Small deviations of iterated processes in space of trajectories

TL;DR

This paper investigates the probabilities of small deviations in iterated processes within the space of trajectories, deriving asymptotic formulas and identifying conditions for their equivalence to the generating processes.

Contribution

It provides new asymptotic results for small deviations of iterated processes and clarifies when these match or differ from the original processes' asymptotics.

Findings

Logarithmic asymptotics for small deviations derived

Conditions identified for asymptotics to match those of generating processes

Differences in asymptotics when conditions are not met

Abstract

We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.