In mathematics, a Bessel process, named after Friedrich Bessel. The n-dimensional Bessel process is the solution to the stochastic differential equation (SDE)
d
X
t
=
d
W
t
+
n
−
1
2
d
t
X
t
{\displaystyle dX_{t}=dW_{t}+{\frac {n-1}{2}}{\frac {dt}{X_{t}}}}
where W is a 1-dimensional Wiener process (Brownian motion)
Contents
Formal definition
The Bessel process of order n is the real-valued process X given (when n ≥ 2) by
X
t
=
‖
W
t
‖
,
{\displaystyle 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
{\displaystyle 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 > 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 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).