分数次布朗运动的高效数值模拟方法

论文总字数：28435字

摘 要

分数次布朗运动在金融、通信、排队论等领域有着重要的应用价值，研究分数次布朗运动的高效数值模拟方法具有重要的意义。本文首先对已有的分数次布朗运动的模拟方法进行综述，主要包括两种类型：精确方法和近似方法。我们分别分析了两种方法的优缺点，然后介绍了一种基于Bessel函数的谱展开近似的新方法，虽然近似方法在模拟分数次布朗运动时比精确方法效率高，能够更好的应用到实际的模型当中去，但是我们不仅希望能够高效地得到一组近似样本，更希望这组样本能够具备分数次布朗运动的某些重要性质，为了比较这些近似算法，我们分别介绍了几种评估样本自相似性参数H的方法，经过测试发现基于Bessel函数的谱展开近似的Hurst指数测量值要比另外两种近似算法的Hurst估计更加接近理论结果。最后我们将利用这种新算法去求解一类由分数次布朗运动驱动的二阶随机微分方程。

关键字:分数次布朗运，Hurst指数估，随机微分方程

Efficient numerical simulation method for fractional Brownian motion

Abstract

Fractional Brownian motion has important application value in the fields of finance, communication and queuing theory. It is of great significance to study the numerical simulation method of fractional Brownian motion. In this paper, we first review some existing simulation methods of fractional Brownian motion, which include two types: precise methods and approximation methods. We separately analyze the advantages and disadvantages of the two methods, and then we introduce a new approximation method based on the Bessel function for spectral expansion approximation. Although the approximation method is more efficient than the exact method in simulating fractional Brownian motion, and it can be better applied to the model in reality. We not only hope to get a set of samples efficiently, but also hope that the samples can have some important properties of fractional Brownian motion. In order to compare these approximation algorithms, we introduce several evaluation methods to estimate self-similarity parameter H of these samples. Test results show that the Hurst indexes of the method based on the Bessel function for spectral expansion approximation is closer to the theoretical value than the Hurst indexes of the other two approximation algorithms. Finally, we will use this new algorithm to solve a class of second-order stochastic differential equations driven by fractional Brownian motion.

KEY WORDS: Fractional Brownian motion, estimation for Hurst index, stochastic differential equation.

目 录

摘要 ……………………………………………………………………………… Ⅰ

Abstract ………………………………………………………………………… Ⅱ

1. 引言 ……………………………………………………………………… 1
2. 分数次布朗运动的模拟方法 …………………………………………… 2

2.1 精确方法 …………………………………………………………… 4

2.1.1 Hosking方法 ……………………………………………… 4

2.1.2 Cholesky方法 ………………………………………………5

2.1.3 Davies and Harte方法 …………………………………… 6

2.2近似方法……………………………………………………………… 8

2.2.1 谱展开近似 ………………………………………………… 8

2.2.2 泊松方法 …………………………………………………… 11

2.2.3 基于Bessel函数的谱展开近似…………………………… 12

1. 近似方法的Hurst指数估计和检验 …………………………………… 16

3.1 Hurst指数估计 ………………………………………………………16

3.1.1 Aggregated variance方法 …………………………………16

3.1.2 Absolute moments方法 …………………………………… 18

3.1.3 Higuchi方法………………………………………………… 19

3.1.4 Periodogram方法 ……………………………………………19

3.2 测试与评估 ……………………………………………………………20

1. 分数次布朗运动驱动的一类二阶随机微分方程数值求解方法………… 23

4.1 Euler格式 ……………………………………………………………24

4.2 差分格式 ………………………………………………………………25

4.3 测试结果比较与分析 …………………………………………………25

1. 文章总结 ……………………………………………………………………27

致谢 …………………………………………………………………………………28

参考文献（References）……………………………………………………………29

第一章 引言

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