On the location of the maximum of a continuous stochastic process

TL;DR

This paper establishes a necessary and sufficient condition for the uniqueness of the maximum's location in a continuous stochastic process and relates its mean position to the process's expected maximum derivative, with applications to Brownian motion.

Contribution

It provides a new criterion for maximum location uniqueness and a formula for its mean, extending understanding of extremal points in stochastic processes.

Findings

Condition for uniqueness of maximum location

Expression for mean location in terms of derivatives

Application to Brownian motion with variable drift

Abstract

In this short note we will provide a sufficient and necessary condition to have uniqueness of the location of the maximum of a stochastic process over an interval. The result will also express the mean value of the location in terms of the derivative of the expectation of the maximum of a linear perturbation of the underlying process. As an application, we will consider a Brownian motion with variable drift. The ideas behind the method of proof will also be useful to study the location of the maximum, over the real line, of a two-sided Brownian motion minus a parabola and of a stationary process minus a parabola.