| With the development of modern society and the deepening of financial field, financial products have become the indispensable part of people’s life, and various mathematical models and analysis tools for dealing with the fi-nancial market transactions, which including investment portfolio analysis, asset pricing and financial risk measure, also emerge in an endless stream. S-ince the nineteen nineties, the worldwide financial crises happened frequently have highlighted the significance of such research to prevent these disaster-s. Backward stochastic differential equation (BSDE) has been well devel-oped in financial mathematics, and Forward-backward stochastic differential equations (FBSDE) is also playing its increasingly important role. In addi-tion, among the fields of the economic and financial, engineering technology, biotechnology and others, we encounter more large sample complicated data, especially the high-dimension data, which bring not only the model specifica-tion great challenges by the curse of dimensionality, and also rich information hidden behind just like the gospel of dimensionality. All these above require us finding effective way managing multivariate and high-dimensional data in order to cope with their applications in statistical modeling and econometric analysis, etc.Backward Stochastic Differential Equation (BSDE) has been well stud-ied and widely applied in mathematical finance. 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 our knowledge, the terminal-dependent statistical inference for such a model has not been ex-plored in the existing literature. This paper proposes two terminal-dependent estimation methods via a terminal control variable model and the integral form of Forward-backward Stochastic Differential Equation (FBSDE). The reasons why we do so are that the resulting models contain terminal condition as model variable and therefore the newly proposed inference procedures in-herit the terminal-dependent characteristic. In this paper, the FBSDE is first rewritten as the regression versions and then the semi-parametric estimation procedures are proposed. Because of the control variable and integral form, the newly proposed regression versions are more complex than the classical ones and thus the inference methods are somewhat different from which de-signed for the OSDE. Even so, the statistical properties of the new methods are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors.To reduce the curse of dimensionality arising from nonparametric esti-mation procedure for multiple nonparametric regression, in this paper we sug-gest a simulationbased two-stage estimation. We first introduce a simulation-based method to decompose the multiple nonparametric regression into two parts. The first part can be estimated with the parametric convergence rate and the second part is small enough so that it can be approximated by or-thogonal basis functions with a small trade-off parameter. Then the linear combination of the first and second step estimators results in a two-stage estimator for multiple regression function. Our method does not need any specified structural assumption on regression function and it is proved that the newly proposed estimation is always consistent even if the trade-off pa-rameter is designed to be small. Thus when the common nonparametric estimator such as local linear smoothing collapses because of the curse of dimensionality, our estimator still works well.Misspecified models have attracted much attention in some fields such as statistics and econometrics. When a global misspecification exists, even the model contains a large number of parameters and predictors, the mis-specification cannot disappear and sometimes it instead goes further away from the true one. Then the inference and correction for such a model are of very importance. In this paper we use the generalized method of moments (GMM) to infer the misspecified model with diverging numbers of param-eters and predictors, and to investigate its asymptotic behaviors, such as local and global consistency, and asymptotic normality. Furthermore, we suggest a semiparametric correction to reduce the global misspefication and, consequently, to improve the estimation and enhance the modeling. The theoretical results and the numerical comparisons show that the corrected estimation and fitting are better than the existing ones.This dissertation consists of five chapters. Its main conclusions and innovations are organized as follows:In Chapter1, after reviewing the FBSDE model, multiple nonparametric model and the misspecified model with diverging numbers of parameters and predictors, we survey briefly various statistical modelings and existing infer-ence methods, and point out their relative merits. We put forward research backgrounds and theory foundations for three kinds terminal-dependent sta-tistical inference for the FBSDE, simulation-based two-stage estimation for multiple nonparametric regression, GMM and misspecification correction for misspecified models with diverging number of parameters.Chapter2investigates the following FBSDE model, expresses the FBSDE as a statistical framework, assuming g(t, Yt, Zt)=bYt+cZt.Let{Xi,Yi,i=1,...,n} be the observed time series data. Since the distribution of ξ is supposed known, we can get its sample as{ξi,1≤i≤m} for m≥1/Δn, then the original model can be approximately rewritten through integral discretization asThen address the proper estimator of g and Zt. We might adopt the N-W kernel nonparametric method to estimate Zt as Its asymptotic property is shown as below.Theorem2.1Besides the conditions (2.1),(2.2) and (2.3), suppose that Xi∈(x0-h, x0+h) is a stationary ρ-mixing Markov process with the p-mixing coefficients satisfying ρ(l)=ρl for0<ρ<1, and has a common probability density p(x) satisfying p(x0)>0. Furthermore, functions p(x) and Zx have continuous two derivatives in a neighborhood of X0. As n'∞, if nh'∞, nh5'0and nhΔ2'0, thenFrom the above, it is simple to deduce the estimator of β=(b, c)τ with common parametric methods. For example, the least square (LS) estimator is obtained by minimizing The following theorem states the result aomost standard as the asymptotic normality with the convergence rate of (n) Theorem2.2Besides the conditions(2.1),(2.2),(2.3)and(2.4), suppose that{Xi,i=1,…,n}is a stationary ρ-mixing Markou process with the ρ-mixing coefficients satisfying ρ(l)=ρ for0<ρ<1and has a common probability density p(x)satisfying p(x0)>0. Furthermore,functions p(x) and Zx have continuous two derivatives in a neighborhood of x0.As n'∞, if nh'∞,nh5'0and nhΔ2'0,then where σ2=Var(ξ/T).In chapter3,we first introduce the terminal control variable model, where m(Xt,ξ)=E(Zt(B+Δ-Bt)|Xt,ξ),ut=Zt(Bt+Δ-Bt)-m(Xt,ξ). When Δ tends to zero quite fast,we can derive the estimator of β by mini-mizing Otherwise,the flollowing estimating equation could be used to derive the estimator of β: Then the estimators have the closed representations of βTC,whose asymp-totic distribution is as follows.Theorem3.1Besides the conditions(2.1),(2.2),(2.3)and(3.1),sup-pose that {Xi,i=1,…,n}is a stationary ρ-mixing Markou process with the ρ-mixing coefficients satisfying ρ(l)=ρl for0<p<1.Furthermore,(Xt,ξ)has a joint probability density pXt,ξ(x0,ξ0),and functions pXt,ξ(x0,ξ0), m(x0,ξ0)and Zx0,ξ0have continuous two derivatives in neighborhood of(x0,ξ0). As n,m'∞and h'0,if nmh2'∞,then By the end of this Chapter we also briefly analyzes the FBSDE mod-el by means of the Bayesian method in the cases including one single risk investment, and K candidates instead, and infer the posterior distributions and major estimation procedures.To reduce the curse of dimensionality arising from nonparametric esti-mation procedures for multiple nonparametric regression without any speci-fied structural assumption on the regression function, in Chapter4we suggest a simulation-based two-stage estimation, then the resultant estimator is and we present its asymptotic behavior as follow,Theorem4.1If E(Y2) and E(fU2(U, σU2)) both exist, the components U(1),...U(d) of U are designed to be independent, fz(z)>0for all z=x+u∈∈X∪U the maximum eigenvalue of Pm(x)P’m(x) is bounded, and r(x) belongs to the Soblev ellipsoid S(β,L) defined in (4.2.13), and can be expressed by a series expansion of the cosine basis functions as above, then Particulary, if βj=β0and mj=O(nδ) for all j and then with ρ=β0d/(2(β0d+1))-log(γU(m)Ld)/(β0d+1) log n).The last Chapter we use the generalized method of moments (GMM) to infer the misspecified model with diverging numbers of parameters and predictors, which means g{x,θ) is globally biased such as, We only consider the estimator of GMM defined by and investigate its asymptotic behaviors, such as local and global consistency, and asymptotic normality.Theorem5.1Suppose the Assumptions5.1-5.2hold. When n tends to infinity, if then there is a local minimizer θn of Q(θ) such thatTheorem5.2Under the Assumptions5.1-5.3, if λminΛ>C for a positive constant C, and (13) holds, then θn satisfies (14).Theorem5.3Suppose the Assumptions5.1-5.4hold. When n tends to infinity, if then where D stands for the convergence in distribution.Furthermore, the additive regression model is defined by finally an corrected version of estimating function valued at θn0is given by consequently, we suggest an corrected estimator as The theoretical results show that the corrected estimation and fitting are better than the existing ones.Theorem5.4Suppose that in additive regression model (16) only the term r(xn2,θn2) is misspecified, r(xn2,θn2) and r0(xn2) have two continuous derivatives with respect to xn2. Then for j=1,2,...,qn and0<xn2<1.Theorem5.5Under the condition of Theorem5.1and Assumption5.5, then there is a local minimizer θn of Q(θ) such that, if thenSimulations are used to illustrate various methods. |
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