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 $X_Delta$ for some $Delta > 0$ in a stationary regime to the Blumenthal-Getoor indexes of the Lévy 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 prove that these rates can not be improved in general.