本文選題：隨機積分　切入點：隨機積分離散化　出處：《浙江大學》2016年博士論文

【摘要】：本文研究了在金融統(tǒng)計和計量經(jīng)濟中涉及隨機積分弱收斂的幾個問題：其一,本文基于Hayashi, Jacod和Yoshida (2011, Annales de l'Institut Henri Poincare 47,1197-1218)提出的隨機采樣的方法,結合隨機分析的一些技巧,精確地得到了隨機積分的離散化誤差鞅刻畫,并得到了隨機積分離散誤差的弱收斂結果.我們推廣了Hayashi等人(2011)的結果至更加一般的問題.同時,我們克服了Fukasawa (2011, The Annals of Applied Probability 21,1436-1465)結果中關于局部有界過程的限制.相比而言,本文中的假設條件更具有一般性,且不過度依賴于過程軌道性質(zhì).作為應用,本文研究了對沖誤差分布和隨機微分方程的近似解.其二,本文借鑒Bandi和Phillips (2003, Econometrica 71,241-283)中對連續(xù)擴散過程漂移系數(shù)估計時采用的雙光滑核估計方法,對帶跳擴散模型的漂移系數(shù)進行了研究.我們得到了漂移函數(shù)雙光滑估計量的漸近分布.本文假設帶跳擴散模型具有非平穩(wěn)性.因此,估計量的漸近分布往往具有隨機方差,利用常規(guī)證明方法很難得到漸近性質(zhì)的證明.本文利用局部鞅時間變換定理,將漂移函數(shù)雙光滑估計量看成一種特殊的離散化的隨機積分,利用隨機過程弱收斂方法得到了漸近分布.我們的結果中漸近分布可認為是由一種隨機積分分布變換得來.此外,相比于一般的局部常數(shù)估計,雙光滑的方法可以有效地減小漸近方差,提高估計的有效性.其三,本文對帶跳擴散模型的擴散系數(shù)的估計進行了研究.由于帶跳擴散模型中跳的存在會對擴散項估計產(chǎn)生很大的影響,為了克服這個困難,我們采用門限核估計的方法構造漂移項的估計量.更重要是,傳統(tǒng)的估計方法很難得到最優(yōu)窗寬,我們考慮時間跨度和采樣間隔同時變化,從而便于最優(yōu)窗寬的研究.我們利用局部鞅時間變換定理等隨機分析技巧,得到了帶跳擴散過程擴散系數(shù)核估計量的精確漸近表示,并且得到了最優(yōu)窗寬.最后,本文對含內(nèi)生變量非線性協(xié)整模型中參數(shù)的最小二乘估計問題進行研究.由于內(nèi)生變量的存在,以往的研究方法很難得到估計量的漸近分布.Liang, Phillips, Wang 和 Wang (2015, Econometric Theory即將發(fā)表)基于α-混合序列樣本進行了研究.由于α-混合系數(shù)在實際中很難刻畫,本文基于非平穩(wěn)ρ-混合序列樣本,利用鞅逼近的方法,將估計量巧妙地轉化為一類特殊的隨機積分.進一步利用隨機積分弱收斂的方法,得到了估計量的漸近分布.相比于Liang等人(2015)中a-混合系數(shù)的假設,本文關于ρ-混合系數(shù)的假設更實用.
[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 偽-mixed sequence samples. Because 偽-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.
【學位授予單位】：浙江大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O211.4

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本文編號：1657272