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

我们考虑了扩散模型的基于观测的估计问题。在观测区间固定小的情况下，得到了估计量的渐近展开式，它们的形式类似于时间序列分析的渐近展开式。对于带跳的扩散过程，在具有遍历性或小噪声性质的强条件下，证明了极大似然估计的渐近正态。提出了检测客户风险因素指数的多重检验方法，证明了该方法在连续观测情况下保持了误发现率，并通过数值实验证明了该方法对离散观测情况下的FDR也具有控制作用。

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.