【作者】 刘涛；

【导师】 王冉；

【作者基本信息】 武汉大学 ， 统计学， 2019， 硕士

【摘要】 随机微分方程的模型已经被广泛地应用在工程学、金融学、生物学等学科中,但是随机微分方程往往都无法得到精确解,此时我们只能求取其近似数值解来代替精确解.因此,随机微分方程的数值解研究比较受到学者的重视.然而现在绝大多数的数值解方法都只能有效解决低维随机微分方程的近似数值求解,当方程维度上升时,现有的数值计算方法的复杂度将会呈指数增加,求取的近似数值解准确性将无法保证.本文将介绍一种深度学习神经网络算法来求取高维随机微分方程的数值解,并以50维的black-scholes方程为例,利用Tensor Flow框架来建立四层神经网络来求取近似数值解.本文第一章首先介绍随机微分方程的起源与发展,然后介绍随机微分方程数值解方法的研究意义和研究现状.第二章对布朗运动进行了介绍,主要介绍了布朗运动的定义和性质,并用R软件模拟了一维标准布朗运动的轨迹;然后介绍了伊藤公式和随机微分方程,并对随机微分方程解的存在唯一性进行了证明.第三章介绍了随机微分方程两种比较常见的数值方法:Euler-Maruyama方法和Milstein方法,并且介绍了两种数值方法的收敛性和稳定性;随后利用R软件模拟一维随机微分方程的近似数值解.第四章介绍了科尔莫哥洛夫偏微分方程和随机微分方程之间的关系,然后介绍了一种利用深度学习神经网络来计算高维随机微分方程数值解的算法,随后以50维的Black-Scholes方程为例,建立4层神经网络模型来计算其近似数值解.更多还原

【Abstract】 The models of stochastic differential equations have been widely used in engineering,finance,biology and other disciplines,but the exact solutions of stochastic differential equations are not often available.At this time,we can only find the approximate numerical solutions to replace the exact solutions.Therefore,the numerical solution of the stochastic differential equation is brought to the attention of the scholars.However,now most of the numerical solution method can effectively solve the low dimensional stochastic differential equation numerical solution,the approximate equation dimension when it rises,the complexity of the existing numerical method will increase exponentially,calculate the approximate numerical solution accuracy will not be able to guarantee.In this paper,a deep learning neural network algorithm is introduced to obtain numerical solutions of high-dimensional stochastic differential equations,and black-scholes equation of 50 dimensions is taken as an example to establish a four-layer neural network using Tensor Flow framework to obtain approximate numerical solutions.In the first chapter,the origin and development of stochastic differential equations are introduced.In the second chapter,we introduces the Brownian motion,mainly introduces the definition and properties of Brownian motion,and simulates the trajectory of onedimensional standard Brownian motion with R software.Then we introduce it? formula and stochastic differential equation,and prove the existence and uniqueness of the solution of stochastic differential equation.In the third chapter,we introduces two common numerical methods for stochastic differential equations: Euler-Maruyama method and Milstein method,and introduces the convergence and stability of the two numerical methods.Then R software is used to simulate the approximate numerical solution of one dimensional stochastic differential equation.In the fourth chapter,we introduces the relationship between kolmogorov’s partial differential equation and stochastic differential equation,and then introduces an algorithm to calculate the numerical solution of high-dimensional stochastic differential equation with deep learning neural network.更多还原