Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models

Abstract: Thresholded Realized Power Variations (TPV) are one of the most popular nonparametric estimators for general continuous-time processes with a wide range of applications. In spite of their popularity, a common drawback lies in the necessity of choosing a suitable threshold for the estimator, an issue which so far has mostly been addressed by heuristic selection methods. We address this crucial issue and propose an objective selection method based on desirable optimality properties of the estimators. Concretely, we develop a well-posed optimization problem which, for a fixed sample size and time horizon, selects a threshold that minimizes the expected total number of jump misclassifications committed by the thresholding mechanism associated with these estimators. We analytically solve the optimization problem under mild regularity conditions on the density of the underlying jump distribution, allowing us to provide an explicit infill asymptotic characterization of the resulting optimal thresholding sequence at a fixed time horizon. The leading term of the optimal threshold sequence is shown to be proportional to the Lévy’s modulus of continuity of the underlying Brownian motion, hence theoretically justifying and sharpening selection methods previously proposed in the literature based on power functions or multiple testing procedures. Furthermore, building on the aforementioned asymptotic characterization, we develop an estimation algorithm, which allows for a feasible implementation of the newfound optimal sequence. Simulations demonstrate the improved finite sample performance offered by optimal TPV estimators in comparison to other common non-optimal alternatives.