Terminal-Dependent Statistical Inference for the Integral Form of FBSDE

Qi Zhang

doi:10.1155/2013/753025

Abstract

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.

Highlights

The Backward Stochastic Differential Equation (BSDE) was first presented by Bismut [1] for the linear case and by Pardoux and Peng [2] for the general case
The solution of a BSDE consists of a pair of adapted processes (Yt, Zt) satisfying
The terminal condition is designed as a random variable with given distribution

Summary

Introduction

The Backward Stochastic Differential Equation (BSDE) was first presented by Bismut [1] for the linear case and by Pardoux and Peng [2] for the general case. This paper only considers the model with generator being parametric structure; that is to say, g can be written in the form of g = g(θ, t, Yt, Zt), where θ is an unknown parameter vector Even so, such a simplified form is widely used in financial markets, and, the proposed methods can be extended to the other complicated forms. Unlike the forward equation, because of the integral, the cumulative error appears not neglectable; the resultant estimation is still asymptotically unbiased for the condition of mixing dependency of Xt attached Another difference from the ordinary model is that the generator contains the unobservable process Zt, and it is necessary to estimate Zt first.

Terminal-Dependent Semiparametric Estimation for the FBSDE

Asymptotic Results

Simulations

Proofs