Integrability and tail estimates for Gaussian rough differential equations

TL;DR

This paper provides explicit tail estimates for the Jacobian of solutions to Gaussian rough differential equations, demonstrating finite moments for a broad class of Gaussian processes including fractional Brownian motion with H>1/4.

Contribution

It introduces new tail estimates for the Jacobian in Gaussian rough differential equations, applicable to a wide class of Gaussian processes, advancing understanding of their regularity properties.

Findings

Jacobian has finite moments of all orders for fractional Brownian motion with H>1/4

Explicit tail estimates derived for the Jacobian of Gaussian rough differential equations

Relevance to open problems in stochastic analysis and rough path theory

Abstract

We derive explicit tail-estimates for the Jacobian of the solution flow for stochastic differential equations driven by Gaussian rough paths. In particular, we deduce that the Jacobian has finite moments of all order for a wide class of Gaussian process including fractional Brownian motion with Hurst parameter H>1/4. We remark on the relevance of such estimates to a number of significant open problems.