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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 365240, 11 pages
http://dx.doi.org/10.1155/2014/365240
Research Article

Terminal-Dependent Statistical Inference for the FBSDEs Models

1China University of Petroleum, Qingdao 266580, China
2Shandong University Qilu Securities Institute for Financial Studies, Shandong University, Jinan 250100, China

Received 12 March 2014; Accepted 27 May 2014; Published 25 June 2014

Academic Editor: Guangchen Wang

Copyright © 2014 Yunquan Song. 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.

Abstract

The original stochastic differential equations (OSDEs) and forward-backward stochastic differential equations (FBSDEs) are often used to model complex dynamic process that arise in financial, ecological, and many other areas. The main difference between OSDEs and FBSDEs is that the latter is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. It is interesting but challenging to estimate FBSDE parameters from noisy data and the terminal condition. However, to the best of our knowledge, the terminal-dependent statistical inference for such a model has not been explored in the existing literature. We proposed a nonparametric terminal control variables estimation method to address this problem. The reason why we use the terminal control variables is that the newly proposed inference procedures inherit the terminal-dependent characteristic. Through this new proposed method, the estimators of the functional coefficients of the FBSDEs model are obtained. The asymptotic properties of the estimators are also discussed. Simulation studies show that the proposed method gives satisfying estimates for the FBSDE parameters from noisy data and the terminal condition. A simulation is performed to test the feasibility of our method.