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

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.