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.

Hartman-Watson Distribution

Definition

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

\int _{0}^{\infty }e^{-u^{2}t/2}\mu _{r}(\mathrm {d} t)={\frac {I_{|u|}(r)}{I_{0}(r)}}\quad

for

u\in \mathbb {R} ,\;r>0

,

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

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

^[3]

Explicit representation

The unnormalized density of the Hartman-Watson distribution is

\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 $r>0,\;t>0$ .

It satisfies the equation

\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 $\mathbb {R} _{+}$ and given by

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

or explicitly

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

r>0,\;t\geq 0

.

Connection to Brownian exponential functionals

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

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

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

for

\;t\geq 0

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

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

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

References

^ Hartman, Philip; Watson, Geoffrey S. (1974). "Normal" Distribution Functions on Spheres and the Modified Bessel Functions". The Annals of Probability. 2 (4). Institute of Mathematical Statistics: 593 -- 607. doi:10.1214/aop/1176996606.
^ Yor, Marc (1980). "Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson". Z. Wahrscheinlichkeitstheorie verw Gebiete. 53: 71–95. doi:10.1007/BF00531612.
^ Yor, Marc (1980). "Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson". Z. Wahrscheinlichkeitstheorie verw Gebiete. 53: 72. doi:10.1007/BF00531612.
^ Yor, Marc (1980). "Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson". Z. Wahrscheinlichkeitstheorie verw Gebiete. 53: 84–85. doi:10.1007/BF00531612.
^ Yor, Marc (1992). "On Some Exponential Functionals of Brownian Motion". Advances in Applied Probability. 24 (3): 509–531. doi:10.2307/1427477.
^ Matsumoto, Hiroyuki; Yor, Marc (2005). "Exponential functionals of Brownian motion, I: Probability laws at fixed time". Probability Surveys. 2. Institute of Mathematical Statistics and Bernoulli Society: 312–347. arXiv:math/0511517. doi:10.1214/154957805100000159.

[1] Hartman, Philip; Watson, Geoffrey S. (1974). "Normal" Distribution Functions on Spheres and the Modified Bessel Functions". The Annals of Probability. 2 (4). Institute of Mathematical Statistics: 593 -- 607. doi:10.1214/aop/1176996606.

[Yor-2] Yor, Marc (1980). "Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson". Z. Wahrscheinlichkeitstheorie verw Gebiete. 53: 71–95. doi:10.1007/BF00531612.

[3] Yor, Marc (1980). "Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson". Z. Wahrscheinlichkeitstheorie verw Gebiete. 53: 72. doi:10.1007/BF00531612.

[4] Yor, Marc (1980). "Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson". Z. Wahrscheinlichkeitstheorie verw Gebiete. 53: 84–85. doi:10.1007/BF00531612.

[5] Yor, Marc (1992). "On Some Exponential Functionals of Brownian Motion". Advances in Applied Probability. 24 (3): 509–531. doi:10.2307/1427477.

[6] Matsumoto, Hiroyuki; Yor, Marc (2005). "Exponential functionals of Brownian motion, I: Probability laws at fixed time". Probability Surveys. 2. Institute of Mathematical Statistics and Bernoulli Society: 312–347. arXiv:math/0511517. doi:10.1214/154957805100000159.

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Langbahn Team – Weltmeisterschaft