Bessel 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 Rn and W is an n-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 n started at zero is BES(n).

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 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. .

References

1. Revuz, D.. (1999). "Continuous Martingales and Brownian Motion". Springer.