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Hartman–Watson distribution
Probability distribution related to Brownian motion From Wikipedia, the free encyclopedia
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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.
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Hartman–Watson distribution
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Definition
The Hartman–Watson distributions are the probability distributions , which satisfy the following relationship between the Laplace transform and the modified Bessel function of first kind:
- for ,
where denoted the modified Bessel function defined as
Explicit representation
The unnormalized density of the Hartman-Watson distribution is
for .
It satisfies the equation
The density of the Hartman-Watson distribution is defined on and given by
or explicitly
- for .
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Connection to Brownian exponential functionals
The following result by Yor ([5]) establishes a connection between the unnormalized Hartman-Watson density and Brownian exponential functionals.
Let be a one-dimensional Brownian motion starting in with drift . Let be the following Brownian functional
- for
Then the distribution of for is given by
where und .[6]
is an alternative notation for a probability measure .
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References
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