Bessel process
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In mathematics, a Bessel process, named after
Friedrich Bessel Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method ...
, is a type of stochastic process.


Formal definition

The Bessel process of order ''n'' is the
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
process ''X'' given (when ''n'' ≥ 2) by :X_t = \, W_t \, , where , , ·, , denotes the Euclidean norm in R''n'' and ''W'' is an ''n''-dimensional
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
(
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
). For any ''n'', the ''n''-dimensional Bessel process is the solution to the
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
(SDE) :dX_t = dW_t + \frac\frac where W is a 1-dimensional
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
(
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
). Note that this SDE makes sense for any real parameter n (although the drift term is singular at zero).


Notation

A notation for the Bessel process of dimension started at zero is .


In specific dimensions

For ''n'' ≥ 2, the ''n''-dimensional Wiener process started at the origin is transient from its starting point: with probability one, i.e., ''X''''t'' > 0 for all ''t'' > 0. It is, however, neighbourhood-recurrent for ''n'' = 2, meaning that with probability 1, for any ''r'' > 0, there are arbitrarily large ''t'' with ''X''''t'' < ''r''; on the other hand, it is truly transient for ''n'' > 2, meaning that ''X''''t'' ≥ ''r'' for all ''t'' sufficiently large. For ''n'' ≤ 0, the Bessel process is usually started at points other than 0, since the drift to 0 is so strong that the process becomes stuck at 0 as soon as it hits 0.


Relationship with Brownian motion

0- and 2-dimensional Bessel processes are related to local times of Brownian motion via the Ray–Knight theorems. The law of a Brownian motion near x-extrema is the law of a 3-dimensional Bessel process (theorem of Tanaka).


References

* *Williams D. (1979) ''Diffusions, Markov Processes and Martingales, Volume 1 : Foundations.'' Wiley. . {{Stochastic processes Stochastic processes