Symmetry group methods for fundamental solutions. (English) Zbl 1065.35016

The authors use Lie symmetry group methods to find fundamental solutions for a class of partial differential equations of the form \(u_t=xu_{xx}+f(x)u_x\), \(x\geq 0\), i.e. to find the kernel function \(p(t,x,y)\), such that \(u(x,t)=\int_0^\infty\varphi(y)p(t,x,y)\,dy\) is a solution of the relevant Cauchy problem with \(u(x,0)=\varphi(x)\).


35A30 Geometric theory, characteristics, transformations in context of PDEs
35K55 Nonlinear parabolic equations
58J70 Invariance and symmetry properties for PDEs on manifolds
35K65 Degenerate parabolic equations
35A08 Fundamental solutions to PDEs
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[1] Bluman, G.; Kumei, S., Symmetries and differential equations, (1989), Springer Berlin · Zbl 0698.35001
[2] Craddock, M., Symmetry groups of partial differential equations, separation of variables and direct integral theory, J. funct. anal., 125, 2, 452-479, (1994) · Zbl 0809.35014
[3] Craddock, M., The symmetry groups of linear partial differential equations and representation theory I, J. differential equations, 116, 1, 202-247, (1995) · Zbl 0845.35020
[4] Craddock, M., The symmetry groups of linear partial differential equations and representation theorythe Laplace and axially symmetric wave equations, J. differential equations, 166, 1, 107-131, (2000) · Zbl 0962.35010
[5] Craddock, M.; Dooley, A., Symmetry group methods for heat kernels, J. math. phys., 42, 1, 390-418, (2001) · Zbl 1063.35516
[6] M. Craddock, E. Platen, Symmetry group methods for fundamental solutions and characteristic functions, Technical Report, University of Technology, Sydney, Research Paper 90, Quantitative Finance Research Group, University of Technology, Sydney, February 2003. · Zbl 1065.35016
[7] Doetsch, G., Introduction to the theory and application of the Laplace transformation, (1970), Springer Berlin · Zbl 0202.12301
[8] Lamberton, D.; Lapeyre, B., Introduction to stochastic calculus applied to finance, (1996), Chapman & Hall London · Zbl 0898.60002
[9] Lie, S., Über die integration durch bestimmte integrale von einer klasse linear partieller differentialgleichung, Arch. math., 6, 328-368, (1881) · JFM 13.0298.01
[10] Lie, S., Zur allgemeinen theorie der partiellen differentialgleichungen beliebeger ordnung, Leipz. berichte, 47, 53-128, (1895)
[11] Lie, S., Vorlesungen über differentialgleichungen mit bekannten infinitesimalen transformationen, (1912), Teubner · JFM 43.0373.01
[12] Longstaff, F.A., A nonlinear general equilibrium model of the term structure of interest rates, J. financial econom., 23, 195-224, (1989)
[13] W. Miller, Symmetry and Separation of Variables, Encylopedia of Mathematics and its Applications, vol. 4, Addison-Wesley, Reading MA, 1974.
[14] P.J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, vol. 107, Springer, New York, 1993. · Zbl 0785.58003
[15] E. Platen, A Minimum Financial Market Model, Trends in Mathematics, Birkhauser, 2001, pp. 293-301. · Zbl 1004.91029
[16] D. Revuz, M. Yor, Continuous Martingales and Brownian Motion, third ed., Grundlehiren der mathematischen Wissenschaften, vol. 293, Springer, Berlin, 1998. · Zbl 0731.60002
[17] Widder, D.V., The Laplace transform, (1966), Princeton University Press Princeton · Zbl 0148.10902
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