Krylov–Bogolyubov theorem: Difference between revisions
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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 | |||
:<math>X_t = \| W_t \|,</math> | |||
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]] | |||
:<math>dX_t = dZ_t + \frac{n-1}{2}\frac{dt}{X_t}</math> | |||
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. | |||
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. | |||
A notation for the Bessel process of dimension ''n' started at zero is BES<sub>0</sub>(n). | |||
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> | |||
The law of a Brownian motion near x-extrema is the law of a 3 dimensional Bessel process (theorem of Tanaka). | |||
==References== | |||
{{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]] | |||
{{probability-stub}} | |||
Revision as of 16:46, 20 April 2013
In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process. The Bessel process of order n is the real-valued process X given by
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
where Z is a 1-dimensional Wiener process (Brownian motion). Note that this SDE makes sense for any real parameter (although the drift term is singular at zero). Since W was assumed to have started from the origin the initial condition is X0 = 0.
For n ≥ 2, the n-dimensional Wiener process is transient from its starting point: with probability one, 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.
A notation for the Bessel process of dimension n' started at zero is BES0(n).
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).
References
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- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. ISBN 0-471-99705-6.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534