The Fourier estimator of spot volatility: Unbounded coefficients and jumps in the price process

We study a Fourier-based estimator of the spot volatility of a semimartingale, originally introduced by Malliavin and Mancino, and extend its asymptotic theory. First, we remove the uniformly bounded volatility assumption and establish almost sure convergence of the estimated Fourier coefficients under mild integrability conditions. We further derive explicit convergence rates that yield almost sure convergence of the reconstructed spot volatility via Fejér inversion. We then extend the analysis to processes with jumps. In this setting, we provide a precise decomposition of the Fourier coefficient estimation error into diffusion-diffusion, diffusion-jump, jump-diffusion, and jump-jump interaction terms, and obtain $L^p$-rates for each component. We show that the estimated coefficients converge almost surely to the sum of the true volatility coefficients and an explicit jump-induced term. Moreover, we prove that an appropriate rescaling of the trigonometric reconstruction yields almost sure pathwise recovery of the quadratic jump process.