Functional representation and functional calculus for controlled paths
Abstract
We study the relation between non-anticipative functional calculus and compatible families of controlled paths. For a non-anticipative functional satisfying horizontal Lipschitz regularity, we show that the iterated vertical derivatives generate compatible higher-order controlled Taylor expansions along Hölder controls, with level-dependent remainder exponents. We derive a rough-integration criterion from these estimates and show that, for γ−Hölder controls with 1/2≥ γ>√2−1, the first-order remainder estimate is recovered. Our main result is a converse representation theorem. We consider non-anticipative functionals G0,…,Gp which satisfy compatible higher-order controlled Taylor estimates along γ-Hölder paths. Under natural continuity and compatibility assumptions, we prove that they can be represented as the iterated vertical derivatives of the base functional: $ G_j=\nabla_ω^jG_0,\qquad j=1,\ldots,p.$ Thus the Gubinelli coefficients of a compatible controlled family are symmetric and uniquely determined by its base functional; in particular, the Gubinelli derivative is identified with the vertical derivative introduced in functional Itô calculus, giving the coefficient hierarchy an intrinsic path-space differential structure. We show that this class of compatible coefficient families is stable under admissible non-anticipative functional composition and derive the corresponding functional chain rule. As an application, we obtain well-posedness for a class of path-dependent rough differential equations with Volterra memory, and identify the Gubinelli derivative of the resulting path-dependent rough coefficient.
Related
- Higher-Order Paracontrolled Calculus for Geometric Rough Paths
- Rough paths below the Young threshold: an exact scale calculus and the locality phase transition at one quarter
- Real-analytic functional calculus and strictly real Banach algebras
- The computable functional calculus
- Rapid decay and functional calculus in C*-probability spaces
BibTeX
@misc{ananova2026functional,
title = {{Functional representation and functional calculus for controlled paths}},
author = {Anna Ananova and Rama Cont},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06228},
eprint = {2609.06228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06228},
}
RIS
TY - UNPB
TI - Functional representation and functional calculus for controlled paths
AU - Anna Ananova
AU - Rama Cont
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.06228
UR - https://arxiv.org/abs/2609.06228
AB - We study the relation between non-anticipative functional calculus and compatible families of controlled paths. For a non-anticipative functional satisfying horizontal Lipschitz regularity, we show that the iterated vertical derivatives generate compatible higher-order controlled Taylor expansions along Hölder controls, with level-dependent remainder exponents. We derive a rough-integration criterion from these estimates and show that, for γ−Hölder controls with 1/2≥ γ>√2−1, the first-order remainder estimate is recovered. Our main result is a converse representation theorem. We consider non-anticipative functionals G_0,…,G_p which satisfy compatible higher-order controlled Taylor estimates along γ-Hölder paths. Under natural continuity and compatibility assumptions, we prove that they can be represented as the iterated vertical derivatives of the base functional: $ G_j=\nabla_ω^jG_0,\qquad j=1,\ldots,p.$ Thus the Gubinelli coefficients of a compatible controlled family are symmetric and uniquely determined by its base functional; in particular, the Gubinelli derivative is identified with the vertical derivative introduced in functional Itô calculus, giving the coefficient hierarchy an intrinsic path-space differential structure. We show that this class of compatible coefficient families is stable under admissible non-anticipative functional composition and derive the corresponding functional chain rule. As an application, we obtain well-posedness for a class of path-dependent rough differential equations with Volterra memory, and identify the Gubinelli derivative of the resulting path-dependent rough coefficient.
ER -