Mathematical Sciences: Bootstrap Asymptotics, Stochastic Differential Equations and Optimal Control, and Non- parametric Estimation for Diffusion Processes

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The principal investigator will continue his study of the probabilistic properties of the resampled sample in the bootstrap methods. The investigation will extend to bootstraping stochastic processes such as Brownian motion. The effects of changing the resampling size will be explored. Resampling from a trimmed empirical rather than the original empirical distribution will be shown to be better. Limit theorems for bootstraping of the U statistics will be found. The probabilistic study of the bootstraping phenomena leads to not only interesting theorems in probability but better bootstraping methods and the validity for the existing methods.