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Nonparametric estimation of integrated diffusion processes has been thoroughly studied, primarily focusing on point-wise convergence. This paper firstly obtains the uniform convergence rates of the Nadaraya-Watson estimators for the coefficients of the integrated diffusion processes. We derive the uniform convergence rates over unbounded support under the shrinking observation interval and long time span assumption. While existing literature suggests that the convergence rate for the diffusion coefficient estimator in continuous-time diffusion processes can reach $(\log n/n)^{2/5}$, which corresponds to the minimax lower bound in sup-norm risk for nonparametric estimation with i.i.d. data. However, we find that the diffusion coefficient estimator of integrated diffusion processes can not attain this rate. Our results are necessary tools for specification testing and semiparametric estimation of certain types of diffusion processes and time series, based on nonparametric estimators, with applications in fields such as finance, geology, and physics.

This paper investigates the nonparametric estimation of the functional coefficients of the FBSDEs with random terminal time, including the local constant and local linear estimators. We provide complete two-dimensional asymptotics in both the time span and the sampling interval, allowing for the precise characterization of their distribution. Moreover, the empirical likelihood (EL) method to construct the data-driven confidence intervals for these estimators is provided. Some numerical simulations investigate the finite-sample properties of the estimators and compare the performance of the EL method and the conventional method in constructing confidence intervals based on asymptotic normality.