统计与数据科学学院知识库

  近年来, 高斯过程的占位时及其有限区间内的局部时都是国内外学者关注的热点问题且在金融风险模型中均有较为广泛的应用. 在金融市场中, 许多期权的定价问题与期权的价格处于某个价格区间有关, 称为与占位时有关的期权; 在风险理论中, 研究以各种随机过程为背景的保险风险模型的剩余价值过程处于某些水平区间的情况, 由此来衡量保险公司的运营状况, 因此研究各种随机过程的局部时和占位时对相关期权及风险理论有重要的作用. 随着局部时理论研究的不断深入及其应用领域的不断扩展, 又产生了自相交局部时的概念. 目前, 对于多重自相交局部时的研究大多基于布朗运动展开, 而对于其它自相似高斯过程多重自相交局部时的研究并不完善. 此外, 关于局部时和占位时的统计推断问题也是统计学研究的重点问题之一. 由于非参数统计方法对总体所要求的条件非常宽泛, 往往具有较好的稳健性, 所以基于离散观测研究占位时和局部时的非参数估计得到了广大学者的关注并且在金融统计领域中也越来越重要.

  本文从自相似高斯过程、局部时、导数型局部时、局部时的统计推断等四个方面对国内外文献进行梳理、归纳与总结, 充分掌握已有研究进展及未来可拓展之处, 进而确定本文研究思路. 在此基础上, 将自相似高斯过程二重自相交局部时的研究推广至多重的情形, 证明其存在性等为后续研究奠定理论基础. 接着, 基于自相似高斯过程多重自相交局部时的存在性结合占位时公式给出自相似高斯过程导数型多重自相交局部时的定义形式, 运用不同的方法证明其相关性质. 最后, 从应用的角度出发, 在分数布朗运动局部时存在的基础上, 给出分数布朗运动局部时和占位时的非参数估计并证明相关性质. 具体结论如下:

      (1)研究自相似高斯过程的多重自相交局部时. 首先利用 Fourier 分析方法结合强局部非确定性研究自相似高斯过程多重自相交局部时的存在性条件, 并且在此基础上证得多重自相交局部时满足指数可积性. 接着, 根据自相似高斯过程的局部非确定性考虑多重自相交局部时分别关于时间变量和空间变量的 Hölder 连续性. 最后, 借助 Malliavin 分析中的 Wiener 混沌展式证明自相似高斯过程多重自相交局部时的平滑性. 特别地, 给出一维分数布朗运动多重自相交局部时的 Wiener 混沌展式, 用混沌展式证明其满足 Meyer-Watanabe 意义下的平滑性, 也将相应的结果推广至 d 维分数布朗运动的情形, 而其它自相似高斯过程多重自相交局部时的平滑性类似可得证. 

      (2)考虑自相似高斯过程导数型多重自相交局部时. 首先, 基于自相似高斯过程多重自相交局部时的存在性结合占位时公式, 给出自相似高斯过程多重自相交局部时关于空间变量的导数定义. 接着,  利用样本配置方法给出自相似高斯过程导数型多重自交局部时在L^p中的存在性条件和其关于空间变量满足的Hölder连续性条件. 最后, 借助混沌展式证明其满足 Meyer-Watanabe 意义下的平滑性. 为方便, 本文仅给出分数布朗运动导数型多重自相交局部时平滑性的证明, 关于其它自相似高斯过程的结论类似可得.

      (3)基于离散观测, 研究分数布朗运动占位时和局部时的非参数估计. 首先介绍分数布朗运动占位时和局部时的 Riemann 和估计, 通过分数布朗运动的特征函数和其满足的局部非确定性来给出 Riemann 和估计L²逼近误差的精确上界. 接着通过矩估计方法结合链式论证证得占位时估计的中心极限定理, 而局部时估计的中心极限定理类似可证得. 最后为提高收敛速率, 简要介绍占位时和局部时的另一种非参数估计方法——条件期望估计. 利用分数布朗运动和条件期望的相关性质, 通过直接计算得到条件期望估计的中心极限定理.

    In recent years, the occupation time of Gaussian process and its local times in finite interval have become the hot topic and have been widely used in financial and risk models. In the financial market, there are many options pricing problems related to the price of options in a certain price range, called options related to the occupation time; In risk theory, it is often necessary to study the residual value process of the insurance risk model in the background of various stochastic processes at certain level intervals, in order to measure the operation of the insurance company. Therefore, the study of local times and occupation time for various stochastic processes plays an important role in studying the options and the risk theory. With the deepening of the study of local times and the expansion of its application areas, the concept of intersection local times has emerged. At present, most of the studies on multiple self-intersection local times are based on Brownian motion, and the studies on multiple self-intersection local times for other self similar Gaussian processes are not perfect.  In addition, the problems of statistical inference for local times and occupation time are one of the key issues in statistical research. Since the non-parametric statistical methods require very broad conditions on the population, they tend to be more robust, the non-parametric estimation of occupation time and local times based on discrete observations has gained much attention and become more and more important in the field of financial statistics.

  This paper combs, summarizes, and summarizes the domestic and international literature from four aspects: self similar Gaussian process, local time, the derivative of local time, and statistical inference of local time. In order to fully grasp the progress of existing research and future expansion, the research idea of this paper is determined. On this basis, the research on the double self-intersection local times of self similar Gaussian process is extended to the multiple case, and its existence is proved, which lays a theoretical foundation for subsequent research. Next, based on the existence of the multiple self-intersection local times for self similar Gaussian process and the formula of the occupation time, the definition of the derivative  of multiple self-intersection local times for self similar Gaussian process is given, and its relevant properties are proved by different methods. Finally, from the perspective of application, the non- parametric estimation of the local times and occupation time of fractional Brownian motion are given, and its relevant properties are proved. The detailed conclusions are as follows:

     (1)Study the multiple self-intersection local times for self similar Gaussian processes.  Firstly,  the existence condition of multiple self-intersection local times for self similar Gaussian processes investigated by using Fourier analysis combined with strong local nondeterminism,  and on this basis it is proved that it satisfies exponential integrability.  And then, the Hölder continuity conditions for multiple self-intersection local times with respect to time and space variables are considered according to the local nondeterminism,  respectively.  Finally, using the Wiener chaos decomposition method in the Malliavin calculus, the smoothness of the multiple self-intersection local times for self similar Gaussian processes is demonstrated.  In particular,  the Wiener chaos decomposition of the multiple self-intersection local times for one-dimensional fractional Brownian motion is given, and its smoothness in the sense of Meyer-Watanabe is proved by the Wiener chaos decomposition. We also extend the corresponding results to the case of d-dimensional fractional Brownian motion, and the smoothness of multiple self-intersection local times for other self similar Gaussian processes can be similarly proved.

     (2)Consider the derivative of multiple self-intersection local times for self similar Gaussian processes. Firstly, we give the definition of the derivative of multiple self-intersection local times for self similar Gaussian processes with respect to the spatial variables by combining with the occupation time formula. Next, a sample configuration method is used to prove the existence of the derivative of multiple self-intersection local times in L^p and to show that it satisfies Hölder continuity with respect to the spatial variables. Finally, it is proved that the derivative of multiple self-intersection local times for self similar Gaussian process satisfies the smoothness in the Meyer-Watanabe sense by means of chaos expansion. For convenience,  only the case of fractional Brownian motion is proved and other self similarity processes can be obtained similarly.

   (3)Based on discrete observations,  nonparametric estimations of occupation time and local times for fractional Brownian motion are investigated.  In this study, we firstly introduce the  Riemann sum estimations for the occupation time and local times for fractional Brownian motion, and give the exact upper bounds of L² approximation by the characteristic function and  local nondeterminism of fractional Brownian motion.  Then, by using the properties of fractional Brownian motion and the moment estimation method combined with the chain argument, the central limit theorem of occupation time estimation can be obtained, and the central limit theorem of local times estimation can be similarly proved. Finally, another nonparametric estimation method, conditional expectation estimation, is briefly introduced to improve the rate of convergence. Using the related properties of fractional Brownian motion and conditional expectation, the central limit theorem of conditional expectation estimation is obtained by direct computation.