Consistency Of Kernel Estimation Of Volatility In Diffusion Process

Volatility is one of the ways to measure the volatility of return on investment in the financial market.It plays an important role in risk aversion and asset investment.However we cannot accurately calculate it because investment returning is a random process.We can only get its estimated value through large sample data.Therefore,it is important for every investor to study the estimator of volatility and its related properties.In many studies of volatility,diffusion process model provides theoretical basis both for option pricing and logarithmic price of financial products.The construction of volatility model and volatility estimator is the focus of current research.There are many models to describe Volatility:ARCH(autoregressive conditional heteroskedas-ticity model),SV(stochastic volatility)model and stochastic diffusion model.In the research of estimators,we use high-frequency data to combine realized volatility with the theory of quadrat-ic variation,which starts the research upsurge of post volatility.Then we put forward various forms of nonparametric estimation:estimating volatility with absolute power variation,measuring volatility with double power variation based on range.There are many researches on the combi-nation of volatility and diffusion function while limited researches are on the time-varying dif-fusion function.Kristensen(2010)combined with the second power variation and gave a kernel method to estimate the instantaneous volatility:(?).They then used the two-step estimation method to estimate the instantaneous volatility in the subse-quent research,which can be applied to both parametric estimation and nonparametric estimation.Although Kanaya and Kristensen(2016)have proved the asymptotic properties of the above es-timators,their conditions are complex,and some conditions are not easy to meet.Therefore,the purpose of this paper is to simplify some of these conditions.For this reason,we use different methods to prove the consistency of the estimator by using moment inequality and other methods,and we get:(?).Our convergence speed is simpler than that of Kanaya and Kristensen(2016)and is optimized for a wider scope of application.At the end of this paper,we test the estimator by numerical simulation:the shorter the time interval we choose,the better the fitting degree of our truth line and simulation line,and the smaller the error.