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| In [[mathematics]], a '''Bessel process''', named after [[Friedrich Bessel]],<!-- Why is it named after him? --> is a type of [[stochastic process]]. The Bessel process of order ''n'' is the [[real number|real-valued]] process ''X'' given by
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| :<math>X_t = \| W_t \|,</math>
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| where ||·|| denotes the [[Norm (mathematics)#Euclidean norm|Euclidean norm]] in '''R'''<sup>''n''</sup> 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]]
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| :<math>dX_t = dZ_t + \frac{n-1}{2}\frac{dt}{X_t}</math>
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| where ''Z'' is a ''1''-dimensional [[Wiener process]] ([[Brownian motion]]). Note that this SDE makes sense for any real parameter <math>n</math> (although the drift term is singular at zero). Since ''W'' was assumed to have started from the origin the initial condition is ''X''<sub>0</sub> = 0.
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| For ''n'' ≥ 2, the ''n''-dimensional Wiener process is [[Markov chain#Recurrence|transient]] from its starting point: [[almost surely|with probability one]], ''X''<sub>''t''</sub> > 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''<sub>''t''</sub> < ''r''; on the other hand, it is truly transient for ''n'' > 2, meaning that ''X''<sub>''t''</sub> ≥ ''r'' for all ''t'' sufficiently large.
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| A notation for the Bessel process of dimension ''n' started at zero is BES<sub>0</sub>(n).
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| 0 and 2 dimensional Bessel processes are related to local times of Brownian motion via the Ray-Knight theorems.<ref>{{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 }}</ref>
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| The law of a Brownian motion near x-extrema is the law of a 3 dimensional Bessel process (theorem of Tanaka).
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| ==References==
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| {{Reflist}}
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| *{{cite book | author=Øksendal, Bernt | title=Stochastic Differential Equations: An Introduction with Applications | publisher=Springer |location=Berlin | year=2003 | isbn=3-540-04758-1}}
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| *Williams D. (1979) ''Diffusions, Markov Processes and Martingales, Volume 1 : Foundations.'' Wiley. ISBN 0-471-99705-6.
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| {{Stochastic processes}}
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| [[Category:Stochastic processes]]
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| {{probability-stub}}
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