Branched Itô formula and natural Itô--Stratonovich isomorphism

Abstract

Branched rough paths define integration theories that may fail to satisfy the usual integration by parts identity. The intrinsically-defined projection of the Connes-Kreimer Hopf algebra onto its primitive elements defined by Broadhurst and Kreimer, and further studied by Foissy, allows us to view it as a commutative B?-algebra and thus to write an explicit change- of-variable formula for solutions to rough differential equations. This formula, which is realised by means of an explicit morphism from the Grossman-Larson Hopf algebra to the Hopf algebra of differential operators, restricts to the well-known Itô formula for semimartingales. We establish an isomorphism with the shuffle algebra over primitives, extending Hoffman?s exponential for the quasi shuffle algebra, and in particular the usual Itô-Stratonovich correction formula for semimartingales. We place special emphasis on the one-dimensional case, in which certain rough path terms can be expressed as polynomials in the extended trace path, which for semimartingales restrict to the well-known Kailath-Segall polynomials. We end by describing an algebraic framework for generating examples of branched rough paths, and, motivated by the recent literature on stochastic processes, exhibit a few such examples above scalar 1/4-fractional Brownian motion, two of which are ?truly branched?, i.e. not quasi- geometric.