Nonparametric estimation of forward-backward stochastic differential equations with random terminal time

This paper investigates the nonparametric estimation of the functional coefficients of the forward-backward stochastic differential equations with random terminal time, focusing on both local constant and local linear estimators. We establish the asymptotic properties of these estimators under both long observation time spans and short sampling intervals, providing precise expressions for the bias and variance terms. Moreover, we propose an empirical likelihood method to construct data-driven confidence intervals for these functional coefficients. We conduct numerical simulations to examine the finite-sample properties of the estimators and to compare the performance of the empirical likelihood method with the conventional approach for constructing confidence intervals based on asymptotic normality.