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

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