Controlled fields, rough stochastic calculus, and It\^o-Wentzell-Alekseev-Gr\"obner identities

TL;DR

This paper introduces a calculus for space-time controlled fields in rough stochastic systems, leading to a unified rule for evaluating random fields along rough semimartingales and deriving a rough stochastic Itô-Wentzell formula.

Contribution

It develops a new calculus for controlled fields in rough stochastic systems, unifying composition rules and deriving a rough Itô-Wentzell formula under natural conditions.

Findings

Established a unified composition rule for rough stochastic systems.

Derived a rough stochastic Itô-Wentzell formula.

Connected to recent works on Itô-Alekseev-Gr"obner and diffusion interpolation.

Abstract

We develop a calculus of space-time controlled fields for rough stochastic systems. This approach provides a unified composition rule for evaluating random fields along rough semimartingales and yields a rough stochastic It\^o-Wentzell formula under natural and verifiable regularity assumptions. Our motivation comes from works of Hudde et al. (2024) and, independently, Del Moral and Singh (2022) where the authors established, respectively, It\^o-Alekseev-Gr\"obner, backward It\^o-Wentzell, and diffusion interpolation formulas.