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Статья

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

Stochastic Processes and their Applications. 2013. Vol. 123. No. 1. P. 15-44.

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  L\'evy 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.