Developing statistical asymptotic theory for jump processes and its applications

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Mainly, we have derived the following results concerning statistical inference for stochastic process models with jumps: (1) Asymptotic normality of the Gaussian quasi-likelihood type estimator together with an easy-to-use approximate confidence regions, when the model has general non-linear coefficients and non-Gaussian noise; (2) Model-free asymptotic distribution of a bias-corrected functional of residuals, with applications to noise-normality and diffusion-coefficient misspecification tests, when the model is of a general continuous-time regression type; (3) Asymptotic mixed normality of the least-absolute deviation estimator and the quasi-likelihood estimator based on the small-time stable approximation, when the noise process can be approximately non-Gaussian stable in small time. In particular, the proposed estimator in (3) is much more efficient than that in (1), while the model setting in (3) is more limited compared with that of (1).