Zero-coupon bond pricing模型的最优系统和对称约化

【作者】 张颖；

【导师】 屈长征；

【作者基本信息】 西北大学 ， 基础数学， 2009， 硕士

【摘要】 本文主要是将李群方法应用于金融问题中的数学模型,研究了Zero-couponbond pricing模型(以下简称“ZCB”模型).我们求出ZCB模型所容许的单参李点对称群及其该群相应的伴随表达式,并在此基础上构建了该李点对称群的一维子代数的最优系统.针对所构建的最优系统中的每一个元素,我们对ZCB模型进行了对称约化,得出该模型一些不同类的解.同时我们还将以上方法应用于(2+1)维非线性Sine-Gordon方程,对其进行了对称约化.本文第二章我们得到了金融数学中如下所示的ZCB模型:所容许的李点对称群并且给出该李对称群的交换关系.在本文的第三章我们构建了ZCB模型的李点对称群的伴随表达式,利用该伴随表达式对李点对称群相应的一维李代数进行了分类.构建了其最优系统,然后利用所构建的最优系统中的每个元素对原方程进行了对称约化,得到一些不同形式的解.在第四章我们将本文上两章中所示的方法用于研究(2+1)维非线性Sine-Gordon方程,得到一些低维的微分方程.最后一章我们对全文内容进行总结.更多还原

【Abstract】 Lie group theory is applied to differential equations occurring as a mathematical model in financial problems. The Zero-coupon bond pricing (ZCB for short) model is studied. Its one-paramctcr Lie point symmetries and corresponding group of adjoint representations are obtained. An optimal system of one-dimensional subalgebra is derived and used to construct distinct families of special closed-form solutions of the equation. The method above is also used to study the (2+1)-dimensional nonlinear Sine-Gordon equation, obtaining the corresponding optimal system and some symmetry reductions.An outline of the paper is as follows.In Chapter 2 we determine the Lie symmetries admitted by the ZCB model occurring as a mathematical model in financial problems :and give the communication relations of the corresponding Lie symmetries.In Chapter 3 we construct the adjoint representations of the ZCB Lie group and use it to perform the classification of one-dimensional subalgebras of the ZCB Lie algebra.The optimal system of the obtained symmetry groups is also discussed and many interesting group-invariant solutions are obtained.In Chapter 4 the same method discussed above is used to study (2+1)-dimensional nonlinear sine-Gordon equation,getting some lower dimensional differential equations.Finally in Chapter 5 we conclude.更多还原