Time to reach the maximum for a random acceleration process

TL;DR

This paper derives exact analytical expressions for the distribution of the time at which a random acceleration process reaches its maximum within a fixed interval, considering two boundary conditions, and verifies results with simulations.

Contribution

It provides the first exact formulas for the maximum time distribution in a non-Markovian random acceleration process under different boundary conditions.

Findings

Exact probability density functions for maximum time are obtained.

Results are validated through numerical simulations.

The study enhances understanding of non-Markov stochastic processes.

Abstract

We study the random acceleration model, which is perhaps one of the simplest, yet nontrivial, non-Markov stochastic processes, and is key to many applications. For this non-Markov process, we present exact analytical results for the probability density $p(t_m|T)$ of the time $t_m$ at which the process reaches its maximum, within a fixed time interval $[0,T]$. We study two different boundary conditions, which correspond to the process representing respectively (i) the integral of a Brownian bridge and (ii) the integral of a free Brownian motion. Our analytical results are also verified by numerical simulations.