The argmin process of random walks, Brownian motion and L\'evy processes

TL;DR

This paper studies the argmin process of Brownian motion, random walks, and Lévy processes, establishing its Markovian properties, invariant measures, and path decompositions, with applications to extrema analysis.

Contribution

It introduces the argmin process for Brownian motion, proves its Markov and Feller properties, derives transition kernels, and extends results to random walks and Lévy processes, including path decompositions.

Findings

The argmin process of Brownian motion is stationary with arcsine invariant measure.

The argmin process is a Markov process with explicit transition kernels.

Brownian extrema of fixed length form a delayed renewal process.

Abstract

In this paper we investigate the argmin process of Brownian motion $B$ defined by $\alpha_t:=\sup\left\{s \in [0,1]: B_{t+s}=\min_{u \in [0,1]}B_{t+u} \right\}$ for $t \geq 0$. The argmin process $\alpha$ is stationary,with invariant measure which is arcsine distributed. We prove that $(\alpha_t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot)$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and L\'evy processes are derived. We also consider Brownian extrema of a given length. We prove that these extrema form a delayed renewal process with an explicit path construction. We also give a path decomposition for Brownian motion at these extrema