  This paper investigates the Gaussian quasi-likelihood estimation of an
exponentially ergodic multidimensional Markov process, which is expressed as a
solution to a L\'{e}vy driven stochastic differential equation whose
coefficients are known except for the finite-dimensional parameters to be
estimated, where the diffusion coefficient may be degenerate or even null. We
suppose that the process is discretely observed under the rapidly increasing
experimental design with step size $h_n$. By means of the polynomial-type large
deviation inequality, convergence of the corresponding statistical random
fields is derived in a mighty mode, which especially leads to the asymptotic
normality at rate $\sqrt{nh_n}$ for all the target parameters, and also to the
convergence of their moments. As our Gaussian quasi-likelihood solely looks at
the local-mean and local-covariance structures, efficiency loss would be large
in some instances. Nevertheless, it has the practically important advantages:
first, the computation of estimates does not require any fine tuning, and hence
it is straightforward; second, the estimation procedure can be adopted without
full specification of the L\'{e}vy measure.
