An operatorial approach of Black-Scholes partial differential equation

Abstract

This paper proposes an operatorial approach for the well-known Black-Scholes partial differential equation. Thus, we consider the Black-Scholes partial differential equation as a differential equation of the form $u'(t) = Au(t)$, where the operator $A$ acts on the Hilbert function space $L^2_{\mathbb{R}_+}$. The technique used in our approach is related to the theory of strongly continuous one-parameter semigroups.