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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 753025, 13 pages
Research Article

Terminal-Dependent Statistical Inference for the Integral Form of FBSDE

1School of Mathematics, Shandong University, Jinan 250100, China
2College of Mathematics, Qingdao University, Qingdao 266071, China

Received 14 June 2013; Revised 4 September 2013; Accepted 22 September 2013

Academic Editor: Vasile Dragan

Copyright © 2013 Qi Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Backward Stochastic Differential Equation (BSDE) has been well studied and widely applied. The main difference from the Original Stochastic Differential Equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. However, to the best of knowledge, the terminal-dependent statistical inference for such a model has not been explored in the existing literature. This paper is concerned with the statistical inference for the integral form of Forward-Backward Stochastic Differential Equation (FBSDE). The reason why I use its integral form rather than the differential form is that the newly proposed inference procedure inherits the terminal-dependent characteristic. In this paper the FBSDE is first rewritten as a regression version, and then a semiparametric estimation procedure is proposed. Because of the integral form, the newly proposed regression version is more complex than the classical one, and thus the inference methods are somewhat different from those designed for the OSDE. Even so, the statistical properties of the new method are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors of the proposed estimators.