Lie symmetry analysis of differential equations in finance. (English) Zbl 0929.35006

Gazizov, R. K.; Ibragimov, N. H.

Nonlinear Dyn. 17, No. 4, 387-407 (1998).

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.

Reviewer: V.A.Yumaguzhin (Pereslavl’-Zalesskij)

MSC:

35A30	Geometric theory, characteristics, transformations in context of PDEs
91B24	Microeconomic theory (price theory and economic markets)
58J70	Invariance and symmetry properties for PDEs on manifolds

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