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

TL;DR

This paper analyzes the first-passage and first-exit times of a Bessel-like stochastic process, providing analytical and numerical results for different boundary conditions, with applications across various scientific fields.

Contribution

It introduces a detailed analysis of a Bessel-like process with singular drift, deriving explicit formulas for first-passage and first-exit time distributions for different boundary types.

Findings

Analytical expressions for first-passage time densities.

Numerical validation of theoretical results.

Distinct behaviors observed at different boundary conditions.

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 $dX_t = (nD/X_t) dt + \sqrt{2D} dW_t$, 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.