||This thesis deals with the statistical problems in finance and other dynamical systems which can be appropriately modeled by a semimartingale. It consists of two parts. Firstly, we revisit the estimation problem of integrated volatility (IV). One of the significant features of high frequency data is that there are many transactions within one second, whose effect on estimating the IV is yet to be studied in the literature. We study the effect and proposed an estimator to settle the issue. Assuming that the observations are contaminated by noise and have jumps, we propose an estimator of IV which can thoroughly remove the effect of the noise and jumps. The estimator outperforms the existing approaches. The consistency is proved under very mild conditions and the central limit theorem is established under slightly stronger conditions. Secondly, we focus on model specification problem and fill some gaps in the literature. Two nonparametric procedures are proposed. The first one is to test whether a diffusion is present for discretely observed high frequency data. The second one is to test whether the underlying process has jumps or not when the observations are contaminated by microstructure noise. The asymptotic results for both tests are established. For both parts, we study the finite sample performances in several numerical examples and implement the methods to real asset return data.