Theory of Statistical Inference for Stochastic Processes and its Application to Mathematical Finance

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(1)In the case where degenerate components are involved with a random variable of interest, we proved the validity of its distributional asymptotic expansion, and applied the result to the third order asymptotic expansion of the maximum likelihood estimator for the Ornstein-Uhlenbeck process.(2)As for the prediction problem, by using expansion formula for the generalized Wiener functional, we showed the Stein phenomenon, i.e., the prediction region centered at a shrinkage type estimator is better than that centered at a usual sample mean.(3)For functionals of Markovian process with a mixing property, we derived general and explicit formulas of distributional asymptotic expansion with a error bound, and used them to obtain a third order asymptotic expansion of the M-estimator for a general statistical model. As a typical application, we present third order asymptotic expansions of M or minimum contrast estimators for diffusion processes.(4)In order to approximate closely to the probabilities of mis-discrimination, we calculated asymptotic expansions for Bayes, plug-in, likelihood ratio type discrimination rules, and studied their numerical accuracies by Monte-Carlo experiments to show that our results gives very close approximations.