Statistical Inference For Diffusion Processes With Jumps

The second-order diffusion process overcomes the nondifferentiability of Brow-nian motions and can represent continuous integrated and differentiated pro-cesses. Now these processes are widely used in empirical finance or physics. Nowdays a lot of empirical studies have shown that when economic policy or other big new occurs, there exists the jump in the financial markets. Basic on this fact, we firstly extended the model in Nicolau (2007) to the case with jump——second-order diffusion process with jump. We have systematically done some research work on the inference for the coefficients based on this extended model and some popular nonparametric statistics method.Firstly, the Nadaraya-Watson kernel estimation is a simple and practical nonparametric estimation. Based on infinitesimal moment conditions we propose the Nadaraya-Watson estimators for the coefficients of the second-order diffusion process with jump. Under certain assumptions, we obtain the consistency and asymptotic normality for the estimators.Secondly, relative to the confidence interval constructed by the asymptotic normality, the shape of the confidence interval constructed by the empirical like-lihood method is driven by data. In addition, this method could ignore the asymptotic variance estimation and possess better property for the finite sam-ples. We construct empirical likelihood statistics for the drift coefficient of the second-order diffusion process with jump and prove that they are asymptotically convergent to3/2χ2(1) under mild conditions. Based on them we can construct the confidence intervals for these unknown quantities.Thirdly, the re-weighted estimation not only possesses the better property of the reduced bias similarly as the local linear estimator but also can make the estimator for the estimated nonnegative quantity nonnegative in finite samples. Combined with the local linear method and the empirical likelihood method, we construct the re-weighted estimator for the volatility of the second-order diffusion process with jump. Under certain appropriate conditions we prove the asymptotic normality for the estimator.Finally, based on the discrete observations,We study the local linear estima-for for the drift coefficient of stochastic differential equations driven by α-stable Levy motions. Under regular conditions, we derive the weak consistency and central limit theorem of the estimator.Compared with Nadaraya-Watson esti-mator in Long and Qian. the local linear estimator has a bias reduction whether the kernel function is symmetric or not under different schemes. A Monte-Carlo simulation study demonstrates that the local linear estimator performs better than Nadaraya-Watson estimator, especially on the boundary.