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Volatility occupation times. (English) Zbl 1277.62196

Summary: We propose nonparametric estimators of the occupation measure and the occupation density of the diffusion coefficient (stochastic volatility) of a discretely observed Itô semimartingale on a fixed interval when the mesh of the observation grid shrinks to zero asymptotically. In a first step we estimate the volatility locally over blocks of shrinking length, and then in a second step we use these estimates to construct a sample analogue of the volatility occupation time and a kernel-based estimator of its density. We prove the consistency of our estimators and further derive bounds for their rates of convergence. We use these results to estimate nonparametrically the quantiles associated with the volatility occupation measure.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62G05 Nonparametric estimation
62P05 Applications of statistics to actuarial sciences and financial mathematics
60H30 Applications of stochastic analysis (to PDEs, etc.)
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
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