In mathematics, a Bessel process, named after Friedrich Bessel. The n-dimensional Bessel process is the solution to the stochastic differential equation (SDE)

where W is a 1-dimensional Wiener process (Brownian motion)

Formal definition

The Bessel process of order n is the real-valued process X given (when n≥2) by

where ||·|| denotes the Euclidean norm in Rn and W is an n-dimensional Wiener process (Brownian motion (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., Xt0 for all t0. It is, however, neighbourhood-recurrent for n=2, meaning that with probability1, for any r>0, there are arbitrarily large t with Xt<r; on the other hand, it is truly transient for n>2, meaning that Xt≥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. .