Probability Seminar

Title: The Itô–Stratonovich formula for rough sheets.

Abstract: We present a new approach to developing an Itô–Stratonovich-type formula for rough sheets. Historically, planar integration for irregular paths has been notoriously difficult. The emergence of 'mixed differential' terms in 2D leads to overlapping iterated integrals, the construction of which previously required exhaustive combinatorial structures. One example of such a structure is the substantial 36-element planar signature introduced by K. Chouk and M. Gubinelli (Rough Sheets, arxiv.org/abs/1406.7748, 2014). In this work, we propose a simplified setting for rough sheet calculus that relies on elementary Taylor expansions in a more fundamental way. We claim that this simple trick significantly streamlines the complexity of planar algebraic integration. We illustrate the methodology in three regimes. Firstly, we present a special Rough-Young framework (joint work with S. Tindel, arxiv.org/abs/2606.20908) for paths with ‘’asymmetric'' Hölder regularity: H₁ > 1/3 in direction 1 and H₂ > 1/2 in direction 2. Relying on structured controlled path expansions, we rigorously minimize the set of iterated integrals required in the signature to 4 elements. We extend the classical planar change-of-variable formula to this regime and express it as an explicit limit of Riemann sums. The signature elements required for this are then constructed for the fractional Brownian sheet with 1/3<H₁<1/2 and H₂ > 1/2. Secondly, we consider the fully rough regime, where 1/3< H₁ ,H₂ < 1/2 (joint work with S. Tindel). Here, a symmetric second-order Taylor expansion produces a planar integral with a signature of 10 elements instead of 36 compared to Chouk-Gubinelli’s paper. Third, we demonstrate that the class of controlled sheets is not stable under integration. However, in the Rough-Young regime, the rough sheet integral can be interpreted as a one-parameter infinite-dimensional rough path integral (ongoing work with M. R. Lahwate). This allows us to interpret some stochastic Goursat PDEs as infinite-dimensional RDEs.