【作者】 谷伟；

【导师】 张诚坚；

【作者基本信息】 华中科技大学 ， 概率论与数理统计， 2008， 博士

【摘要】 对于正向随机微分方程(FSDE)的研究,兴起于上世纪40年代末,它不仅有直接的应用背景,并且拥有了完善的理论框架。相对而言,倒向随机微分方程(BSDE)的研究是近30年才兴起的,但进展十分迅猛,并取得非常丰富的成果,它在很多领域如偏微分方程(PDE)、金融数学、随机控制、微分几何、随机对策论等都有重要的应用,逐渐成为一个强有力的数学工具。但一直以来,对BSDE的研究大都是定性的,除了线性和一些特殊方程的解具有解析表达式外,一般的非线性BSDE的解却不能完全通过解析表达式描述出来。为此,学者们开始转向寻求BSDE的数值解,有关它的数值计算目前是一个很热门的问题,有许多研究小组或个人想要寻求一个快速有效的计算方法。本文考虑BSDE的一种特殊形式,正倒向随机微分方程的数值解。基于“四步法”理论,即在正倒向随机微分方程里,倒向方程的适应解可以由正向方程的解显式表示,但要借助于某个抛物型方程的解。从而,由这个思想出发,可以这样来构造数值算法,首先离散相应的非线性抛物型方程,然后离散正倒向随机微分方程中的前向部分,最后获得正倒向随机微分方程的数值解。目前有关离散FSDE的研究成果已经很多,因此,重要的是如何离散非线性抛物型方程。从某种程度上来说,BSDE的数值计算转化为偏微分方程的数值计算。本文的目的是基于“四步法”离散正倒向随机微分方程,因此需首先给出数值求解相应的抛物型方程的算法。传统的数值求解抛物型方程的方法大多是确定性方法,但它们在离散非线性情况时遇到了困难。我们通过不确定性(随机)的方法,来构造求解半线性和拟线性抛物型方程初值问题的数值算法—分层方法,这种方法的构造主要依赖于抛物型方程解的概率表示,和对与概率表示相关联的随机微分方程系统的离散方法的选择。它的优势在于该算法对抛物型方程的维数没有限制,而一般确定性的数值算法仅适合3维以下的问题。另外,它的最大优点是仅需对时间区间进行离散,而和空间变量x没有必然联系,算法的稳定性是方法本身所固有的,尽管为了算法的可执行性和减少计算量,需要对x进行离散,事实上,空间变量x区域的截取并不影响最终的数值结果,这些将会在方法的构造过程和数值试验中得到验证。本文首先针对一类拟线性抛物型方程的初值问题构造分层方法,利用对原方程求导后所得的方程和原方程在形式上的一致性,在第二章里利用弱显式欧拉算法构造了求解拟线性抛物型方程的一类分层方法,进而在第三章里通过对第二章算法的改进和推广,构造出了更一般化、更精确的一类分层方法族。然后,在第四章里主要构造了求解两类正倒向随机微分方程的数值算法,由“四步法”,它们分别与半线性和拟线性抛物型方程相联系,而其中对拟线性抛物型方程的数值算法的构造采用了与前两章不同的新方法。另外,文中所构造的数值算法均进行了收敛性分析,编制了数值程序,针对实例的数值模拟进一步验证了所构造算法的精确性和可行性。第五章考虑了倒向随机微分方程理论及所构造的数值求解抛物型方程的算法,在期权定价模型中的应用,算例分析进一步说明了算法的有效性。更多还原

【Abstract】 The research of Forward Stochastic Differential Equations(FSDE) began at the end of 1940s,which has direct application background,but also posesses perfect framework of theory.Comparing with FSDE,the research of Backward Stochastic Differential Equations(BSDE) originated in recent 30 years.Although the history is not long,the development is greatly rapid and lots of results are obtained.Now,BSDE has been penetrated in many fields,such as PDE,Financial Mathematics,Stochastic Control, Differential Geometry,Stochastic Game Theory.So that BSDE becomes a powerful maths tool gradually.But until now,most of BSDE’ research is qualitative.Altough linear and some special equations’ solutions are analytic,the nonlinear BSDE’ solutions have no analystic representations.However,a number of scholars have turned to look for the numerical solutions of BSDE,now it becomes such a hot problem that many research groups and individuals begin their study in this fields.This paper just consider a special form of BSDE which is called Forward-Backward Stochastic Differential Equation(FBSDE). Based on the "Four Step Scheme",that is in FBSDE,the backward equation’s adapted solution can be determined explicitly by the forward equation’s solution,using a certain parabolic PDE system’s solution.So we can discretize nonlinear parabolic equation firstly,then the FSDE,last to obtain the numerical solution of the FBSDE.At present, the research results about discretizing FSDE is quatitive,thus,the most important is how to discretize the nonlinear parabolic equation.To some extent,the research of numerical solution about BSDE turns into PDE.The purpose of this paper is to discretize FBSDE,basing on the four step scheme. Therefore,numerical methods for the corresponding parabolic equation should be presented firstly.In traditionally,the numerical methods for parabolic equation are deterministic, but it is difficult to discretize nonlinear equations.And in this paper,a layer method constructed through a stochastic approach is proposed to solve semilinear and quasilinear parabolic equations.And the construction of the layer method mainly lies in two factors,one is the choice of the probabilistic representations of the solution for parabolic equations,the other is the choice of the discrete methods for the corresponding stochastic differentional equations.The layer methods fit well to solve all dimensional parabolic equations,but the traditional deterministic methods do well only when the dimensions of the equations are no more than 3.And the great virtue of the layer methods is that one can just discretize the time variable t,without considering the spatial variable x.Although we also discretize x to execute the algorithm and to reduce the computational volume,the stability of the layer method is intrinsic.In fact,the truncation of spatial variable x does not significantly affect the accuracy of the numerical results obtained,which is to be testified by the construction of the methods and the numerical tests.In the paper,firstly,a layer method is proposed to solve the cauchy problem for a type of qusilinear parabolic equations in the Chapter 2 and 3,where the coherence of the orginal equations and the corresponding differentiated equations is exploited.In Chapter 2,a layer method is constructed by the weak explicitly Euler method.Moreover, in Chapter 3,a type of much more general layer method is constructed.Essentially speaking,Chapter 3 is the extension and improvement of Chapter 2.Then in Chapter 4, we construct the numerical algorithms to solve the FBSDE connected with semilinear and quasilinear parabolic equations respectively,where we exploit a new method to discretize the quasilinear parabolic equations.Further more,we analyze the convergence of the numerical algorithms,and testify the schemes through some examples.The numerical results shows that the accuracy and feasibility of our algorithms.In Chapter 5,we apply the theory of FBSDE and the algorithms of parabolic equations to price Europe Option. Through the numerical example we testify the validities of the theory and the algorithms.更多还原