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

2025-04-08

AI Summary

This paper establishes an Itô formula for non-anticipative (non-prospective) functionals defined on càdlàg rough paths with jumps, and derives their path-signature-based Taylor expansions. Methodologically, it introduces the Marcus transformation—previously unexploited in non-anticipative functional calculus—to handle vertical jump perturbations, thereby resolving the non-commutativity of second-order vertical derivatives (e.g., for signature functionals); it integrates rough path theory, Marcus-type integration, and the Dupire derivative framework into a unified functional Itô calculus. Key contributions are: (1) the first Itô formula for non-anticipative functionals on càdlàg rough paths; (2) existence and convergence of path-signature-based Taylor expansions for sufficiently regular non-anticipative functionals; and (3) unification of the Friz–Zhang (2018) and Dupire (2009) functional Itô formulas, extending the Dupire–Tissot-Daguette (2022) result to the càdlàg rough path setting.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Economics, Online Markets and Human Computation: The sharing economy

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).

Problem

• Extends Itô-formula for non-anticipative càdlàg rough paths
• Incorporates jump perturbations via Marcus transformation
• Derives functional Taylor expansions using path signatures

Innovation

• Functional Itô-formula for non-anticipative rough paths
• Marcus transformation handles vertical jump perturbations
• Functional Taylor expansion via path's signature