Derivatives Along a Curve and the Functional Stochastic Calculus

TL;DR

This paper introduces a new notion of functional derivative along a curve to extend the functional stochastic calculus for path-dependent functionals, providing deeper structural insights.

Contribution

It develops a path-dependent directional extension of functional derivatives, broadening the applicability of stochastic calculus to more complex functionals.

Findings

New concept of derivatives along a curve introduced

Extended the functional stochastic calculus framework

Provided structural insights into path-dependent functionals

Abstract

Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent directional extensions. Our results then focus on a comprehensive exploration of these derivatives and the insights they provide on the structure of functionals.