No abstract

Diffusions, Discrete-time observations, High-frequency data, Nonsynchronicity, Quadratic variation, Realized volatility,

An asymptotic distribution theory of the nonsynchronous covariation process for continuous semimartingales is presented. Two continuous semimartingales are sampled at stopping times in a nonsynchronous manner. Those sampling times possibly depend on the history of the stochastic processes and themselves. The nonsynchronous covariation process converges to the usual quadratic covariation of the semimartingales as the maximum size of the sampling intervals tends to zero. We deal with the case where the limiting variation process of the normalized approximation error is random and prove the convergence to mixed normality, or convergence to a conditional Gaussian martingale. A class of consistent estimators for the asymptotic variation process based on kernels is proposed, which will be useful for statistical applications to high-frequency data analysis in finance. As an illustrative example, a Poisson sampling scheme with random change point is discussed.

We propose a methodology for evaluating the hedging errors of derivative securities due to the discreteness of trading times or the observation times of market prices, or both. Utilizing a weak convergence approach, we derive the asymptotic distributions of the hedging errors as the discreteness disappears in several situations. First, we examine the hedging error due to discrete-time trading when the true strategy is known, which generalizes the result of Bertsimas, Kogan, and Lo (2000) to continuous Itô processes. Then we consider a data-driven strategy, when the true strategy is unknown. This strategy is free of parametric model assumptions, therefore it is expected to serve as a benchmark for the evaluation of parametric strategies. Finally, we consider a case study of the Black-Scholes delta-hedging strategy when the volatility is unknown in the proposed framework. The results obtained give us a prospect for further developments of the framework under which various parametric strategies could be compared in a unified manner.

The nonlinear equation for the rotational vibration of a pair of spur gears has a restriction that the analytical solution of the equation cannot be obtained. In this paper, the linear equation of vibration is derived theoretically and its physical model, i.e., the linear model of vibration is presented. The analytical solution of the linear equation, which is derived by analytical method, agrees well with the numerically calculated result by the nonlinear equation. By analyzing the analytical solution of the linear equation in detail, we clarified the relation between the waveforms of the vibration and the profile error of gear pairs, and also found that the effect of the contact ratio to the vibration is large and complex. The equivalent error, accounting for effects of the static load, the time-varying stiffness, and the profile error of gear pairs, is proposed in this paper. It can be considered as promising for evaluating the profile error, because the vibration of gear pairs is excited mainly by the equivalent error. Finally, for confirming the above results, the vibration of two tested gear pairs has been measured by an experimental set-up for this purpose.