Branched It\^o formula and natural It\^o-Stratonovich isomorphism

Branched rough paths, defined as paths with values in the character group of the Connes-Kreimer Hopf algebra $\mathcal{H}_\mathrm{CK}$, constitute integration theories that may fail to satisfy the usual integration by parts identity. Using known results on the primitive elements of $\mathcal{H}_\mathrm{CK}$ we can view it as a commutative cofree Hopf algebra (i.e. a commutative $\mathbf{B}_\infty$-algebra) and thus write an explicit change-of-variable formula for solutions to rough differential equations. This formula restricts to the well-known It\^o formula in the very special case of semimartingales. In addition, we establish an isomorphism between $\mathcal{H}_\mathrm{CK}$ and the shuffle algebra over its primitives, which extends Hoffman's exponential for the quasi-shuffle algebra, and can therefore be viewed as a far-reaching generalisation of the usual It\^o-Stratonovich correction formula for semimartingales. Indeed, this can be stated as a characterisation of the algebra structure of any commutative $\mathbf{B}_\infty$-algebra. Compared to previous approaches, this transformation has the key property of being natural in the decorating vector space. We study the one-dimensional case more closely, by introducing the branched analogue of the Kailath-Segall polynomials and Dol\'eans-Dade exponential, and conclude with some examples of branched rough path lifts of a stochastic process which are not quasi-geometric.