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

We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small L'{e}vy noises. We do not impose any moment condition on the driving L'{e}vy 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 $\epsilon \to 0$ and $n\to \mathrm{\infty }$ 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 L'evy process.