Estimation of the instantaneous volatility and detection of volatility jumps

Concerning price processes, the fact that the volatility is not constant has been observed for a long time. So we deal with models as $dX_t=\mu_tdt+\sigma_tdW_t$ where $\sigma$ is a stochastic process. Recent works on volatility modeling suggest that we should incorporate jumps in the volatility process. Empirical observations suggest that simultaneous jumps on the price \underline{and} the volatility \cite{BarShep1,ConTan} exist. The hypothesis that jumps occur simultaneously makes the problem of volatility jump detection reduced to the prices jump detection. But in case of this hypothesis failure, we try to work in this direction. Among others, we use Jacod and Ait-Sahalia' recent work \cite{jac1} giving estimators of cumulated volatility $\int_0^t|\sigma_s|^pds$ for any $p\geq 2.$ This tool allows us to deliver an estimator of instantaneous volatility. Moreover we prove a central limit theorem for it. Obviously, such a theorem provides a confidence interval for the instantaneous volatility and leads us to a test of the jump existence hypothesis. For instance, we consider a simplest model having volatility jumps, when volatility is piecewise constant: $\sigma_t=\sum_{i=0}^{N_t-1} \sigma_i \1_{[\tau_i,\tau_{i+1}[}(t).$ The jump times are $\tau_i, i\geq 1,$ and $\sigma_i$ is a $\F_{\tau_{i}}$-measurable random variable. Another example is studied: $\sigma_t=|Y_t|$ where $(Y_t)$ is a solution to a L\'evy driven SDE, with suitable coefficients.