Zero-coupon bond pricing模型的最优系统和对称约化
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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.
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.
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.
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.
引文
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[5] Stephani H.. Differential equations-their solutions using Symmetries[M]. Cambridge: Cambridge University Press, 1989
[6] Vinogradov A. M.. Symmetries of Partial Differential Equations: Conservation Laws, Applications, Algorithms[M]. Dordrecht: Kluwer, 1989
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[10] Ovsiannikov L. V.. Group Analysis of Differential Equations[M]. New York: Academic Press, 1982
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[13] Ibragimov N. H.. CRC Handbook of Lie Group Analysis of Differential Equations[M].Boca Raton: CRC Press, 1995
[14] Reid G. J.. Algorithms for reducing a system of partial differential equations to standard form, determining the dimension of its solutions space and calculating its Taylor series solution[J]. Eur. J. Appl.Math.,1991,V2(191):293-318
[15] Ibragimov N. H. Elementary Lie group analysis and ordinary differential equations[M]. New York: Wiley,1999
[16] Yang Xuanliu, Zhang Shunli, Qu Changzheng. Symmetry Breaking for Black-Scholes Equations.Commun.Theor.Phys.,2007,V47:995-1000
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[19] Boyer C. D., Sharp R. T., Wintcritz P.. Symmetry breaking interactions for the time dependent Schrodinger equation[J]. J.Math.Phys.,1976,V17(8):1439-1443
[20] Beckers J., Patera J., Winteritz P.. Subgroups of the Euclidean group and symmetry breaking in nonrelativistic quantum mechanics[J].J.Math.Phys.,1977,V18(1):72-78
[21] Winteritz P.. Subgroups of Lie group and Breaking Group Theoretical Methods in Physics[M].New York: Acdcmic press, 1977, V5: 540-572
[22] Qu Changzheng. Allowed transformations and symmetry classes of variable coefficient Burgers equations[J]. IMA,J.Appl.Math., 1995, V54(3): 203-225
[23] Patera J., Winteritz P.. A new basis for the representation of the rotation group[J]. J.Math.Phys., 1973, V14(8): 1130-1156
[24] Macky G.. Induced Respresentations of Groups Quantum Mechanics[M]. New York: Benjamin, 1969
[25] Gaeta G.. Bifurcation and symmetry breaking[J]. Phys.Rep.. 1990, V189: 1-87
[26] Gaeta G.. Reduction and Equivariant Branching Lemma: Dynamical systems, evolution PDEs, and gauge theories[J]. Acta.Appl.Math., 1992, V28: 43-68
[27] Boyer C, Kalnins E. G., Miller W. Jr.. J.Math.Phys., 1925, V16: 499-510
[28] Chou Kaiseng, Qu Changzhang. Optimal System Group and Classification of (1+2)Dimensional Heat Equation[J]. Acta.Appl.Math., 2004, V83: 257-287
[29] Azad H., Mustafa M. T.. symmetry analysis of wave equation on sphere[J], J.Math.Anal.Appl., 2007, V333: 1180-1188
[30] Patera J., Winteritz P., Zassenhaus. Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaregroup[J]. J.Math.Phys.(1975).V16: 1597-1623
[31] Patera J.. Winteritz P., Zassenhaus. Quantum numbers for particles in de Sitter space[J]. J.Math.Phys., 1976, V17: 977-982
[32] Patera J., Winteritz P., Zassenhaus. Continuous subgroups of the funda- mental groups of physics. II. The similitude group[J]. J.Math.Phys., 1975, V17(8): 717-1615
[33] Chou Kaiseng, Li Guanxin, Qu Changzheng. A Note on Optimal Systems for the Heat Equation[J]. J.Math.Anal.Appl., 2001, V261: 741-751
[34] Maluleke G.H., Mason D.P.. Optimal system and group invariant solutions for a nonlinear wave equation[J]. Commun.Nonlin.Sci.Numer.Simulat., 2004, V9: 93-104
[35] Sinkala W.,Leach P.G.J.,O'Hara J.G.. An optimal system and group invariant solutions of the .. .[J]. Appl.Math.Comput., 2008
[36] Clarkson P. A., Kruskal M. D.. New similarity solutions of the Boussinesq equations[J]. J.Math.Phys., 1989, V30: 2201-2213
[37] Cieciura G., Grundland A.. A certain class of solutions of the nonlinear wave cquation[J]. J.Math.Phys., 1984, V25(12): 3460-3469
[38] Morales-Molina L., Mertens F. G.. Sanchez A.. Soliton ratchets out of point-like inhomogeneities[J]. Eur.Phys.J., 2001, V19: 107-123
[39] Lou S Y,Huang G X and Ni G J, Phys.Lett.A 146(1990),11.
[40] Friedberg R., Lee T. D.. Fermion-field nontopological solitons. II. Models for hadrons[J]. Phys.Rev.D., 1977, V15: 1694-1710
[41] Lie S.. Theorie der transformationsgruppen I[J]. Mathematische Annalen, 1880, V16(4): 441-528
[42] Gazizov R. K., Ibragimov N. H.. Lie symmetry analysis of differential equations in finance[J]. Nonlin.Dyn.. 1998,V17: 387-407
[43] Cox J. C, Ingersoll J. E., Ross S. A.. The theory of the term structure of interest ratcs[J]. Econometrica, 1985, V53: 385-407
[44] Vasicek O.. An equilibrium characterisation of the term structure[J]. Journal of Financial Economics, 1977,V5: 177-188
[45] Ho T. S., Lee S.. Term structure movements and pricing interest rate contingent claims. Journal of Finance, 1986. V42: 1129-1142
[46] Chan K. C, Karolyi G. A., Longstaff F.. Sanders A. B.. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance, 1992,V47: 1209-1227
[47] Ahn D., Gao B.. A parametric nonlinear model of term structure dynamics. Review of Financial Studies, 1999. V12: 721-762
[48] Goard J.. New solutions to the bond-pricing equation via Lie's classical method. Mathematical and Computer Modelling, 2000, V32: 299-313
[49] Kenneth Johnpillai I.. Scott W.McCue, James M. Hill. Lie group symmetry analysis for granular media stress equations[J]. J.Math.Anal.Appl., 2005, V301: 135-157
[50] Dresner L.. Applications of Lie's Theory Ordinary Partial Differential Equations[M]. Philadelphia: IOP Publishing Ltd., 1999
[51] Makinde O. D., Moitsheki R. J., Tau B. A.. Similarity reductions of equations for river pollution,Applied Mathematics and Computation, 2007, V188: 1267-1273
[52] Edwards M. P., Hill J. M.. Selvadurai A. P. S.. Lie group symmetry analysis of transport in porous media with variable transmissivity[J]. J.Math.Anal.Appl., 2008, V341: 906-921
[2] Bluman G. W., Cole J. D.. The general similarity solution of the heat equation[J]. J. Math. Mech., 1969, V18: 1025-1042
[3] Fushchich W. I., Shtelen W. M., Serov N. I, Symmetry analysis and exact solutions of equations of nonlinear mathematical physics[M]. Dordrecht: Kluwer Academic Publishers, 1993
[4] Rogers C, Ames W. F.. Nonlinear Boundary Value problem in science and engineering[M]. Boston: Acdemic press, 1989
[5] Stephani H.. Differential equations-their solutions using Symmetries[M]. Cambridge: Cambridge University Press, 1989
[6] Vinogradov A. M.. Symmetries of Partial Differential Equations: Conservation Laws, Applications, Algorithms[M]. Dordrecht: Kluwer, 1989
[7] Ovsiannikov L. V.. Group Properties of Differential Equations[M]. Russian: Novosibirsk, 1962
[8] Bluman G. W., Cole J. D.. Similarity Method for Differential Equations[M]. New York: Springer-Verlag. 1989
[9] Olver P. J.. Application of Lie Groups to Differential Equation[M]. Second Edition. New York: Springer, 1993
[10] Ovsiannikov L. V.. Group Analysis of Differential Equations[M]. New York: Academic Press, 1982
[11] Ibragimov N. H..Transformation Groups Applied to Mathematical Physics[M]. Boston: D.Reidel,1985
[12] Bluman G.W.,Kumei S..Symmetries and Differential Equation[M]. Berlin:Springer, 1989
[13] Ibragimov N. H.. CRC Handbook of Lie Group Analysis of Differential Equations[M].Boca Raton: CRC Press, 1995
[14] Reid G. J.. Algorithms for reducing a system of partial differential equations to standard form, determining the dimension of its solutions space and calculating its Taylor series solution[J]. Eur. J. Appl.Math.,1991,V2(191):293-318
[15] Ibragimov N. H. Elementary Lie group analysis and ordinary differential equations[M]. New York: Wiley,1999
[16] Yang Xuanliu, Zhang Shunli, Qu Changzheng. Symmetry Breaking for Black-Scholes Equations.Commun.Theor.Phys.,2007,V47:995-1000
[17] 刘式适，刘式达．物理学中的非线性方程[M]．北京：北京大学出版社，2000
[18] 楼森岳，唐晓艳．非线性数学物理方法[M].北京：科学出版社，2006
[19] Boyer C. D., Sharp R. T., Wintcritz P.. Symmetry breaking interactions for the time dependent Schrodinger equation[J]. J.Math.Phys.,1976,V17(8):1439-1443
[20] Beckers J., Patera J., Winteritz P.. Subgroups of the Euclidean group and symmetry breaking in nonrelativistic quantum mechanics[J].J.Math.Phys.,1977,V18(1):72-78
[21] Winteritz P.. Subgroups of Lie group and Breaking Group Theoretical Methods in Physics[M].New York: Acdcmic press, 1977, V5: 540-572
[22] Qu Changzheng. Allowed transformations and symmetry classes of variable coefficient Burgers equations[J]. IMA,J.Appl.Math., 1995, V54(3): 203-225
[23] Patera J., Winteritz P.. A new basis for the representation of the rotation group[J]. J.Math.Phys., 1973, V14(8): 1130-1156
[24] Macky G.. Induced Respresentations of Groups Quantum Mechanics[M]. New York: Benjamin, 1969
[25] Gaeta G.. Bifurcation and symmetry breaking[J]. Phys.Rep.. 1990, V189: 1-87
[26] Gaeta G.. Reduction and Equivariant Branching Lemma: Dynamical systems, evolution PDEs, and gauge theories[J]. Acta.Appl.Math., 1992, V28: 43-68
[27] Boyer C, Kalnins E. G., Miller W. Jr.. J.Math.Phys., 1925, V16: 499-510
[28] Chou Kaiseng, Qu Changzhang. Optimal System Group and Classification of (1+2)Dimensional Heat Equation[J]. Acta.Appl.Math., 2004, V83: 257-287
[29] Azad H., Mustafa M. T.. symmetry analysis of wave equation on sphere[J], J.Math.Anal.Appl., 2007, V333: 1180-1188
[30] Patera J., Winteritz P., Zassenhaus. Continuous subgroups of the fundamental groups of physics. I. General method and the Poincaregroup[J]. J.Math.Phys.(1975).V16: 1597-1623
[31] Patera J.. Winteritz P., Zassenhaus. Quantum numbers for particles in de Sitter space[J]. J.Math.Phys., 1976, V17: 977-982
[32] Patera J., Winteritz P., Zassenhaus. Continuous subgroups of the funda- mental groups of physics. II. The similitude group[J]. J.Math.Phys., 1975, V17(8): 717-1615
[33] Chou Kaiseng, Li Guanxin, Qu Changzheng. A Note on Optimal Systems for the Heat Equation[J]. J.Math.Anal.Appl., 2001, V261: 741-751
[34] Maluleke G.H., Mason D.P.. Optimal system and group invariant solutions for a nonlinear wave equation[J]. Commun.Nonlin.Sci.Numer.Simulat., 2004, V9: 93-104
[35] Sinkala W.,Leach P.G.J.,O'Hara J.G.. An optimal system and group invariant solutions of the .. .[J]. Appl.Math.Comput., 2008
[36] Clarkson P. A., Kruskal M. D.. New similarity solutions of the Boussinesq equations[J]. J.Math.Phys., 1989, V30: 2201-2213
[37] Cieciura G., Grundland A.. A certain class of solutions of the nonlinear wave cquation[J]. J.Math.Phys., 1984, V25(12): 3460-3469
[38] Morales-Molina L., Mertens F. G.. Sanchez A.. Soliton ratchets out of point-like inhomogeneities[J]. Eur.Phys.J., 2001, V19: 107-123
[39] Lou S Y,Huang G X and Ni G J, Phys.Lett.A 146(1990),11.
[40] Friedberg R., Lee T. D.. Fermion-field nontopological solitons. II. Models for hadrons[J]. Phys.Rev.D., 1977, V15: 1694-1710
[41] Lie S.. Theorie der transformationsgruppen I[J]. Mathematische Annalen, 1880, V16(4): 441-528
[42] Gazizov R. K., Ibragimov N. H.. Lie symmetry analysis of differential equations in finance[J]. Nonlin.Dyn.. 1998,V17: 387-407
[43] Cox J. C, Ingersoll J. E., Ross S. A.. The theory of the term structure of interest ratcs[J]. Econometrica, 1985, V53: 385-407
[44] Vasicek O.. An equilibrium characterisation of the term structure[J]. Journal of Financial Economics, 1977,V5: 177-188
[45] Ho T. S., Lee S.. Term structure movements and pricing interest rate contingent claims. Journal of Finance, 1986. V42: 1129-1142
[46] Chan K. C, Karolyi G. A., Longstaff F.. Sanders A. B.. An empirical comparison of alternative models of the short-term interest rate. Journal of Finance, 1992,V47: 1209-1227
[47] Ahn D., Gao B.. A parametric nonlinear model of term structure dynamics. Review of Financial Studies, 1999. V12: 721-762
[48] Goard J.. New solutions to the bond-pricing equation via Lie's classical method. Mathematical and Computer Modelling, 2000, V32: 299-313
[49] Kenneth Johnpillai I.. Scott W.McCue, James M. Hill. Lie group symmetry analysis for granular media stress equations[J]. J.Math.Anal.Appl., 2005, V301: 135-157
[50] Dresner L.. Applications of Lie's Theory Ordinary Partial Differential Equations[M]. Philadelphia: IOP Publishing Ltd., 1999
[51] Makinde O. D., Moitsheki R. J., Tau B. A.. Similarity reductions of equations for river pollution,Applied Mathematics and Computation, 2007, V188: 1267-1273
[52] Edwards M. P., Hill J. M.. Selvadurai A. P. S.. Lie group symmetry analysis of transport in porous media with variable transmissivity[J]. J.Math.Anal.Appl., 2008, V341: 906-921