Terminal-Dependent Statistical Inference for the FBSDEs Models

Yunquan Song

doi:10.1155/2014/365240

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.

Highlights

Since 1973, when the world’s first options exchange opened in Chicago, a large number of new financial products have been introduced to meet the customer’s demands from the derivative markets
Black and Scholes [1] provided their celebrated formula for option pricing and Merton [2] proposed a general equilibrium model for security prices
The terminal condition is designed as a random variable with given distribution

Summary

Introduction

Since 1973, when the world’s first options exchange opened in Chicago, a large number of new financial products have been introduced to meet the customer’s demands from the derivative markets. These methods cannot be directly employed to infer the BSDE and FBSDE because the two models are related to a terminal condition. Zhang and Lin [9] proposed two terminal-dependent estimation methods via terminal control variable for the integral form models of FBSDE They only considered the parametric form of the generator g in their paper. Some problems arise naturally, including how to correct the bias of the model and how to construct the constraint-dependent estimation To solve these problems, we will use remodeling method to draw terminal condition into differential equation, similar but not the same as IV, called quasi-instrumental variable methods; in other words, the terminal condition ξ enters into the equation as a control variable.

Model and Method

Asymptotic Results

Simulation Studies

Proofs