带有稳定新息的条件异方差模型的统计推断及其应用

Based on asymptotic theory and point theory, this project mainly investigates statistical inference and their applications for two classes of conditional heteroscedastic models with alpha stable innovations. One is the generalized autoregressive conditional heteroskedastic model with alpha-stable innovations (alpha-stable GARCH), the other is the double autoregressive model with alpha-stable innovations (alpha-stable DAR). For each, under either stationarity or non-stationarity, we will discuss the consistency and limit distribution of the unified maximum likelihood estimation with the estimation of parameters in innovations, and develop new tools on model diagnostic checking. In applications, the project will apply theoretical results obtained into return modelling of financial assets and risk management. The significance of the project is that it theoretically further enriches the class of conditional heteroscedastic models, which overcomes the constraints on moments of innovations on traditional conditional heteroscedastic models, so that it is better to model and forecast the volatility of the retrun of financial assets, and it will provide more precise pricing on financial derivatives and measure on value-at-risk in practice. It proposes a new quantitative index and serves financial institutions better.

本项目以渐近理论和点过程理论为主要工具，研究两类条件异方差模型的统计推断及其应用，一类是带有稳定新息的广义自回归条件异方差模型，一类是带有稳定新息的双自回归模型。对每一类模型，在平稳和非平稳的条件下，统一讨论它的最大似然估计的相合性及渐近分布，以及同时估计新息序列所涉及的参数，发展新的模型诊断工具。在应用方面，本项目将获得的理论成果应用到金融资产收益率建模以及风险管理。本研究的意义在于，理论上，进一步丰富条件异方差模型类，克服传统的异方差模型对新息序列矩的要求限制，使得条件异方差模型在实际中能更好的模拟和预测金融资产收益率的波动率；应用上，此类模型能够提供对金融衍生品更准确的定价以及对风险估值更精确的度量，为风险管理提供新的数量指标，为金融机构提供更优质的服务。

条件异方差模型在理论与应用方面有着非常重要的意义，尤其是重尾的异方差模型。本项目着重研究两类异方差模型，GARCH模型和DAR模型。在概率结构方面，从泛函的观点出发，研究了两类模型的样本路径结构，得到了深刻的结果，即重整化之后的风险过程弱收敛到Brown运动的指数泛函。对于无漂移的两类模型，同样考虑了其重整化路径结构，并且完整地讨论了其统计推断和模型诊断，并把这些成果推广到门限DAR模型。对这些理论结果，给出了大量的应用实例，主要包括股票收益、股指等。对于alpha-稳定的GARCH模型和DAR模型，已经证明其参数的估计依然是普通的收敛率，这一点令人感到意外。在非平稳的情况下，常数项依然是不可估的。这些结果填补了这两类模型在理论研究中的空白，具有理论与实际应用的意义。