Arbitrary-order Frechet derivatives of exponential and logarithmic functions in real and complex Banach algebras: Applications to stochastic functional differential equations

Abstract

In this paper we derive explicit closed-form recursion-free formulae for arbitrary-order Frechet derivatives of exponential and logarithmic functions in unital Banach algebras (complex or real). These computations are obtained via Bochner integrals for Banach-algebra-valued functions with respect to the standard Lebesgue measure. As an application, we demonstrate how our results can help approximately solve stochastic functional differential equations.