非确定性倒向随机微分方程的多水平蒙特卡罗解法

发布时间：2018-04-20 03:23

本文选题：倒向随机微分方程 + 数值解法　；参考：《山东大学》2017年硕士论文

【摘要】：多水平蒙特卡罗方法作为蒙特卡罗方法的一种改进由Stefan Heinrich[1]和Michael Giles 提出.针对需要进行时间或者空间划分的一类问题,与传统的蒙特卡罗方法相比,多水平蒙特卡罗方法可以在保持误差阶的情况下降低运算复杂度.Weidong Zhao,Lifeng Chen和Shige Peng[3]在2006年给出了针对如下形式倒向随机微分方程:的数值解法—θ-格式.该方法利用倒向随机微分方程自身性质进行数值求解,实现了误差收敛率的提高.若上述形式的倒向随机微分方程(0.1)其终端条件yT中还包含一个与布朗运动独立的随机变量ω,则带有随机变量ω的非确定性倒向随机微分方程,其形式为当t = 0时,W0 = 0,y0(ω)是一个随机变量.对此类问题我们所关注的是t=0时y0(ω)的期望E[y0(ω)].近似期望一般采用蒙特卡罗方法,而在本文中我们将采用多水平蒙特卡罗方法.数值实验结果表明,对此类问题采用多水平蒙特卡罗方法能够在保持误差阶的情况下降低运算复杂度,利用本文所提出的改进方法可以实现更快速地求解E[y0(ω)].由于某些实际的金融问题中存在着终端条件不确定的情况,因此这种针对非确定性倒向随机微分方程(0.2)的多水平蒙特卡罗解法在金融领域具有一定的实际应用价值.本文分为六个章节.第一章是绪论部分,在本部分我们将粗略地介绍本论文所运用的方法.第二章我们将对蒙特卡罗方法进行详细介绍.第三章我们将介绍多水平蒙特卡罗方法.第四章的内容为倒向随机微分方程的数值解法,我们将重点介绍如何利用θ-格式来求解倒向随机微分方程.第五章我们将介绍非确定性倒向随机微分方程的多水平蒙特卡罗方法.对于带有随机变量ω的非确定性倒向随机微分方程,我们将利用多水平蒙特卡罗方法对其解法进行改进.第六章为数值实验结果及结论.
[Abstract]:As an improvement of Monte Carlo method, multilevel Monte Carlo method is proposed by Stefan Heinrich [1] and Michael Giles. For a class of problems that require time or space partitioning, compared with the traditional Monte Carlo method, The multilevel Monte Carlo method can reduce the computational complexity with preserving the error order. Weidong Zhaofeng Chen and Shige Peng [3] gave a numerical solution to backward stochastic differential equations in 2006-胃-scheme. By using the properties of backward stochastic differential equations, the convergence rate of errors is improved. If the above form of backward stochastic differential equation is 0.1) the terminal condition YT also contains a random variable 蠅 independent of the Brownian motion, then the uncertain backward stochastic differential equation with the random variable 蠅 The form is that W 0 = 0 y 0 (蠅) is a random variable when t = 0. We are concerned with the expectation E [y _ 0 (蠅)] of y _ 0 (蠅) when t = 0. The approximate expectation is that the Monte Carlo method is generally used, and in this paper we will adopt the multilevel Monte Carlo method. The numerical results show that the multi-level Monte Carlo method can reduce the computational complexity while preserving the error order, and the improved method proposed in this paper can be used to solve E [y0 (蠅)] more quickly. Due to the uncertainty of terminal conditions in some practical financial problems, this multi-level Monte Carlo solution for uncertain backward stochastic differential equations (0.2) has certain practical application value in the field of finance. This paper is divided into six chapters. The first chapter is the introduction. In this part, we will introduce the methods used in this paper. In the second chapter, we will introduce the Monte Carlo method in detail. In chapter 3, we introduce the multilevel Monte Carlo method. In chapter 4, the numerical solution of backward stochastic differential equations is presented. We will focus on how to solve backward stochastic differential equations by 胃-scheme. In chapter 5, we introduce the multilevel Monte Carlo method for uncertain backward stochastic differential equations. For uncertain backward stochastic differential equations with random variables 蠅, we will improve its solution by using multilevel Monte Carlo method. The sixth chapter is the numerical experimental results and conclusions.
【学位授予单位】：山东大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8

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