Parametric estimation for discretely observed stochastic processes with jumps

Abstract

We consider a two dimensional stochastic process (X, Y ), which may have jump components and is not necessarily ergodic.There is an unknown parameter θ within the coefficients of (X, Y ).The aim of this paper is to estimate θ from a regularly spaced sample of the process (X, Y ).When the dynamic of X is known, an estimator is constructed by using a moment-based method.We show that our estimators will work if the Blumenthal-Getoor index of the jump part of Y is less than 2. What is perhaps the most interesting is the rate at which the estimators converge: it is 1/ √ n (as when the underlying processes are not contaminated by jumps) when that index is not greater than 1.When the dynamic of X is unknown, we introduce a spot volatility estimator-based approach to estimate θ.This approach can work even if the sample is contaminated by microstructure noise.