摘要
近年来,对倒向随机微分方程的研究得到了广泛的关注.它不仅与非线性偏微分方程有着密切的联系,更一般地,非线性半群,随机控制问题等也与此密不可分.同时,在数理金融中,未定权益的估价和定价理论也可以通过一个线性倒向随机微分方程表示.对倒向方程而言,投资组合的动态变量Y_t可由一个生成元为f的倒向方程来刻画,Z_t对应未定权益投资组合.特别地,当生成元f又同时是一个扩散过程的函数时,相应的方程称为正倒向随机微分方程.
基于正倒向随机微分方程的结构形式,我们提出一类新的模型,形式如下:这里X_t满足方程不引起混淆起见,仍然称函数f为生成元函数.这个模型与一般的倒向方程及正向方程都有所不同.与倒向方程相比,我们的模型不考虑终端条件;与正向随机微分方程相比,我们的模型中漂移项不仅含有扩散项函数,而且漂移项还与某个扩散方程有关,这一点是正向方程所不具备的.因而论文中考虑的问题是具有创新性的.
本学位论文中所考虑的是关于一类由正倒向随机微分方程衍生的模型(1)的半参数统计推断问题.当模型中生成元为线性函数,以及生成元受到不等式约束时,我们考虑了方程的半参数估计及假设检验问题.在这种情形下的倒向方程估计问题不同于非参数情形下的估计问题(参见Chen and Lin(2009)([97]),Yang and Yang(2006)([18])).如果生成元具有参数结构,我们的问题就变成一个半参数估计问题.虽然在参数估计过程中,有一个非参估计代入,我们仍然得到了参数估计的标准形式的渐进正态性.由此,我们可以进一步地考虑关于参数的假设检验问题.我们主要考虑了生成元具有线性结构时参数的假设检验问题,所借助的主要工具为渐进正态及经验似然方法.特别是对于经验似然,虽然在估计函数里有非参估计代入,我们仍然证明了所构造的经验似然比统计量是渐进χ~2分布的.对于有约束模型,我们试图得到在不等式约束下的统计结果.文中所得到的结果可以看作是现有结果的改善与深入,其中的渐进结果及假设检验和有约束下模型的估计结果都是全新的.本学位论文共分五章,包含如下部分的内容:
第一章主要介绍了正倒向方程的发展及我们提出的模型的相关知识,并给出了我们的模型与一般的倒向方程和正向方程的不同之处,从而意味着我们得到的结果是具有创新性和建设性的.很多在一般随机微分方程中常用的估计方法也做了简要介绍,回顾了参数模型,非参数模型及半参数模型中的估计方法,然后介绍了平稳过程及混合相依过程的定义,这对于考察我们模型的渐进统计性质是必要的.
在第二章中,我们主要考虑了如下形式模型的半参数估计及其渐进性质:这里X_t为几何布朗运动,满足方程:这里u,σ均为未知参数.
我们的目的是要在有非参估计(?)_t代入时,得到参数β= (c,μ)的半参数估计以及考虑估计的渐进性质.所得到的非参数及半参数估计都是简单可实现的,虽然有一个非参插入,但是我们在不需要高阶核,undersmoothing及纠偏方法时,仍然得到了估计以最优速度渐进正态性的结果.
将模型(2)离散化,观测时问间隔记为△,这里我们假设得到的是高频数据.假设在某时刻t_0,有观测数据从而形成n对综合数据由此,我们可以给出Z~2(x_0)的N-W核估计:
定理2.3.1设{X_(iΔ)~*,i = 0, ...,n - 1}是来自于平稳ρ混合马氏过程的观测序列,其混合系数满足ρ(l) =ρ~l,0 <ρ< 1.假设{X_(iΔ)~*,i = 0, ...,n - 1}同分布且其密度函数p(x)有界.对于p(·)支撑内的一个内点x_0,p(x_0)>0,Z~2(x_0)>0,同时假设p(·),Z(·)在x_0的邻域内有二阶连续可微函数.在条件(A2)的假设之下,当n→∞,s.t.nh→∞,及nhΔ~2→0时,我们有
(a) (?)~2(x_0)的渐进偏差为渐进方差为
(b)若进一步假设nh~5→0,则对于(?)~2(x_0),我们有如下的渐进正态性
这里μ_2 =∫_(-1)~1 u~2K(u)du, v_0 =∫_(-1)~1K~2(u)du.
上述定理给出了非参数估计(?)_t的渐进偏差,渐进方差及渐近正态性.下面将给出参数估计(?)的渐进正态性结果.
定理2.3.2除了定理2.3.1的条件之外,进一步假设条件(A1)-(A2)成立,以及当n→∞时,nΔ→∞,我们有这里V = E[Z_t~2]Σ~(-1),
第三章包含两方面的结果.首先,我们考虑了基于渐进正态性质下置信域的构造.这种方法里估计方差中包含了多个需要估计的统计量,在一定程度上会降低置信域的覆盖率,因而在第二部分,我们转而考虑经验似然方法下的置信域.这种方法不需估计未知参数.在两种情形之下,我们给出了置信域的构造,并通过模拟比较了两种方法下的置信域的覆盖率和置信域长度,得出结论投影经验似然方法更具有优势.
以下的两个定理都是基于渐进正态下置信域的构造.
定理3.2.1假设附录中给出的条件(A0)-(A3)满足.若μ取真值μ_0,那么当n→∞时,s.t.nhΔ~2→∞,nh~5→0及nΔ→∞,枢轴量(?)渐进于正态分布,即这里P(U 定理3.2.2假设附录中给出的条件(A0)-(A3)满足.若c取真值c_0,那么当n→∞时,nhΔ~2→∞,nh~5→0及nΔ→∞,枢轴量(?)渐进正态分布,即这里P(U 为了避免在估计方差中对未知项的估计,我们提出用经验似然方法构造置信域.经验似然方法是1988年由Owen([69])提出的.经验似然方法在构造置信域方面有许多突出的优点.例如:无需对未知参数进行估计,无需对渐进方差进行估计,置信域的形状由数据自行决定、域保持性、变换不变形、Barlett纠偏性以及无需构造枢轴统计量等.应用经验似然的经典证明方法,我们得到如下定理.
定理3.3.1假设附录中给出的条件(A0)-(A5)满足.若β_0为β的真值,则经验对数似然函数(?)(β_0)渐进χ_2~2分布,即这里P(χ_2~2≤c_α) = 1 -α,
重新记经验对数似然函数(?)(β_0)为(?)(c,μ).以下的两个定理是基于投影经验似然方法得到的.
定理3.3.2假设附录中给出的条件(A0)-(A5)满足.若μ_0为μ的真值,则L(μ_0)(?)渐进χ~2分布,即这里P(χ_1~2≤c_α) = 1-α,(?)为μ_0固定时,使(?)(c,μ_0)取得最小值的c.
定理3.3.3假设附录中给出的条件(A0)-(A5)满足.若c_0为c的真值,则T(c_0)渐进χ~2分布.即这里P(χ_1~2≤c_α) = 1-α,及T(c)(?)这里(?)为使(?)(c,μ)达到最小值的μ.
第四章,我们考虑了不等式约束下模型的统计推断问题,其中包含了两个方面的情况:一种是生成元为线性函数的情况,此时的约束相应地也是线性的.约束是基于倒向方程的生存性质给出的,不等式记为:Z(y,z)β≤Χ(y,z),这里Z(y,z)和Χ(y,z)是关于y,z的函数,记号可见第4.2节;另一种情况下,我们并不对生成元的函数作出限制,相应的约束为非线性的.
在第一种情况下,我们给出了不等式约束下的Z_t和参数β的估计.此时的问题转化为二次规划问题,相应的结果也是成立的.进一步,关于参数β,我们有如下的定理.
定理4.2.1如果不等式约束是正确的,那么存在足够大的样本容量n≥n_0,使得在此样本下,不等式约束下的最小二乘(ICLS)估计(?)退化为无约束下的最小二乘估计(?).
如果Z_t已知,我们有如下的定理:
定理4.2.2若原假设H_0:Z(y,z)β≤Χ(y,z)成立,似然比统计量LR的分布有如下的性质:对所有的c>0,这里F_(m,n)为自由度为m,n的F-分布，A=(?)，U=(?),这里ω(·)为权重.
实际上,在我们的问题中,Z_t是未知的不可观测的量,只要我们选取Z_t的无偏估计代入,对于上面的似然比统计量,会有类似的结果.
如果生成元没有假定特定的形式,那么相应的问题就转化为非线性不等式约束下的估计问题.我们采取的方法是利用不等式约束下的最小二乘方法得到转换数据,进而通过非参数平滑估计生成元.
假设对原始数据(?)应用约束最小二乘得到转换数据{m_i}_(i=1)~n.那么由这些转换数据及对数凸核函数得到的局部线性估计会满足约束条件.
The theory of Backward Stochastic Differential Equations (BSDEs for short) has been considered with great interest in the last several years not only because of its connections with the non-linear partial differential equations and more generally the theory of non-linear semi-groups, but also stochastic control problems. At the same time, in mathematical finance, the theory of the hedging and pricing of a contingent claim is typically expressed in terms of a linear BSDE.
The dynamic value of replicating portfolio Y_t is given by a BSDE with a generator f, with Z_t corresponding to the hedging portfolio. Particularly, when the generator function is connected with another stochastic process characterized by diffusion process, the equation is called Forward Backward Stochastic Differential Equation (short as FBSDE).
Based on the form of FBSDE, we propose a class of model generated from FBSDE, which has the form as followswhere X_t is a stochastic differential equationWithout any confusion, we still call the function f as generator funcion. Additionally, the function Y_t and Z_t are assumed to be connected with X_t.
Note that our model is different from the ordinary stochastic differential equation (short as OSDE). The drift term not only contains diffusion term but also is connected with some diffusion process, which is not shared by OSDE. In addition, our model has the same representation as BSDE except the terminal condition.
This dissertation focuses on the semi-parametric inference for the model (1) generated from FBSDE. We consider the estimation and hypothesis test problem when the generator function is linear and that of under inequality constraints. For the estimation problem in our model, our estimation procedure is different from that of nonparametric estimation (see Yang and Yang(2006)([97]), Chen and Lin (2009)([18]) for details). When the generator function f has a parametric form, the resulting model is semi-parametric model. Though there is a nonparametric plug-in, the standard normality of the parametric estimator is obtained. Then, the corresponding hypothesis test problem is coming into existence. We construct the confidence regions for the coefficients of the linear generator function with two different tools: asymptotic normality and profile empirical likelihood(short as EL). Especially for the profile EL method, though there is a plug-in nonparametric estimator in the estimating equation, the empirical log-likelihood ratio statistic still tends to a standardχ~2 variable in distribution. For model (1) with constraints on f, we try to get the statistical inference results under inequality restrictions. These obtained results represent extensions and improvements of existing results, some of which are thoroughly new advancements in the area of statistical inference for the related fields. This dissertation consists of four chapters, whose main contents are described as follows:
In Chapter one, we give some introductions about the (Forward) Backward Stochastic Differential Equation and the proposed model. Additionally, the differences between our model and FBSDE, OSDE are illustrated, which mean that our results are meaningful and constructive. Some fundamental estimating methods applied in ordinary stochastic differential equations (OSDEs) are introduced, which include an overview of estimation on parametric, nonparametric and semi-parametric models and the optimal convergence rates of the estimators. Then, we give the definition of stationary and mixing process, which is necessary to the asymptotic behavior for our model.
In Chapter two, we consider the following proposed model:where X_t is a simple Geometric Brownian motion satisfying with u andσthe unknown parameters.
Our aim is to give the semi-parametric estimation and asymptotic property of the parameterβ= (c,μ) with nonparametric plug-in estimator (?)_t. Both the nonparametric and parametric estimators are computationally feasible and the asymptotic properties are standard in the sense of normality. Although there is a plug-in nonparametric estimator in parametric estimation, the high order kernel, under-smoothing and bias correction are not required.
Discretizing the model (2) with a sampling intervalΔtending to zero, given the initial calendar time point t_0, we have the observationsthen form n pairs of synthetic datafrom which we can give the N-W kernel estimator of Z~2(x_0),
Theorem 2.3.1 Let {X_(iΔ)~*,i = 0, ...,n - 1} be a sequence of observation on a stationaryρ-mixing Markov process with coefficient satisfyingρ(l) =ρ~l,0 <ρ< 1. Assume that {X_(iΔ)~*,i = 0, ...,n - 1} have a common and bounded density function p(x). For any given x_0 in the interior of the support of p(·), p(x_0) > 0, Z~2(x_0) > 0, p(·),Z(·) is second order continuous differentiable function in a neighborhood of x_0. Under the condition (A2), then as n→∞, such that nh→∞, and nhΔ~2→0,
(a) The asymptotic bias of (?)~2(x_0) is given bythe asymptotic variance is
(b) Assume further that nh~5→0, then we have the asymptotic normality:
whereμ_2 =∫_(-1)~1 u~2K(u)du, v_0 =∫_(-1)~1K~2(u)du.
The theorem gives the bias, variance and asymptotic normality of the nonparametric estimator (?)_t. Next is the asymptotic normality for the parametric estimator (?).
Theorem 2.3.2 In addition to the condition of Theorem 2.3.1, under the condition (A1)-(A2) in Appendix and nΔ→∞as n→∞, we havewhere V = E[Z_t~2]Σ~(-1),
Chapter three include two main contents. Firstly, we consider the confidence intervals based on the asymptotic normality. However, there contain several unknown statistics in estimated variance, which will slow down the convergence accuracy of confidence intervals. Then, we turn to another method, profile empirical likelihood method, which is free of parametric estimation. In both cases, we give the construction of confidence intervals, and compare the two methods in terms of coverage accuracy and average length of confidence intervals with simulations.
The following two theorems are concerned with the confidence intervals based on asymptotic normality.
Theorem 3.2.1 Assume that conditions (A0)-(A3) given in Appendix hold. Ifμ_0 is the true value ofμthen as n→∞, nhΔ~2→∞, nh~5→0 andnΔ→∞, the pivotal quantity (?) has an asymptotic standardnormal distribution. That is,with P(U Theorem 3.2.2 Assume that conditions (A0)-(A3) given in Appendix hold. If c_0 is the true value of c, then as n→∞, nhΔ~2→∞, nh~5→0, and nΔ→∞, the pivotal quantity (?) has an asymptotic standard normal distribution. That iswith P(U < c_α) = 1 -α, where U -N(0,1) and
To be avoid of estimating unknown terms in estimated variance, we propose empirical likelihood introduced by Owen (1988) ([69]). Many advantages of the empirical likelihood over the the normal approximated-based method have been shown in the literature. In particular, it does not impose prior constraints on the shape of region; it does not require the construction of a pivotal quantity and the region is range preserving and transformation respecting; see for example Hall and La Scala (1990)([44]). Applying the classical procedure in EL, we obtain the following theorem for the parameterβ.
Theorem 3.3.1 Assume that conditions (A0)-(A5) given in Appendix hold. Ifβ_0 is the true value ofβ, then the empirical log-likelihood function (?)(β_0) has an asymptoticχ_2~2 distribution. That iswith P(χ_2~2≤c_α) = 1 -α, and
Rewrite the empirical log-likelihood function (?)(β) as (?)(c,μ), then we have the following two theorems based on the profile EL method.
Theorem 3.3.2 Assume that conditions (A0)-(A5) given in Appendix hold, ifμ_0 is the true value ofμ, then (?) has an asymptoticχ_1~2 distribution. That iswith P(χ_1~2≤c_α) = 1-α, and (?) the optimal value minimizing (?)(c,μ_0) for fixedμ_0.
Theorem 3.3.3 Assume that conditions (A0)-(A5) given in Appendix hold. if c_0 is the true value of c, then T(c_0) has an asymptoticχ_1~2 distribution. That iswith P(χ_1~2≤c_α) = 1-α,, and (?) with (?) be the optimalμwhich minimizes (?)(c,μ).
In Chapter four, we consider the statistical inference problem for our proposed model under inequality constraints, which include two kind of cases. One is that of the linear generator function with inequality constraint. The inequality constraint is based on the viability property of BSDE, which is denoted by Z(y,z)β≤Χ(y,z), here Z(y,z) andΧ(y,z) are functions with respect to y, z as stated in section 4.2. In the other case, we do not impose any functional form on the generator function, and the corresponding restriction is a nonlinear one.
About the first case, we propose the estimation procedure for unknown Z_t and the parameterβunder inequality constraints. The problem can be reduced to the quadratic program problem, therefore the parrelled results are satisfied.
Theorem 4.2.1 If the prior belief on the inequality constraint is correct, there exists a sufficiently large samples n≥n_0 such that the inequality constrained least squares (ICLS) estimator vector (?) on such large samples reduces to the unrestricted LS estimator vector (?).
In addition, if we pretend that Z_t is known, the theorem of likelihood ratio test statistic for the parameterβis satisfied.
Theorem 4.2.2 Under the null hypothesis H_0 : Z(y,z)β≤Χ(y,z), the distribution of LR, the likelihood ratio statistic, has the following property: for all c > 0,withω(·) the weights, F_(m,n) the F- distribution with freedom m,n, A =(?),U=(?).
However, in our problem Z_t is unobservable. We can choose an unbised estimator (?)_t to replace Z_t, from which we can obtain the similar results as Theorem 4.2.2.
When the generator function has no specific form, the corresponding problem is that of under nonlinear inequality constraints. We combine inequality restraints with nonparametric regression to estimate the generator function within a local polynomial estimate procedure. We are able to implement the method in such a way that the locally polynomial estimator will always produce estimators satisfying the constraints, which is also possible with some of the other methods, but in our case turns out to require no modification to the estimator, only its application to some transformed data.
Assume that the transformed data {m_i}_(i=1)~n result from applying the constrained least squares algorithm to the original data (?). Then the locally linear estimator obtained from the transformed data and a log-concave kernel function satisfies the required constraints in sample.
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