Abstract
The non-linear sewing lemma constructs flows of rough differential equations from a broad class of approximations called almost flows. We consider a class of almost flows that could be approximated by solutions of ordinary differential equations, in the spirit of the backward error analysis. Mixing algebra and analysis, a Taylor formula with remainder and a composition formula are central in the expansion analysis. With a suitable algebraic structure on the non-smooth vector fields to be integrated, we recover in a single framework several results regarding high-order expansions for various kinds of driving paths. We also extend the notion of driving rough path. We introduce as an example a new family of branched rough paths, called aromatic rough paths modeled after aromatic Butcher series.
Highlights
Introduced at the end of the 1990s by T
Lyons [39], the theory of rough paths defines pathwise solutions to stochastic differential equations driven by Brownian paths and more generally by a large class irregular paths, random or deterministic
Generalizations to branched rough paths with trees of order n lead to similar controls [10, 22]. Another approximation is given by setting F rxs,tspaq “ z1s,t where zs,t is the solution to the ordinary differential equation (ODE)
Summary
Introduced at the end of the 1990s by T. Generalizations to branched rough paths with trees of order n lead to similar controls [10, 22] Another approximation is given by setting F rxs,tspaq “ z1s,t where zs,t is the solution to the ordinary differential equation (ODE). We extend the notion of a driving rough paths as a path with values in the truncated algebra Wďn derived from quotienting graded algebra W up to a given order n The latter result may be generalized by replacing Wďn to a quotient of W. Aromatic trees exhibit some universal properties [40], so that we could think they lead to the broadest natural class of branched rough paths The relationship between these different constructions will be subject to future work.
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