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 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)


 * $$dX_t = dW_t + \frac{n-1}{2}\frac{dt}{X_t}$$

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 $n$ started at zero is $BES0(n)$.

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., Xt &gt; 0 for all t &gt; 0. It is, however, neighbourhood-recurrent for n = 2, meaning that with probability 1, 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).