Abelian theorems for stochastic volatility models with application to the estimation of jump activity of volatility

Abstract

In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models (X; V ); where both the state process X and the volatility process V may have jumps. Our results relate the asymptotic behavior of the characteristic function of X for some > 0 in a stationary regime to the Blumenthal-Getoor indexes of the Levy processes driving the jumps in X and V . The results obtained are used to construct consistent estimators for the above Blumenthal-Getoor indexes based on low-frequency observations of the state process X. We derive the convergence rates for the corresponding estimator and show that these rates can not be improved in general.