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 met ...
, 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 ...
process ''X'' given (when ''n'' ≥ 2) by
:
where , , ·, , denotes the
Euclidean norm
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
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 i ...
(
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 ...
(SDE)
:
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 i ...
(
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
(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
ECHELON, originally a secret government code name, is a surveillance program (signals intelligence/SIGINT collection and analysis network) operated by the five signatory states to the UKUSA Security Agreement:Given the 5 dialects that us ...
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