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 terminaldependent 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 ForwardBackward 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 terminaldependent 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.
The Backward Stochastic Differential Equation (BSDE) was first presented by Bismut [
The study history of the BSDE was relatively short but progressed rapidly. In addition to the interesting mathematical nature, its extensive applications gained more and more attentions; see for example Peng [
In terms of the backward equation, within a complete market it serves to characterize the dynamic value of replicating portfolio
For the FBSDE defined above, the statistical inference was investigated initially by Su and Lin [
As well the FBSDE could turn to the integral form:
It is worth mentioning that the key point of the method is the use of the integral equation rather than the differential equation. This change leads to a completely new work among the existing researches. Unlike the forward equation, because of the integral, the cumulative error appears not neglectable; nevertheless, the resultant estimation is still asymptotically unbiased for the condition of mixing dependency of
The paper is organized as follows. In Section
I consider the integral form of the standard FBSDE:
To recast the model (
It seems that formula (
Given the initial calendar time point
In this section I assume
This is the statistical version of (
I now turn to estimating unknown parameter vector
Concerning inference of
To this end, consider the FBSDE model (
Denote
By (
From the above, it is simple to deduce the estimator of
The following two theorems are concerned with asymptotic properties of the estimators deduced in the previous section.
First of all, I lead in several conditions.
with
The continuous kernel function
As
where the matrix
with
The condition (a) is commonly used for the weakly dependent process; see for example Rosenblatt [
Besides the conditions (a), (b), and (c), suppose that
The proof is presented in Section
In addition to the condition of Theorem
The proof is also presented in Section
As is mentioned in Section
For arbitrary
Set
Let
Consider that the semiparametric models in Section
Then, for the flexibility of modeling the above case, a nonlinear semiparametric model can be defined as
Before estimating nonlinear model (
After plugging the estimator
In this section I investigate the finitesample behaviors by simulation. Despite Theorems
Consider CoxIngersollRoss (CIR) process:
Denote parameters
I present the true curves and the NW nonparametric estimation curves for
Parameter  True value  Mean  MSE 


0.12  0.1404  0.0089 

0.05  0.0503 


−3.9062  −4.5826  4.6295 
The real lines are the true curves of
In this part I consider the case that the terminal time is far away from the last observed time, as mentioned in the Supplement Section. The distance between
Parameter  True value  Mean  MSE 


0.12  0.1470  0.0319 

0.05  0.0490 


−3.9062  −3.8360  9.2929 
The real lines are the true curves of
I turn to the nonstationary case in this part. Obviously when forward process
I consider a simple FBSDE as
Firstly, let
Finally, I choose
Parameter  True value  Mean  STD 


0.05  0.0473  0.0348 

5  5.7788  7.3388 
Parameter  True value  Mean  STD 


0.05  0.0580  0.0217 

1  0.9959  7.6034 
Parameter  True value  Mean  STD 


0.05  0.0557  0.0421 

0.4167  0.4227  6.5970 
The real lines are the true curves of
The real lines are the true curves of
The real lines are the true curves of
Denote
Note that
From Lemma 1 of Politis and Romano [
I present the basic results for
From
On the other hand, the expectation of
By condition (d), the variance is bounded uniformly for
This paper was supported by NBRP (973 Program 2007CB814901) of China, NNSF Project (10771123) of China, RFDP (20070422034) of China, and NSF Projects (Y2006A13 and Q2007A05) of Shandong Province of China.