Bessel process

In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process.

Formal definition

The Bessel process of order ''n'' is the real-valued 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 ( Brownian motion).
For any ''n'', the ''n''-dimensional Bessel process is the solution to the stochastic differential equation (SDE)

:$dX_t = dW_t + \frac\frac$
where W is a 1-dimensional Wiener process ( Brownian motion). 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. .

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Stochastic processes