First-passage and first-exit times of a Bessel-like stochastic process

Abstract

We study a stochastic process $X_t$ related to the Bessel and the Rayleigh processes, with various applications in physics, chemistry, biology, economics, finance and other fields. The stochastic differential equation is $d{X}_t=(nD/X_t)dt+\sqrt{2D}d{W}_t$, where $W_t$ is the Wiener process. Due to the singularity of the drift term for $X_t=0$, different natures of boundary at the origin arise depending on the real parameter $n$: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behaviour is observed in the case of a regular boundary.

15 pages, 6 figures, submitted to Physical Review E