Rough differential equations with unbounded drift term

TL;DR

This paper investigates rough differential equations with unbounded drift, establishing conditions for solution flows and semiflows, and deriving bounds, tail estimates, and large deviation results, with implications for stochastic analysis.

Contribution

It extends the theory of rough differential equations to include unbounded drifts, providing existence, bounds, and probabilistic estimates for solutions.

Findings

Solution flow exists if drift grows at most linearly

Semiflow exists under one-sided growth conditions

Derived tail estimates and large deviation principles for Gaussian-driven equations

Abstract

We study controlled differential equations driven by a rough path (in the sense of T. Lyons) with an additional, possibly unbounded drift term. We show that the equation induces a solution flow if the drift grows at most linearly. Furthermore, we show that the semiflow exists assuming only appropriate one-sided growth conditions. We provide bounds for both the flow and the semiflow. Applied to stochastic analysis, our results imply "strong completeness" and the existence of a stochastic (semi)flow for a large class of stochastic differential equations. If the driving process is Gaussian, we can further deduce (essentially) sharp tail estimates for the (semi)flow and a Freidlin-Wentzell-type large deviation result.