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.
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. In the same year, Black and Scholes [
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 a result, their methods fail to cover the full problems given in the FBSDE. Zhang and Lin [
This paper intends to explore the method to fulfill the terminal-dependent inference: quasi-instrumental variable methods. It is worth mentioning that the key point of our method is the use of the terminal condition information rather than neglecting it. This change leads to a completely new work among the existing researches. The key technique in our method is the use of quasi-instrumental variable which is similar but not the same as instrumental variable (IV). It is known that IV is widely employed in applied econometrics to achieve identification and carry out estimation and inference in the model containing endogenous explanatory variables or panel data; see Hsiao [
Through the backward equation (
We use the nonparametric form of the generator
Note that
The remainder of the paper is organized as follows. In Section
In this section, we propose a nonparametric estimator with the help of quasi-instrumental variable.
We begin the following original model by combining (
In this section, we present the statistical structure of FBSDEs by taking advantage of quasi-instrumental variable and obtain the consistent asymptotically normal estimators of
To construct terminal-dependent estimation for the generator
Before estimating the model function
For each
After plugging the estimator
As was shown in the nonparametric instrumental variables estimator of Hall and Horowitz [
Let
It is assumed that the support of
Let
To construct an estimator of
In this section, we study the asymptotic properties of our proposed estimators. All proofs are presented in Appendix.
To give the asymptotic results of
with
The continuous kernel function
Condition (a) is commonly used for weakly dependent process; see, for example, Kolmogorov and Rozanov [
Besides conditions (a), (b), and (c), let
The asymptotic result in Theorem
This section gives conditions under which the HH estimator of the generator
Define
The data
The distribution of
The constants
The tuning parameters
Consider
Let Assumptions
Let Assumptions
As was shown in the remark given in the previous section, even the conditional mean of error of the model is nonzero, and the newly proposed estimation is consistent because of the mixing dependency; for details see the proof of Theorem
In this section, we investigate the finite-sample behaviors by simulation.
We consider a simple FBSDE as
Firstly, let
The real lines are the true curves of
According to the theory of mathematical finance, we represent a European call option by the following FBSDEs model:
We use the Euler scheme to generate the price series of the stock as
The price series by Black Scholes formula is part of the solution of the FBSDEs above at discrete time points; that is,
We produce the true curve of the drift coefficient by
Let
The simulated curve and the estimated curves of
The simulated curve and the estimated curves of
Denote
From Lemma
Theorem
Let Assumptions
Define
Assumption
Let Assumptions
Define
Now consider
To analyze
Now consider
Now combine (
This completes the proof.
The author declares that there is no conflict of interests regarding the publication of this paper.