One-Dimensional Diffusions That Eventually Stop Down-Crossing

TL;DR

This paper provides explicit criteria based on the coefficients of a one-dimensional diffusion process to determine whether it will eventually cease making down-crossings of a certain length, with specific asymptotic conditions on the drift.

Contribution

It introduces a precise criterion for the almost sure cessation of down-crossings in transient diffusions, linking the behavior to the coefficients of the diffusion operator.

Findings

Diffusions with drift exceeding a specific logarithmic threshold eventually stop down-crossing.

Diffusions with drift below a certain logarithmic threshold continue to make down-crossings indefinitely.

Explicit asymptotic conditions on the drift determine the long-term crossing behavior.

Abstract

Consider a diffusion process corresponding to the operator $L=\frac12a\frac{d^2}{dx^2}+b\frac d{dx}$ and which is transient to $+\infty$. For $c>0$, we give an explicit criterion in terms of the coefficients $a$ and $b$ which determines whether or not the diffusion almost surely eventually stops making down-crossings of length $c$. As a particular case, we show that if $a=1$, then the diffusion almost surely stops making down-crossings of length $c$ if $b(x)\ge\frac1{2c}\log x+\frac\gamma c\log\log x$, for some $\gamma>1$ and for large $x$, but makes down-crossings of length $c$ at arbitrarily large times if $b(x)\le\frac1{2c}\log x+\frac1c\log\log x$, for large $x$.