The Lie symmetry algebra of the Longstaff-Schwartz model

This study uses Lie’s theory of symmetries to compute the symmetry group of a class of partial differential equations parameterized by four constants: $u_t=-((a-bx)u_x+(d-ey)u_y+\frac{x}{2}{u}_{xx}+\frac{y}{2}{u}_{yy})$; under the various conditions on the constants $a,b,d$ and $e$, we deduce the largest and smallest Lie algebra of symmetries, and we also determined the structure of these algebras.