Branched Itô Formula and natural Itô-Stratonovich isomorphism

Abstract: Branched rough paths define integration theories that may fail to satisfy the integration by parts identity. The projection of the Connes-Kreimer Hopf algebra (\(\mathcal{H}_{\mathrm{CK}}\)) onto its primitive elements defined by Broadhurst-Kreimer and Foissy, allows us to view \(\mathcal{H}_{\mathrm{CK}}\) as a commutative \(\mathbf{B}_\infty\)-algebra and thus to write an explicit change-of-variable formula for solutions to rough differential equations (RDEs), which restricts to the well-known Itô formula for semimartingales. When compared with Kelly’s approach using bracket extensions, this formula has the advantage of only depending on internal structure. We proceed to define an isomorphism between \(\mathcal{H}_{\mathrm{CK}}\) and \(\operatorname{Sh}(\mathcal{P})\) (the shuffle algebra over primitives), which we compare with the previous constructions of Hairer-Kelly and Boedihardjo-Chevyrev: while all three allow one to write branched RDEs as RDEs driven by geometric rough paths taking values in a larger space, the key feature of our isomorphism is that it is natural when \(\mathcal{H}_{\mathrm{CK}}\) and \(\operatorname{Sh}(\mathcal{P})\) are viewed as covariant functors \(\mathsf{Vec}\to\mathsf{Hopf}\). Our natural isomorphism extends Hoffman’s exponential for the quasi shuffle algebra, and in particular the usual Itô-Stratonovich correction formula for semimartingales. Special emphasis is placed on the 1-dimensional case, in which certain rough path terms can be expressed as polynomials in the trace path indexed by \(\mathcal{P}\), which for semimartingales restrict to the well-known Kailath-Segall polynomials.

This talk is based on joint work with E. Ferrucci and C. Bellingeri.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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