Excursions of diffusion processes and continued fractions

TL;DR

This paper demonstrates how the excursions of one-dimensional diffusion processes can be analyzed using continued fractions derived from Riccati equations, providing new insights into their probabilistic structure.

Contribution

It introduces a novel approach to solving Riccati equations associated with diffusion excursions using infinite continued fractions, linking them to probabilistic properties.

Findings

Continued fraction solutions to Riccati equations for diffusion processes

Probabilistic interpretation of continued fraction expansions

Applications to diffusions in deterministic and random environments

Abstract

It is well-known that the excursions of a one-dimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in terms of an infinite continued fraction. We examine the probabilistic significance of the expansion. To illustrate our results, we discuss some examples of diffusions in deterministic and in random environments.