随机积分的弱收敛及应用

发布时间：2018-03-24 07:16

本文选题：随机积分　切入点：随机积分离散化　出处：《浙江大学》2016年博士论文

【摘要】：本文研究了在金融统计和计量经济中涉及随机积分弱收敛的几个问题：其一,本文基于Hayashi, Jacod和Yoshida (2011, Annales de l'Institut Henri Poincare 47,1197-1218)提出的随机采样的方法,结合随机分析的一些技巧,精确地得到了随机积分的离散化误差鞅刻画,并得到了随机积分离散误差的弱收敛结果.我们推广了Hayashi等人(2011)的结果至更加一般的问题.同时,我们克服了Fukasawa (2011, The Annals of Applied Probability 21,1436-1465)结果中关于局部有界过程的限制.相比而言,本文中的假设条件更具有一般性,且不过度依赖于过程轨道性质.作为应用,本文研究了对冲误差分布和随机微分方程的近似解.其二,本文借鉴Bandi和Phillips (2003, Econometrica 71,241-283)中对连续扩散过程漂移系数估计时采用的双光滑核估计方法,对带跳扩散模型的漂移系数进行了研究.我们得到了漂移函数双光滑估计量的渐近分布.本文假设带跳扩散模型具有非平稳性.因此,估计量的渐近分布往往具有随机方差,利用常规证明方法很难得到渐近性质的证明.本文利用局部鞅时间变换定理,将漂移函数双光滑估计量看成一种特殊的离散化的随机积分,利用随机过程弱收敛方法得到了渐近分布.我们的结果中渐近分布可认为是由一种随机积分分布变换得来.此外,相比于一般的局部常数估计,双光滑的方法可以有效地减小渐近方差,提高估计的有效性.其三,本文对带跳扩散模型的扩散系数的估计进行了研究.由于带跳扩散模型中跳的存在会对扩散项估计产生很大的影响,为了克服这个困难,我们采用门限核估计的方法构造漂移项的估计量.更重要是,传统的估计方法很难得到最优窗宽,我们考虑时间跨度和采样间隔同时变化,从而便于最优窗宽的研究.我们利用局部鞅时间变换定理等随机分析技巧,得到了带跳扩散过程扩散系数核估计量的精确渐近表示,并且得到了最优窗宽.最后,本文对含内生变量非线性协整模型中参数的最小二乘估计问题进行研究.由于内生变量的存在,以往的研究方法很难得到估计量的渐近分布.Liang, Phillips, Wang 和 Wang (2015, Econometric Theory即将发表)基于α-混合序列样本进行了研究.由于α-混合系数在实际中很难刻画,本文基于非平稳ρ-混合序列样本,利用鞅逼近的方法,将估计量巧妙地转化为一类特殊的随机积分.进一步利用随机积分弱收敛的方法,得到了估计量的渐近分布.相比于Liang等人(2015)中a-混合系数的假设,本文关于ρ-混合系数的假设更实用.
[Abstract]:In this paper, several problems involving weak convergence of stochastic integrals in financial statistics and econometrics are studied. Firstly, based on the methods of random sampling proposed by Hayashi, Jacod and Yoshida 2011, Annales de l'Institut Henri Poincare 477-1218), this paper combines some techniques of stochastic analysis. The discretization error martingale characterization of stochastic integral is obtained accurately, and the weak convergence result of discrete error of stochastic integral is obtained. We generalize the result of Hayashi et al. 2011 to a more general problem. We overcome the limitation of the local bounded process in the results of Fukasawa 2011, The Annals of Applied Probability 21n 1436-1465). By comparison, the assumptions in this paper are more general and do not depend too much on the properties of the process orbit. In this paper, we study the distribution of hedging errors and the approximate solutions of stochastic differential equations. Secondly, we use the double smooth kernel estimation method used in the estimation of drift coefficients for continuous diffusion processes using Bandi and Phillips's 2003, Econometrica 71241-283. In this paper, we study the drift coefficient of the diffusion model with jump. We obtain the asymptotic distribution of the double smooth estimator of drift function. In this paper, we assume that the diffusion model with hopping is nonstationary. Therefore, the asymptotic distribution of the estimator often has random variance. It is difficult to obtain the asymptotic property by using the conventional proof method. In this paper, the double smooth estimator of drift function is regarded as a special discrete stochastic integral by using the local martingale time transformation theorem. The asymptotic distribution is obtained by using the weak convergence method of stochastic processes. In our results, the asymptotic distribution can be considered as a transformation of a stochastic integral distribution. In addition, compared with the general local constant estimation, The double smooth method can effectively reduce the asymptotic variance and improve the validity of the estimation. In this paper, the estimation of diffusion coefficient of the diffusion model with jump is studied. In order to overcome this difficulty, the existence of jump in the model has a great influence on the estimation of diffusion term. We use the threshold kernel estimation method to construct the estimation of drift term. More importantly, the traditional estimation method is difficult to obtain the optimal window width. We consider the time span and sampling interval change simultaneously. By using random analysis techniques such as the local martingale time transformation theorem, we obtain the exact asymptotic representation of the kernel estimator of diffusion coefficient in the diffusion process with hopping, and obtain the optimal window width. In this paper, the problem of least square estimation of parameters in nonlinear cointegration model with endogenous variables is studied. It is very difficult to obtain asymptotic distribution of estimators. Phillips, Wang and Wang 2015, Econometric Theory to be published in the past) based on 伪-mixed sequence samples. Because 伪-mixing coefficients are difficult to characterize in practice. In this paper, based on the samples of non-stationary 蟻 -mixed sequences, the estimator is subtly transformed into a special kind of stochastic integral by means of martingale approximation, and the method of weak convergence of stochastic integral is further used. The asymptotic distribution of the estimator is obtained. Compared with the assumption of a-mixing coefficient in Liang et al. 2015, the assumption of 蟻 -mixing coefficient is more practical in this paper.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O211.4

【参考文献】