Branched Itô formula and natural Itô-Stratonovich isomorphism

Abstract

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 Itô 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 Itô-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 Doléans-Dade exponential, and conclude with some examples of branched rough path lifts of a stochastic process which are not quasi-geometric.

arXiv comment55 pages, 1 figure

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• A Survey on the Munthe-Kaas-Wright Hopf Algebra