Developing statistical asymptotic theory for jump processes and its applications

主要得到了以下关于跳跃随机过程模型统计推断的结果：(1)当模型具有一般的非线性系数和非高斯噪声时，高斯拟似然估计的渐近正态以及一个易于使用的近似置信域：(2)当模型是一般的连续时间回归类型时，残差的有偏校正泛函的无模型渐近分布，并应用于噪声正态和扩散系数误规范检验；(3)当噪声过程可以在小时间内近似非高斯稳定时，基于小时间稳定近似的最小绝对偏差估计和拟似然估计的渐近混合正态。特别地，(3)中提出的估计量比(1)中提出的估计量效率高得多，而(3)中的模型设置比(1)中的模型设置要有限得多。

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).