Phase transition for extremes of a stochastic volatility model with long-range dependence

Olivier Durieu, Yizao Wang

Published 2020-05-11Version 1

We consider a stochastic volatility model that both the volatility and innovation processes have power-law marginal distributions, with tail indices $\alpha,\alpha'>0$, respectively. In addition, the volatility process is taken as the heavy-tailed Karlin model, a recently investigated model that has long-range dependence characterized by a memory parameter $\beta\in(0,1)$. We establish extremal limit theorems for the empirical random sup-measures of the model, and reveal a phase transition: volatility-dominance regime $\alpha<\alpha'\beta$, innovation-dominance regime $\alpha>\alpha'\beta$, and critical regime $\alpha = \alpha'\beta$. The most intriguing case is the critical regime $\alpha = \alpha'\beta$, where the limit is the logistic random sup-measure. As for the proof, we actually establish the same phase-transition phenomena for the so-called Poisson--Karlin model with multiplicative noise defined on generic metric spaces, and apply a Poissonization method to establish the limit theorems for the volatility model as a consequence.