随机波动率模型的统计推断及数值解

The project is mainly concentrated on the numeric methods for SDEs with stochastic evaluation times and the nonparametric estimation of the stochastic volatility models. The project mainly contains the following three parts. The first part is to consider the asymptotic error for random grid approximations of multidimensional SDES, generalizing the results under equidistance sampling to random evaluation times, also study how the evaluation times can be chosen to make the approximation error have minimal standard deviation. In the second part, we will study the nonparametric estimation of integrated volatility of volatility in the simultaneous presence of microstructure noise and jump, and prove the asymptotic property of estimator. We will also analysis the performance of the estimator by simulation.The project will explore some new estimations and limit theory means, which will provide effective methods to the statistical inference and limit theorem of the associated stochastic process

主要研究随机采样下的随机微分方程数值解以及微结构噪声下随机波动率模型的非参数估计。内容包括：（1）在随机采样下多维随机微分方程数值解的渐近误差，将等距采样下的相关结果推广到随机采样情形，并设计在所研究的随机采样框架下渐近标准差最小的采样方案；（2）研究存在微结构噪声或价格跳跃情况下随机波动率模型波动率过程积分波动率的非参数估计，证明估计量的渐近性质，模拟分析估计量估计效果。本项目的研究，将会探索出一些新的估计思想和极限理论手段，将为相关随机过程的极限理论和统计推断提供新的有效方法