Terminal-Dependent Statistical Inferences for FBSDE

Backward stochastic differential equation (BSDE) has been well studied and widely applied in mathematical finance. The main difference from the original stochastic differential equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which plays key roles in certain financial and ecological circumstances. However, to the best of our knowledge, the terminal-dependent statistical inference for such model has not been explored in the existing literature. This article proposes two terminal-dependent estimation methods via terminal control variable the integral form models of forward-backward stochastic differential equation (FBSDE). We take these measures because the resulting models contain terminal condition as model variable, and therefore, the corresponding estimators inherit the terminal-dependent characteristic. In this article, the FBSDE is first rewritten as regression versions and then two semi-parametric estimation approaches are proposed. Because of the control variable and integral form, our regression versions are more complex than the classical ones, and the inference methods are somewhat different from which designed for the OSDE. Even so, the statistical properties of the terminal-dependent methods are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors.