Exact simulation and importance sampling of diffusion process.

• Other Authors: Huang, Zhengyu.
• Format: Others
• Language: English; Chinese
• Published: 2012
• Subjects: Diffusion processes; Sampling (Statistics); Simulation method; Financial engineering > Statistical methods
• Online Access: http://library.cuhk.edu.hk/record=b5549472; http://repository.lib.cuhk.edu.hk/en/item/cuhk-328171

随着全球金融市场的日益创新和不断加剧的竞争，金融产品也变得越来越结构复杂。这些复杂的金融产品，从定价，对冲到风险管理，都对相应的数学技术提出越来越高的要求。在目前运用的技术中，蒙特卡洛模拟方法由于其广泛的适用性而备受欢迎。本篇论文对于在金融工程和工业界都受到广泛关注的两个问题进行研究：局部化以及对于受布朗运动驱动的随机微分方程的精确抽样；布朗河曲，重要性抽样已经对于扩散过程极值的无偏估计。 === 第一篇文章考虑了使用蒙特卡洛模拟方法产生随机微分方程的样本路径。离散化方法是此前普遍使用的近似产生路径的方法：这种方法很容易实施，但是会产生抽样偏差。本篇文章提出一种模拟方法，可用于随机微分方程路径的精确抽样。一个至关重要的发现是：随机微分方程的概率分布可以被分解为两部分的乘积，一部分是标准布朗运动的概率分布，另外一部分是双重随机的泊松过程。基于这样的分解和局部化技术，本篇文章提出一种接受-拒绝算法。数值试验可以验证，这种方法的均方误差-计算时间的收敛速度可以达到O(t⁻¹[superscript /]²)，优于传统的离散化方法。更进一步的优点是：这种方法可以对带边界的随机微分方程进行精确抽样，而带边界的微分方程正是传统离散方法经常遇到困难的情形。 === 第二篇文章研究了如何计算基于扩散过程极值的泛函。传统的离散化方法收率速度很慢。本篇文章提出了一种基于维纳测度分解的无偏蒙特卡洛估计。运用重要性抽样技术和对于布朗运动路径的威廉分解，本篇文章将对于一般性扩散过程的极值的抽样化简为对于两个布朗河曲的抽样。数值试验部分也验证了本篇文章所提方法的准确性和计算上的高效率。 === With increased innovation and competition in the current financial market, financial product has become more and more complicated, which requires advanced techniques in pricing, hedging and risk management. Monte Carlo simulation is among the most popular ones due to its great °exibility. This dissertation contains two problems recently arises and receives much attention from both the financial engineering and simulation communities: Localization and Exact Simulation of Brownian Motion Driven Stochastic Differential Equations; And Brownian Meanders, Importance Sampling and Un-biased Simulation of Diffusion Extremes. === The first essay considers generating sample paths of stochastic differential equations (SDE) using the Monte Carlo method. Discretization is a popular approximate approach to generating those paths: it is easy to implement but prone to simulation bias. This essay presents a new simulation scheme to exactly generate samples for SDEs. The key observation is that the law of a general SDE can be decomposed into a product of the law of standard Brownian motion and the law of a doubly stochastic Poisson process. An acceptance-rejection algorithm is devised based on the combination of this decomposition and a localization technique. The numerical results corroborates that the mean-square error of the proposed method is in the order of O(t⁻¹[superscript /]²), which is superior to the conventional discretization schemes. Furthermore, the proposed method also can generate exact samples for SDE with boundaries which the discretization schemes usually find difficulty in dealing with. === The second essay considers computing expected values of functions involving extreme values of diffusion processes. The conventional discretization Monte Carlo simulation schemes often converge very slowly. In this paper, we propose a Wiener measure decomposition-based approach to construct unbiased Monte Carlo estimators. Combined with the importance sampling technique and the celebrated Williams' path decomposition of Brownian motion, this approach transforms the task of simulating extreme values of a general diffusion process to the simulation of two Brownian meanders. The numerical experiments show the accuracy and efficiency of our Poisson-kernel unbiased estimators. === Detailed summary in vernacular field only. === Detailed summary in vernacular field only. === Detailed summary in vernacular field only. === Huang, Zhengyu. === Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. === Includes bibliographical references (leaves 107-115). === Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. === Abstract also in Chinese. === Abstract --- p.i === Acknowledgement --- p.iv === Chapter 1 --- Introduction --- p.1 === Chapter 1.1 --- Background --- p.1 === Chapter 1.2 --- SDEs and Discretization Methods --- p.4 === Chapter 1.3 --- The Beskos-Roberts Exact Simulation --- p.15 === Chapter 1.4 --- Major Contributions --- p.19 === Chapter 1.5 --- Organization --- p.26 === Chapter 2 --- Localization and Exact Simulation of SDEs --- p.27 === Chapter 2.1 --- Main Result: A Localization Technique --- p.27 === Chapter 2.1.1 --- Sampling of ζ --- p.33 === Chapter 2.1.2 --- Sampling of Wζ^(T-t) --- p.35 === Chapter 2.1.3 --- Sampling of the Bernoulli I --- p.38 === Chapter 2.1.4 --- Comparison Involving Infinite Sums --- p.40 === Chapter 2.2 --- Discussions --- p.43 === Chapter 2.2.1 --- One Extension: SDEs with Boundaries --- p.43 === Chapter 2.2.2 --- Simulation Efficiency --- p.45 === Chapter 2.2.3 --- Extension to Multi-dimensional SDE --- p.48 === Chapter 2.3 --- Numerical Examples --- p.52 === Chapter 2.3.1 --- Ornstein-Uhlenbeck Mean-Reverting Process --- p.52 === Chapter 2.3.2 --- A Double-Well Potential Model --- p.56 === Chapter 2.3.3 --- Cox-Ingersoll-Ross Square-Root Process --- p.56 === Chapter 2.3.4 --- Linear-Drift CEV-Type-Diffusion Model --- p.62 === Chapter 2.4 --- Appendix --- p.62 === Chapter 2.4.1 --- Simulation of Brownian Bridges --- p.62 === Chapter 2.4.2 --- Proofs of Main Results --- p.64 === Chapter 2.4.3 --- The Oscillating Property of the Series --- p.71 === Chapter 3 --- Unbiased Simulation of Diffusion Extremes --- p.79 === Chapter 3.1 --- A Wiener Measure Decomposition --- p.79 === Chapter 3.2 --- Brownian Meanders and Importance Sampler of Diffusion Extremes --- p.81 === Chapter 3.2.1 --- Exact Simulation of (θT, KT, WT) --- p.83 === Chapter 3.2.2 --- Simulating Importance Sampling Weight --- p.84 === Chapter 3.3 --- Some Extensions --- p.88 === Chapter 3.3.1 --- Variance Reduction --- p.88 === Chapter 3.3.2 --- Double Barrier Options --- p.90 === Chapter 3.4 --- Numerical Examples --- p.94 === Chapter 3.5 --- Appendix --- p.98 === Chapter 3.5.1 --- Brownian Bridges and Meanders --- p.98 === Chapter 3.5.2 --- Proofs of Main Results --- p.101 === Bibliography --- p.107