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

TL;DR

This paper studies the argmin process of random walks and Lévy processes, proving their Markov property and deriving transition kernels in specific cases, contributing to the understanding of their probabilistic structure.

Contribution

It establishes the Markov property for the argmin process of these stochastic processes and derives explicit transition kernels in certain scenarios.

Findings

Argmin processes of random walks and Lévy processes are Markovian.

Explicit transition kernels are obtained for special cases.

Provides foundational results for future analysis of argmin processes.

Abstract

In this paper we consider the argmin process of random walks and L\'evy processes. We prove that they enjoy the Markov property, and provide their transition kernels in some special cases.