First-passage and escape problems in the Feller process

TL;DR

This paper thoroughly investigates the first-passage and escape problems of the Feller process, a positive diffusion process with applications in neuroscience and finance, providing new insights into its level crossing properties.

Contribution

It offers a detailed analysis of first-passage and escape phenomena specific to the Feller process, extending understanding beyond its well-known general properties.

Findings

Derived explicit solutions for first-passage times

Analyzed escape probabilities and rates

Enhanced understanding of level crossing in the Feller process

Abstract

The Feller process is an one-dimensional diffusion process with linear drift and state-dependent diffusion coefficient vanishing at the origin. The process is positive definite and it is this property along with its linear character that have made Feller process a convenient candidate for the modeling of a number of phenomena ranging from single neuron firing to volatility of financial assets. While general properties of the process are well known since long, less known are properties related to level crossing such as the first-passage and the escape problems. In this work we thoroughly address these questions.