Weak Convergence To Stochastic Integrals For Econometric Applications

This paper studies some problems about the weak convergence of stochastic integral involved in financial statistics and econometric.Firstly, Based on the method of random sampling scheme put forwarded by Hayashi, Jacod and Yoshida (2011, Annales DE l’Institut Henri Poincar e 47,1197-1218). Combine this sampling scheme with Some method of stochastic analysis. We obtain the weak convergence result of the discretization error of the stochastic integral. We generalize the The results in Hayashi ea,al (2011) to more general problems. At The same time, we overcame the limitation of local bounded martingale in Fukasawa ((2011), The Annals of Applied aim-listed Probability 21,1436-1465). By contrast, The hypothesis in this paper is more general, and not overly dependent on the properties of the path of process. As an application, the hedge error distribution and the approximate solution of the stochastic differential equation are studied in this paper.Secondly, Based on the double smooth nuclear estimation method, which was used to estimate the drift coefficient of continuous diffusion process in Bandi and Phillips (2003, Econometrica 71,241-283), we studied the drift coefficient of jump diffusion model. We got the asymptotic distribution of double smoothing esti-mator of drift coefficient. We assume the jump diffusion model is non-stationary. Therefore, estimate the asymptotic distribution of often with random variance, using conventional method is hard to get the proof of asymptotic properties. In this paper, using local martingale time transformation theorem, the double s-moothed estimator of drift function similar to a special kind of discretization of stochastic integral, using the weak convergence method of stochastic process get the asymptotic distribution. The asymptotic distribution in our result is consid-ered by a random distribution of integral transformation. In addition, compared with general local constant estimation, double smoothing method can effectively reduce the asymptotic variance, improve the effectiveness of the estimate.Thirdly, We estimate the diffusion coefficient of jump diffusion model. The jump in the jump diffusion model will have great influence on diffusion term estimation, we adopt the method of threshold kernel estimating estimator to overcome the effective of jump. More important is that the traditional estimation method is hard to get the optimal bandwidth, we consider the time span and sampling interval changes at the same time, to facilitate the optimal bandwidth research. We use time transformation theorem of local martingale and some other stochastic analysis technique, the estimator of diffusion coefficient of kernel estimator’s asymptotic property, and got the optimal bandwidth.Finally, the least squared estimators of coefficients in the nonlinear co-integration model, which including endogenous variable, are studied here. As a result of the existence of endogenous variable, previous methods are difficult to get the asymptotic distribution of the estimator. Liang, Phillips, Wang and Wang (2015, Econometric, in press) studied this problem based on the samples of α-mixing, α-mixing coefficient is hard to describe in practice, we based on non-stationary ρ-mixing series, using the method of martingale approach, the estimator neatly into a special kind of stochastic integral. Furthermore, using the weak convergence of stochastic integral method, we obtained the asymptotic distribution of estimators. Compared with the α-mixing coefficient assumptions in Liang et,al (2015), ρ-hypothesis is more practical here.