# The arctangent law for a certain random time related to a one-dimensional diffusion

## TL;DR

This paper generalizes a known result for Brownian motion to a broader class of one-dimensional diffusion processes, analyzing the distribution of the first time after a given point when the process exceeds its previous maximum.

## Contribution

It extends the arctangent law for the first exceeding time from Brownian motion to general one-dimensional diffusions.

## Key findings

- Derived the distribution of the first exceeding time for general diffusions.
- Established the arctangent law in a broader diffusion context.
- Generalized Papanicolaou's result beyond Brownian motion.

## Abstract

For a time-homogeneous, one-dimensional diffusion process $X(t),$ we investigate the distribution of the first instant, after a given time $r,$ at which $X(t)$ exceeds its maximum on the interval $[0,r],$ generalizing a result of Papanicolaou, which is valid for Brownian motion.