The Dupire derivatives and Fr\'echet derivatives on continuous pathes

TL;DR

This paper explores the relationship between Dupire derivatives and Fréchet derivatives for non-anticipative functionals on continuous paths, establishing their coherence and providing a theoretical foundation for their connection.

Contribution

It introduces the definition of Fréchet derivatives for non-anticipative functionals and proves their coherence with Dupire derivatives on continuous paths.

Findings

Dupire derivatives and Fréchet derivatives are coherent on continuous paths

Extended Fréchet derivatives align with Dupire derivatives

Provides a theoretical basis for derivatives on path spaces

Abstract

In this paper, we study the relation between Fr\'echet derivatives and Dupire derivatives, in which the latter are recently introduced by Dupire [4]. After introducing the definition of Fr\'echet derivatives for non-anticipative functionals, we prove that the Dupire derivatives and the extended Fr\'echet derivatives are coherent on continuous pathes.