Asymptotic Properties Of Kernel Estimation Of Instantaneous Volatility Of High Frequency Data

In the era of new information economy,big data analysis is a solid foundation for sustain-able economic development.High-frequency data collection,data collation,and data analysis can better reduce global information gaps in financial areas such as asset portfolios,e-commerce,and securities markets.In theory,high-frequency data samples are closer to the dynamic continuous?time asset price model.Because massive amounts of high-frequency financial data are constantly changing,people need a research method that can accurately display high-frequency data in fi-nancial market operations.In recent years,the instantaneous volatility is a more sophisticated intra-day risk measurement method than the traditional volatility measure,and has caused new attention in the field of high-frequency data research.Since Genon-Catalot,Laredo and Picard(1992)considered deterministic,smooth volatili-ty processes and used wavelet methods to estimate instantaneous volatility,It has attracted many scholars to conduct various research on high frequency data.Andersen and Bollerslev(1998)pro-posed using intraday high frequency data to calculate the integral volatility.Barmdorff-Nielsen and Shephard(2002)demonstrated that the quadratic variation is a consistent estimate of the in-tegral volatility when the asset price process obeys the diffusion model,and gave the central limit theorem.Later,many scholars proposed various volatility estimates and studied the theoretical properties of these volatility estimates.Recently,Kristensen(2010)linked the integral volatility to the estimate of instantaneous volatility and proposed a kernel estimator of instantaneous volatility,and studied the asymptot-ic normality of the instantaneous volatility kernel estimate.This paper continues to study the instantaneous volatility kernel estimation proposed by Kristensen(2010),and proves the weak consistency,strong consistency and asymptotic normality of the instantaneous volatility kernel es-timation under weak conditions.Compared with the conditions used by Kristensen(2010),the conditions used in the conclusions of this paper are greatly reduced.The specific weaken condi-tions are as follows.(1)For the drift process and the volatility process,their variation sums are not required to have a certain convergence speed;(2)The Lipschitz condition in real space is reduced to Lipschitz condition in space with mathematical expectation as norm;(3)This article removes the extra requirement that the tails of the kernel function and its derivatives converge to zero at some rate.