Bessel process

Image:BesselProcess1D.svg

The Bessel process of order n is the real-valued process X given (when n ≥ 2) by

:$X\_t\; =\; \backslash |\; W\_t\; \backslash |,$

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)

:$dX\_t\; =\; dW\_t\; +\; \backslash frac\{n-1\}\{2\}\backslash frac\{dt\}\{X\_t\}$

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).

A notation for the Bessel process of dimension {{mvar|n}} started at zero is {{math|BES{{sub|0}}(n)}}.

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.{{cite book |first=D. |last=Revuz |first2=M. |last2=Yor |title=Continuous Martingales and Brownian Motion |publisher=Springer |location=Berlin |year=1999 |isbn=3-540-52167-4 }}

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

{{Reflist}}

• {{cite book | author=Øksendal, Bernt | title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer |location=Berlin | year=2003 | isbn=3-540-04758-1}}
• Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. {{ISBN|0-471-99705-6}}.

{{Stochastic processes}}

Category:Stochastic processes