Almost sure limit theorems for the maxima of stochastic volatility models

Abstract

In this paper, we proved the almost sure limit theorems for the maxima of stochastic volatility models with both light-tailed and heavy-tailed noises, where the volatility sequence is a log-Gaussian linear process. For the light-tailed noise case, we assume that the autocorrelation function rn of the Gaussian linear process satisfies rnln⁡n(ln⁡ln⁡n)1+ε=O(1) for some ε>0. For the heavy-tailed noise case, we assume that the Gaussian linear process is strong mixing with mixing rate γ(n) satisfying γ(n)(ln⁡ln⁡n)1+ε=O(1) for some ε>0.