Functional It\^o-formula and Taylor expansions for non-anticipative maps of c\`adl\`ag rough paths

Abstract

We derive a functional It\^o-formula for non-anticipative maps of rough paths, based on the approximation properties of the signature of c\`adl\`ag rough paths. This result is a functional extension of the It\^o-formula for c\`adl\`ag rough paths (by Friz and Zhang (2018)), which coincides with the change of variable formula formulated by Dupire (2009) whenever the functionals' representations, the notions of regularity, and the integration concepts can be matched. Unlike these previous works, we treat the vertical (jump) pertubation via the Marcus transformation, which allows for incorporating path functionals where the second order vertical derivatives do not commute, as is the case for typical signature functionals. As a byproduct, we show that sufficiently regular non-anticipative maps admit a functional Taylor expansion in terms of the path's signature, leading to an important generalization of the recent results by Dupire and Tissot-Daguette (2022).