Abstract
In 1971, B. Kent Harrison and Frank B. Estabrook introduced a method to determine the symmetries of partial differential equations (PDEs). These last years, the determination of the symmetries of PDEs in Mathematical Physics, in Mathematical Biology and in Financial Mathematics has proved useful. The computations effected in all these cases let appear a remarkable degree of similarity between them. So with the same aim in mind, we develop a general framework for the computation of the symmetries with this method, we give properties of isovectors for a rather general type of PDE’s and some results on the Lie algebra itself. Finally we present three examples for which all the results we exposed hold.
Highlights
We shall denote by G the isovector algebra of (E); this is the set of vector fields N ∈ T M such that
∂q because M t = M q = 0 and N (M q) − M (N q) only depends on t and q
L[N,N ] = 0 and [N, N ] ∈ H; H is a subalgebra of G
Summary
We shall take as state space M := J × O × Rn+1, the generic point of which will be denoted by (t, q, u, A, B1, . . . , Bn−1). We shall denote by G the isovector algebra of (E); this is the set of vector fields N ∈ T M such that. H is a subalgebra of G, J is an abelian ideal of G and the sum J H is direct; in particular. J is an ideal of G according to the same reasoning as Theorem 3. 2. Examples (i) We can apply the method and these result to find symmetries of the Black-Scholes equation. Examples (i) We can apply the method and these result to find symmetries of the Black-Scholes equation This is the most famous equation in Mathematical Finance:
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