Least squares estimators for discretely observed stochastic processes driven by small Lévy noises

AMS 2010 subject classifications: primary 62F12 62M05 secondary 60G52 60J75 Keywords: Asymptotic distribution of LSE Consistency of LSE Discrete observations Least squares method Stochastic processes Parameter estimation Small Lévy noises a b s t r a c t We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Lévy noises. We do not impose any moment condition on the driving Lévy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient ε → 0 and n → ∞ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to the jump part of the Lévy process. Moreover, we briefly remark that our methodology can be easily extended to the more general case of semi-martingale noises.