Uniform Convergence Rates Of The Modified N-W Estimators For Jump-Diffusion Processes

Jump-diffusion model is one of the most popular models in the field of financial mathematics.It can describe the fluctuation law of stock prices and portfolio asset values.Its statistical inference is still a hot issue in statistics,financial mathematics and econometrics.Many statisticians have considered the nonparametric estimation for diffusion model and jump-diffusion model based on discrete-time observations.The most typical one is the Nadaraya-Waston estimator.From previous work,we can find that scholars have paid more attention to the pointwise convergence of the estimators,such as the consistency and the asymptotic(mixed)normality.Meanwhile,noting that consistent convergence results of nonparametric kernel estimations for time series models,continuous-time processes and stochastic volatility models have been studied,this consistent convergence result can be widely used in the study of semi-parameter estimation.However,the jump-diffusion model based on high-frequency financial data leaves a gap in this regard.Therefore,this paper studies the uniform convergence of the modified N-W estimators for jump-diffusion model.Under the assumption of two-dimensional asymptotics,where the time span T tends to infinity and the sampling interval Δtends to 0,we first obtain the uniform convergence rates of the related coefficient for estimators over an unbounded support on the processes.Our proofs use the covering-number technique and the exponential concentration inequalities of martingales.Our results are not limited to compact sets,but can also consider infinite sets,which have global significance.It is believed that our results can be widely used in the research of semi-parameter estimation and model testing of diffusion processes and jump-diffusion processes.