Abstract

In the early 1990’s, the Malliavin calculus was first applied to statistics in the studies of the higher order inference for stochastic differential equations. This technique was successfully used in statistical inference, as well in finance for option pricing, and today the YUIMA offers fast computation of any order of expansion formula. With the Malliavin calculus, the martingale expansion and the expansion of functionals of a continuous-time Markov process were succeeded in 90’s with statistical applications. They are purely distributional methods, extending the classical theory of Edgeworth expansions for independent models. The martingale expansion was extended to the non-ergodic cases including applications, such as the realized volatility, to volatility estimation in high frequency financial data analysis. Recently further essential extension of the theory took place, i.e., expansion of general Wiener functionals, and asymptotic expansion of Skorohod integrals. By these new engines, it became possible to approach precise approximation of functionals of a fractional Brownian motion. In this talk, we discuss some of very recent developments toward higher-order statistics for stochastic processes: expansion of variations with anticipative weights, and precise approximation for an estimator of the Hurst coefficient. An application to the realized volatility of a fractional diffusion is mentioned.