Terminal-Dependent Statistical Inference for the FBSDEs Models

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 FBSDEparameters fromnoisy 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.


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.In the same year, Black and Scholes [1] provided their celebrated formula for option pricing and Merton [2] proposed a general equilibrium model for security prices.Since then, modern finance has adopted stochastic differential equations as its basic instruments for portfolio management, asset pricing, risk management, and so on.Among these models, the backward stochastic differential equations (BSDEs for short) are a desirable choice for hedging and pricing an option.Its general form is as follows: = − (,   ,   )  +     ,   = ,  ∈ [, ] , ( where  is the generator,   is a Brownian motion, and  is a R-valued Borel function as the terminal condition.Usually the terminal condition is designed as a random variable with given distribution.If  meets certain conditions, the BSDE has a unique solution. In terms of the backward equation, within a complete market, it serves to characterize the dynamic value of replicating portfolio   with a final wealth  and a special quantity   that depends on the hedging portfolio.In particular, while the generator consists of diffusion process, the corresponding equation is proved to be a forward-backward stochastic differential equation (FBSDE), which can be expressed as   = − (,   ,   ,   )  +     ,   = , (2) where   satisfies the following ordinary stochastic differential equation (OSDE): =  (,   )  +  (,   )   ,  ∈ [, ] .
Compared to the OSDE that contains an initial condition, the solution of the FBSDE is affected by the terminal condition   = (  ).As is well known, there exist a number of parametric and nonparametric methods to deal with estimation and test for the OSDE.However, these methods cannot be directly employed to infer the BSDE and FBSDE because the two models are related to a terminal condition.Forward-backward stochastic differential equations are used in biology systems, mathematical finance, insurance, real estate, multiagent, and network control.See Antonelli [3], Wang et al. [4], Zhang and Li [5], and so on.
For the FBSDE defined above, the statistical inference was investigated initially by Su and Lin [6] and Chen and Lin [7].Furthermore, by financial and ecological problems, a relevant statistical model was proposed by Lin et al. [8].However, they did not take the terminal condition into account in the inference procedure.In the framework of the FBSDE mentioned above, the terminal condition is additional, which is not nested into the equation.Thus, there is an essential difficulty to use the terminal condition to refine the inference procedure.
As a result, their methods fail to cover the full problems given in the FBSDE.Zhang and Lin [9] proposed two terminal-dependent estimation methods via terminal control variable for the integral form models of FBSDE.However, they only considered the parametric form of the generator  in their paper.
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 [10] for an overview of the relevant statistical inference and econometric interpretation and see Hall and Horowitz [11] for recent work on nonparametric instrumental variable estimation.
Through the backward equation ( 2) of FBSDE, we get a regression model.To use the terminal condition information, we put the terminal condition as a quasi-instrumental variable and introduce it into our model.However, when a constraint is appended artificially, the original model may change to be biased in the sense of (    |   , ) ̸ = 0, because the constraint condition influences the increase trend of wealth so that     may deviate from the original center zero; in other words, due to the constraint, the trajectory of   may departure from the original expectation so that     cannot be regarded as error.Therefore, 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  enters into the equation as a control variable.This remodeling method takes advantage of the terminal information naturally, and the estimator performs quite well.
We use the nonparametric form of the generator  in model (2) because the correct FBSDEs model for a specific topic can neither be provided automatically by financial market nor be derived from theory of mathematical finance, and in lack of prior information about the structure of a model, nonparametric inference can provide a flexible as well as robust description of a data-generating process.Even in some cases when parametric models are available, nonparametric methods are still employed to avoid the model misspecification that may lead to large errors in option pricing and other problems from financial market.So we adopt the nonparametric form that can endow the model (2) with flexibility and robustness.
Note that   is usually unobservable and  cannot be completely specified in the financial market.The problems of interest are therefore to give both proper estimations of the generator  and the process   based on the observed data (  ,   ) and the terminal expectation .
The remainder of the paper is organized as follows.In Section 2, the FBSDE is rebuilt as a nonparametric model that contains the terminal condition as a quasi-instrumental variable.Consequently, a terminal-dependent estimation procedure is proposed.Next we discuss the asymptotic properties of the newly proposed estimations in Section 3. Simulation study is proposed in Section 4 to illustrate our methods.The proofs of the theorems are presented in Appendix.

Model and Method
In this section, we propose a nonparametric estimator with the help of quasi-instrumental variable.

Model and Its Statistical Version.
We begin the following original model by combining (2)-(3): where   is the standard Brownian motion and  is a Rvalued Borel function.Here the generator  is a function of ,   ,   , and   .For the FBSDEs model ( 4), only one of the backward components,   , and the forward components,   , can be observed.Another backward component   is totally unobservable.Furthermore, the adapted process   and terminal condition could be indicated as a function of   .
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  and   based on observed data {  ,   } and the terminal condition .(4).To construct terminaldependent estimation for the generator  and process   , the key technique is how to plug the terminal condition into the equation.When  is plugged into the model, we call it the quasi-IV, similar but not the same as IV.Evidently, the property of Brownian motion shows that (    |   ) = 0, but (    |   , ) ̸ = 0, which means drawing the terminal control directly into the equation as the condition should not be encouraged at the cost of model bias.Rewriting the first equation of (4) enables us to construct an unbiased model:

Remodeling for Model
where (  , ) = (    |   , ),   =     − (  , ), and (  |   , ) = 0.The newly defined model ( 5), together with the second equation in (4), can be thought of as a quasi-IV FBSDE.Because the equation in ( 5) contains the terminal condition , we can construct the terminal-dependent estimation.From the above definitions, we see that, by bias correction, the original model changes to be an additive nonparametric model with nonparametric components −(,   ,   ,   ) and (  , ).It shows that when terminal condition is regarded as a quasi-IV and then appended to the model, the result model is unbiased and changes to be nonparametric additive model.

Estimation for 𝑍
At any time point  ∈ [, ],  ,  , denoting   and satisfying the initial condition (, ), is a determined function of  ,  .As was shown by Su and Lin [6] and Chen and Lin [7], we can adopt a difference-based method to approximate  2 as It shows that the numerical approximation error to  2  converges to zero at rate of order   (Δ).

Estimation for Generator
Since m( 0 ,  0 ) and Ẑ2  0 , are the consistent estimator of ( 0 ,  0 ) and  2  0 , , respectively, we use them instead of (  , ) and   in the above model and we get Because Ẑ is function of   and   , for simplicity of presentation, we denote (,   ,   , Ẑ ) = (  ,   ).Thus, the model becomes Let Y  = (( +Δ −   ) − m(  , ))/Δ, X  =   , Z  =   , W = , and U  =   / √ Δ; the model becomes It is assumed that the support of (X, Z, W) is contained in [0, 1] 3 .This assumption can always be satisfied by, if necessary, carrying out monotone increasing transformations of X, Z, and W. For example, one can replace X, Z, and W by Φ(X), Φ(Z), and Φ(W), where Φ is the normal distribution function.We take (Y , X, Z, W, U) to be a vector, where Y and U are scalars, X and W are supported on [0, 1], and Z is supported on [0, 1].The model is where (Y  , X  , Z  , W  , U  ), for  ≥ 1, are independent and identically distributed as (Y , X, Z, W, U).Thus, X and Z are endogenous and exogenous explanatory variables, respectively.Data (Y  , X  , Z  , W  , U  ), for 1 ≤  ≤ , are observed.
Let  XZW denote the density of (X, Z, W), write  Z for the density of Z, and, for each  1 ,  2 ∈ [0, 1]  , and put It may be proved that, for each  for which  −1  exists,  (, ) where  W|Z denotes the expectation with respect to the distribution of W conditional on Z.In this formulation, ( −1   XZW )(, , W) denoted the result of applying  −1  to the function  XZW (⋅, , W) and evaluating the resulting function at .

Asymptotic Results
In this section, we study the asymptotic properties of our proposed estimators.All proofs are presented in Appendix.

Asymptotic results of Ẑ𝑠 .
To give the asymptotic results of Ẑ , we need the following conditions.
(c) The continuous kernel function (⋅) is symmetric about 0, with a support of interval [−1, 1], and Condition (a) is commonly used for weakly dependent process; see, for example, Kolmogorov and Rozanov [12], Bradley and Bryc [13], Lu and Lin [14], and Su and Lin [6].Condition (b) is also reasonable because, as is shown by (10),   can be regarded as the deviation between the adjacent two observations.Condition (c) is standard for nonparametric kernel function.
holds except, possibly, on a set of (, ) values whose Lebesgue is 0.
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 8. Furthermore, because of the terminal condition, the asymptotic variance is larger than that without the use of the terminal condition.

Simulation Studies
In this section, we investigate the finite-sample behaviors by simulation.
Example 10.We consider a simple FBSDE as where   is Geometric Brownian motion for modeling stock price satisfying while the riskless asset is the same as formula (31);   =  0 .
Example 11.According to the theory of mathematical finance, we represent a European call option by the following FBSDEs model: Here {  } 0≤≤ and {  } 0≤≤ are the price processes of the stock and the option, respectively, and  is the striking price at the expiration date .{  } 0≤≤ follows the geometric Brownian motion as We use the Euler scheme to generate the price series of the stock as where {  } −1 =0 is an i.i.d.series with standard normality.

Figure 1 :
Figure 1: The real lines are the true curves of   and function (), respectively, and the dashed ones are estimated curves for them in Example 10.

Figure 2 :Figure 3 :Proof of Theorem 1 .
Figure 2: The simulated curve and the estimated curves of   in Example 11.