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Concept
Bessel process
Summary
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 (when n ≥ 2) by
:X_t = | W_t |,
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 + \frac{n-1}{2}\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).
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 started at the origin is transient from its starting point: with probability one, i.e., Xt
Official source
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