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

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.


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
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