Bessel process

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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 by

X_t = \| W_t \|,

where ||·|| denotes the Euclidean norm in Rn and W is an n-dimensional Wiener process (Brownian motion) started from the origin. The n-dimensional Bessel process is the solution to the stochastic differential equation

dX_t = dZ_t + \frac{n-1}{2}\frac{dt}{X_t}

where Z 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). Since W was assumed to have started from the origin the initial condition is X0 = 0.


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 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 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).


  1. Revuz, D.; Yor, M. (1999). Continuous Martingales and Brownian Motion. Berlin: Springer. ISBN 3-540-52167-4.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  • Øksendal, Bernt (2003). Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer. ISBN 3-540-04758-1.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
  • Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. ISBN 0-471-99705-6.