Laplace approximation for rough differential equation driven by fractional Brownian motion

TL;DR

This paper establishes Laplace approximation asymptotics for solutions of rough differential equations driven by fractional Brownian motion with Hurst parameter between 1/4 and 1/2, as a small parameter approaches zero.

Contribution

It provides the first Laplace asymptotics for rough differential equations driven by fractional Brownian motion in the specified Hurst range.

Findings

Derived Laplace asymptotics for fractional Brownian motion-driven equations

Extended rough path analysis to fractional Brownian motion with H in (1/4, 1/2)

Established asymptotic behavior as the small parameter tends to zero

Abstract

We consider a rough differential equation indexed by a small parameter $\varepsilon>0$. When the rough differential equation is driven by fractional Brownian motion with Hurst parameter $H$ ($1/4<H<1/2$), we prove the Laplace-type asymptotics for the solution as the parameter $\varepsilon$ tends to zero.