In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process.
Contents

Formal definition 1

Notation 2

In specific dimensions 3

Relationship with Brownian motion 3.1

References 4
Formal definition
The Bessel process of order n is the realvalued process X given by

X_t = \ W_t \,
where · denotes the Euclidean norm in R^{n} and W is an ndimensional Wiener process (Brownian motion) started from the origin. The ndimensional Bessel process is the solution to the stochastic differential equation

dX_t = dZ_t + \frac{n1}{2}\frac{dt}{X_t}
where Z is a 1dimensional Wiener process (Brownian motion). Note that this SDE makes sense for any real parameter n (although the drift term is singular at zero). Since W was assumed to have started from the origin the initial condition is X_{0} = 0.
Notation
A notation for the Bessel process of dimension n' started at zero is BES_{0}(n).
In specific dimensions
For n ≥ 2, the ndimensional Wiener process is transient from its starting point: with probability one, X_{t} > 0 for all t > 0. It is, however, neighbourhoodrecurrent 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 so strong that the process becomes stuck at 0 as soon as it hits 0.
Relationship with Brownian motion
0 and 2dimensional Bessel processes are related to local times of Brownian motion via the RayKnight theorems.^{[1]}
The law of a Brownian motion near xextrema is the law of a 3dimensional Bessel process (theorem of Tanaka).
References

^ Revuz, D.; Yor, M. (1999). Continuous Martingales and Brownian Motion. Berlin: Springer.

Øksendal, Bernt (2003). Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer.

Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. ISBN 0471997056.
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