Bessel process explained

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 Rⁿ 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} + (n -1)/(2) (dt)/(X _{t}) 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.^[1]

The law of a Brownian motion near x-extrema is the law of a 3-dimensional Bessel process (theorem of Tanaka).

References

• Book: Øksendal, Bernt . Stochastic Differential Equations: An Introduction with Applications . Springer . Berlin . 2003 . 3-540-04758-1.
• Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. .

Notes and References

1. Book: Revuz, D. . M. . Yor . Continuous Martingales and Brownian Motion . Springer . Berlin . 1999 . 3-540-52167-4 .