Bessel process

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

Formal definition

Three realizations of Bessel Processes.

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.

Notation

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., 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 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).

References

↑ Revuz, D.; Yor, M. (1999). Continuous Martingales and Brownian Motion. Berlin: Springer. ISBN 3-540-52167-4.

Øksendal, Bernt (2003). Stochastic Differential Equations: An Introduction with Applications. Berlin: Springer. ISBN 3-540-04758-1.
Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. ISBN 0-471-99705-6.

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Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
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