Nonparametric inference for the spectral measure of a bivariate pure-jump semimartingale

We develop a nonparametric estimator for the spectral density of a bivariate pure-jump Itô semimartingale from high-frequency observations of the process on a fixed time interval with asymptotically shrinking mesh of the observation grid. The process of interest is locally stable, i.e., its Lévy measure around zero is like that of a time-changed stable process. The spectral density function captures the dependence between the small jumps of the process and is time invariant. The estimation is based on the fact that the characteristic exponent of the high-frequency increments, up to a time-varying scale, is approximately a convolution of the spectral density and a known function depending on the jump activity. We solve the deconvolution problem in Fourier transform using the empirical characteristic function of locally studentized high-frequency increments and a jump activity estimator.