Statistical inference for stochastic differential equations from discrete observation and its applications

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We considered statistically asymptotic inference for volatility parameters of stochastic differential equations from high frequency data observed on the fixed interval. By using the polynomial type large deviation inequality for the statistical random field, we showed the asymptotic mixed normality of maximum likelihood type estimator and Bayes type estimator of the volatility parameter and the moments of convergences of the estimators. Under nh^p ->0, where h is the discretization step size, n is the sample size and p is an integer value greater than 2, adaptive maximum likelihood estimators of both drift and volatility parameters for discretely observed ergodic diffusion processes were obtained, and their asymptotic normality and moment convergence wereproved. Moreover, we treated adaptive estimators of drift and volatility parameters for misspecified ergodic diffusion processes from discrete observations and their asymptotic properties were shown.