Uniform convergence of the positive semi-definite modification of Malliavin-Mancino estimator for the spot volatility process

This paper establishes a uniform convergence result for a positive semi-definite modification of the Malliavin-Mancino Fourier estimator, designed for estimating the spot volatility process from high-frequency, non-synchronized data. The estimator, referred to as the PDF estimator, was previously introduced and its consistency in L 2 was established. This work strengthens that result by proving the estimator's uniform convergence in probability together with its convergence rate. Under a set of assumptions concerning the regularity of the volatility process and the relationship between tuning parameters and observation frequency, we demonstrate that the proposed estimator converges uniformly to the true spot volatility process at any order strictly smaller than n − 1 / 7 against the number of observations n in the general non-synchronized setting, provided the underlying process is a semimartingale. When the observations are in addition synchronous and equally spaced, the discretization error becomes orthogonal on the grid and the same analysis yields any order strictly smaller than n − 1 / 4 , which matches the optimal rate known for classical pointwise estimators; a global estimator thus attains, in the uniform topology, the accuracy hitherto established only pointwise. Since there has been no order estimates for the uniform convergence of a global estimator like Malliavin-Mancino's, these findings provide a stronger theoretical guarantee for the method, ensuring that the entire estimated volatility path closely approximates the true path as the observation density increases, which is a crucial property for applications in financial econometrics and risk management.