Bessel process

In mathematics, a Bessel process, named after Friedrich Bessel. 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)

In mathematics, a Bessel process, named after Friedrich Bessel. 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) ==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 R n and W is an n-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 started at zero is . ==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., 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    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. The law of a Brownian motion near x-extrema is the law of a 3-dimensional Bessel process (theorem of Tanaka). ==References== Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. .

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