Time to reach the maximum for a random acceleration process

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.

17 pages, 5 figures Typo in Eq. (B.11) corrected

References in corpus (3)

• On the time to reach maximum for a variety of constrained Brownian motions
• Hitting probability for anomalous diffusion processes
• Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

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