mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...

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 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 Euclidean s ...

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 o ...

(

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 pr ...

(SDE) :

dX_t = dW_t + \frac\frac

where W is a 1-dimensional

(

). 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 In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. ...

, 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