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

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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.