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Zbl 0924.60039
Doney, R.A.; Warren, J.; Yor, M.
Perturbed Bessel processes.
(English)
[A] Azéma, Jacques (ed.) et al., Séminaire de probabilités XXXII. Berlin: Springer. Lect. Notes Math. 1686, 237-249 (1998). ISBN 3-540-64376-1

Let $B_t$, $t\geq 0$, $B_0=0$, be a Brownian motion. The $\alpha$-perturbed Bessel process of dimension $d>1$ starting from $a\geq 0$ is defined as follows: It is a non-negative process $R_{d,\alpha} (t)$, $t\geq 0$, such that $$R_{d,\alpha}(t)= a+B_t+ \frac{d-1}{2} \int_0^t \frac{ds}{R_{d,\alpha}(s)}+ \alpha(M_t^R-a),$$ where $M_t^R= \sup_{0\leq s\leq t}R_{d,\alpha}(s)$. (The modification of this definition for the case $d=1$ is presented as well.) The authors show that a perturbed Bessel process of dimension $d\geq 1$ can be represented in terms of an ordinary Bessel process of dimension $d$ by means of space-time transformation. Ray-Knight theorems on local time for the perturbed Bessel process of dimension $d=3$ and some properties of these processes are presented. The authors consider also the problem of extending the definition of the perturbed Bessel process to dimension $0< d<1$.
[N.S.Bratijchuk (Ky{\" i}v)]
MSC 2000:
*60H10 Stochastic ordinary differential equations
60J05 Markov processes with discrete parameter

Keywords: Brownian motion; local time; Bessel process

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