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) 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 XΔ for some Δ>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 convergence rates for the corresponding estimator and show that these rates cannot be improved in general.