A stochastic Taylor-like expansion in the rough path theory

TL;DR

This paper develops a Taylor-like expansion within rough path theory for Itô maps with varying roughness levels, extending previous results and applying them to analyze Laplace asymptotics of Brownian rough paths.

Contribution

It introduces a novel Taylor-like expansion in rough path theory applicable to different roughness levels, including cases where the roughness parameter exceeds 2.

Findings

Established a Taylor-like expansion for Itô maps in rough path theory.

Extended the expansion to cases with roughness parameter greater than 2.

Applied the expansion to derive Laplace asymptotics for Brownian rough paths.

Abstract

In this paper we establish a Taylor-like expansion in the context of the rough path theory for a family of It ^{o} maps indexed by a small parameter. We treat not only the case that the roughness $p$ satisfies $[p]=2$, but also the case that $[p] ge 3$. As an application, we discuss the Laplace asymptotics for It^{o} functionals of Brownian rough paths.