On the asymptotics of supremum distribution for some iterated processes

TL;DR

This paper investigates the asymptotic behavior of the supremum distribution of certain iterated stochastic processes, especially focusing on Gaussian processes composed with other stochastic processes, including fractional Brownian motion.

Contribution

It provides new asymptotic results for the supremum distribution of iterated Gaussian processes, including fractional Brownian motion, as the threshold tends to infinity.

Findings

Derived asymptotic formulas for supremum probabilities as thresholds grow large.

Analyzed the behavior of supremum distributions over different time scales.

Illustrated results with iterated fractional Brownian motion.

Abstract

In this paper, we study the asymptotic behavior of supremum distribution of some classes of iterated stochastic processes $\{X(Y(t)) : t \in [0, \infty)\}$, where $\{X(t) : t \in \mathbb{R} \}$ is a centered Gaussian process and $\{Y(t): t \in [0, \infty)\}$ is an independent of $\{X(t)\}$ stochastic process with a.s. continuous sample paths. In particular, the asymptotic behavior of $\mathbb{P}\left(\sup_{s \in [0,T]} X(Y(s)) > u\right)$ as $u \to \infty$, where $T > 0$, as well as $\lim_{u\to\infty} \mathbb{P}\left(\sup_{s \in [0, h(u)]} X(Y(s)) > u\right)$, for some suitably chosen function $h(u)$ are analyzed. As an illustration, we study the asymptotic behavior of the supremum distribution of iterated fractional Brownian motion process.