倒向随机微分方程及其金融应用

发布时间：2018-03-03 22:11

本文选题：倒向随机微分方程　切入点：未定权益　出处：《中国科学技术大学》2017年硕士论文　论文类型：学位论文

【摘要】：1973年,Bismut首次引入了倒向随机线性微分方程用来研究随机最优控制问题,从1990年开始,Pardoux和Peng推广了 Bismut提出的倒向随机微分方程,并在理论和应用方面做了大量开创性工作。在研究过程中,数学家和经济学家发现了倒向随机微分方程在随机分析、PDE、金融数学等很多方面具有重要的理论和应用价值,并做了大量深入的研究。现在倒向随机微分方程已成为随机分析、金融、控制等领域的重要分支。而在金融学中,一些未定权益的对冲或定价问题最终可转化为一系列倒向随机微分方程的求解问题。本文主要介绍倒向随机微分方程(BSDE)以及它在经济方面的应用,内容框架大致如下:第二章主要给出了自然存在于未定权益的定价和对冲问题中的倒向随机微分方程(BSDE)的一些例子,并引入了递归效用的定义。第三章将讲述BSDE的几个重要结论。如两个BSDE的解的价差的先验估计,Pardoux和Peng在BSDE的解的存在性和唯一性定理,比较定理、解的连续性和可微性等重要性质。第四章主要研究生成元为凹函数的倒向随机微分方程的一些性质。文中会证明此种BSDE的解可以写成随机控制问题的形式。然后,给出了金融领域中非线性倒向随机微分方程的应用求解,如递归效用,约束条件下的欧式期权的定价问题。
[Abstract]:In 1973, Bismut first introduced backward stochastic linear differential equations to study stochastic optimal control problems. Since 1990, Bismut and Peng have extended the backward stochastic differential equations proposed by Bismut. In the course of the research, mathematicians and economists found that backward stochastic differential equations have important theoretical and practical value in many aspects of stochastic analysis, such as PDE, financial mathematics, etc. And has done a lot of in-depth research. Now backward stochastic differential equation has become an important branch of stochastic analysis, finance, control and so on. And in finance, Some hedging or pricing problems of undetermined rights can eventually be transformed into a series of backward stochastic differential equations. This paper mainly introduces the backward stochastic differential equation (BSDE) and its economic application. The content framework is as follows: in the second chapter, we give some examples of the backward stochastic differential equation (BSDE), which naturally exists in the pricing and hedging problems of undetermined equity. In chapter 3, several important conclusions of BSDE are introduced, such as the existence and uniqueness theorems of the price difference of two BSDE solutions and the existence and uniqueness theorems of Peng's solutions in BSDE. In Chapter 4th, we mainly study some properties of backward stochastic differential equations whose generator is concave function. In this paper, we prove that the solution of this kind of BSDE can be written into the form of stochastic control problem. In this paper, the application of nonlinear backward stochastic differential equations in the field of finance is given, such as recursive utility, pricing of European options under constraints.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O211.63

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