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Zbl 0929.35006
Gazizov, R.K.; Ibragimov, N.H.
Lie symmetry analysis of differential equations in finance.
(English)
[J] Nonlinear Dyn. 17, No.4, 387-407 (1998). ISSN 0924-090X; ISSN 1573-269X/e

The Black-Schole equation $u_t+\frac{1}{2} A^2x^2u_{xx}+ Bxu_x-Cu=0$ and Jacobs-Jones equation $u_t=\frac{1}{2} A^2x^2u_{xx}+ ABCxyu_{xy}+ \frac{1}{2}B^2y^2u_{yy} +(Dx\ln\frac{y}{x}- Ex^{3/2})u_x+ (Fy\ln\frac{G}{y}- Hyx^{1/2})u_y-xu$ are investigated in this paper. For the equations, algebras of classical symmetries are calculated; particular solutions are found with the help of symmetries. The most general transformation of the Black-Schole equation to the heat equation is obtained. The fundamental solution of the Cauchy problem for this equation is found.
[V.A.Yumaguzhin (Pereslavl'-Zalesskij)]
MSC 2000:
*35A30 Geometric theory for PDE, transformations
91B24 Price theory and market structure
58J70 Invariance and symmetry properties

Keywords: fundamental solution of the Cauchy problem; Lie group classification and symmetry analysis; group theoretical modeling; invariant solution; Black-Schole equation; Jacobs-Jones equation; algebras of classical symmetries

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