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. 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.
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 formula_1, which satisfy the following relationship between the Laplace transform and the modified Bessel function of first kind:
formula_2 for formula_3,
where formula_4 denoted the modified Bessel function defined as
formula_5
Explicit representation.
The "unnormalized density" of the Hartman-Watson distribution is
formula_6
for formula_7.
It satisfies the equation
formula_8
The density of the Hartman-Watson distribution is defined on formula_9 and given by
formula_10
or explicitly
formula_11 for formula_12.
Connection to Brownian exponential functionals.
The following result by Yor () establishes a connection between the unnormalized Hartman-Watson density formula_13 and Brownian exponential functionals.
Let formula_14 be a one-dimensional Brownian motion starting in formula_15 with drift formula_16. Let formula_17 be the following Brownian functional
formula_18 for formula_19
Then the distribution of formula_20 for formula_21 is given by
formula_22
where formula_23 und formula_24.
formula_25 is an alternative notation for a probability measure formula_26.