Statistical Asymptotic Theory for Stochastic Processes and Its Application to Actuarial Science

研究了扩散模型基于观测值的估计问题。在观测值区间固定的情况下，得到了估计量的渐近展开式，其形式与时间序列分析的渐近展开式相似。 对于带跳的扩散过程，在遍历性或小噪声性的强条件下，证明了极大似然估计的渐近性。提出了一种检测客户风险因素指标的多重检验方法，证明了该方法在连续观测情况下保持错误发现率，并通过数值实验证明了该方法在离散观测情况下也能控制FDR。

We considered the estimating problem based on observations for the diffusion models. In the case where the intervals of observations are fixed small, the asymptotic expansions for estimators are obtained and they have similar form to that for time series analysis. As for the diffusion processes with jumps, the asymptotic normalities for the maximum likelihood estimators are proved under some strong conditions with ergodicity or small noise property. We proposed the multiple testing procedure to detect the customers risk factor indices, proved that it keeps false discovery rate in the continuous observation case, and show with numerical experiments that it also controls the FDR for discrete observations.