutely continuous probability distribution which arises in the study of Brownian functionals. 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.^[1] Important contributions to the distribution, such as an explicit form of the density in integral representation and a connection to Brownian exponential functionals, came from Marc Yor.^[2]

In financial mathematics, the distribution is used to compute the prices of Asian options with the Black-Scholes model.

The Hartman-Watson distributions are the probability distributions ${\displaystyle (\mu _{r})_{r>0}}$, which satisfy the following relationship between the Laplace transform and the modified Bessel function of first kind:

${\displaystyle \int _{0}^{\infty }e^{-u^{2}t/2}\mu _{r}(\mathrm {d} t)={\frac {I_{|u|}(r)}{I_{0}(r)}}\quad }$ for ${\displaystyle u\in \mathbb {R} ,\;r>0}$,

where ${\displaystyle I_{\nu }(r)}$ denoted the modified Bessel function defined as

${\displaystyle I_{\nu }(t):=\sum _{n=0}^{\infty }{\frac {({\frac {t}{2}})^{2n+\nu }}{\Gamma (n+\nu +1)n!}}.}$^[3]

The unnormalized density of the Hartman-Watson distribution is

${\displaystyle \vartheta (r,t):={\frac {r}{(2\pi ^{3}t)^{1/2}}}e^{\pi ^{2}/2t}\int _{0}^{\infty }e^{-x^{2}/2t-r\cosh(x)}\sinh(x)\sin \left({\frac {\pi x}{t}}\right)\mathrm {d} x}$

for ${\displaystyle r>0,\;t>0}$.

It satisfies the equation

${\displaystyle \int _{0}^{\infty }e^{-u^{2}t/2}\vartheta (r,t)\mathrm {d} t=I_{|u|}(r)\quad {\text{für}}\;\;r>0.}$^[4]

The density of the Hartman-Watson distribution is defined on ${\displaystyle \mathbb {R} _{+}}$ and given by

${\displaystyle f_{r}(t)={\frac {\vartheta (r,t)}{I_{0}(t)}}\quad {\text{für}}\;\;r>0,\;t\geq 0}$

or explicitly

${\displaystyle f_{r}(t)={\frac {r}{(2\pi ^{3}t)^{1/2}}}{\frac {\exp \left(\pi ^{2}/2t\right)\int _{0}^{\infty }\exp \left(-x^{2}/2t-r\cosh(x)\right)\sinh(x)\sin \left({\frac {\pi x}{t}}\right)\mathrm {d} x}{\sum \limits _{n=0}^{\infty }2^{-2n}t^{2n}/(n!)^{2}}}\quad }$ for ${\displaystyle r>0,\;t\geq 0}$.

The following result by Yor (^[5]) establishes a connection between the unnormalized Hartman-Watson density ${\displaystyle \vartheta (r,t)}$ and Brownian exponential functionals.

Let ${\displaystyle (B_{t}^{(\mu )})_{t\geq 0}:=(B_{t}+\mu t)_{t\geq 0}}$ be a one-dimensional Brownian motion starting in ${\displaystyle 0}$ with drift ${\displaystyle \mu \in \mathbb {R} }$. Let ${\displaystyle A^{(\mu )}:=(A_{t}^{\mu })_{t\geq 0}}$ be the following Brownian functional

${\displaystyle A_{t}^{(\mu )}=\int _{0}^{t}\exp \left(2B_{s}^{(\mu )}\right)\mathrm {d} s\quad }$ for ${\displaystyle \;t\geq 0}$

Then the distribution of ${\displaystyle (A_{t}^{(\mu )},B_{t}^{(\mu )})}$ for ${\displaystyle t>0}$ is given by

${\displaystyle P\left(A_{t}^{(\mu )}\in \mathrm {d} u,B_{t}^{(\mu )}\in \mathrm {d} x\right)=e^{\mu x-\mu ^{2}t/2}\exp \left(-{\frac {1+e^{2x}}{2u}}\right)\vartheta (e^{x}/u,t){\frac {1}{u}}\mathrm {d} u\mathrm {d} x}$

where ${\displaystyle u>0}$ und ${\displaystyle x\in \mathbb {R} }$.^[6]

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