Precise Laplace approximation for mixed rough differential equation

TL;DR

This paper develops a precise Laplace approximation for solutions to mixed rough differential equations driven by mixed fractional Brownian motion, involving large deviation principles and detailed Hessian analysis.

Contribution

It introduces a novel Laplace approximation method for mixed rough differential equations, overcoming challenges in Hessian analysis specific to mixed fractional Brownian motion.

Findings

Established Schilder-type large deviation principle for mixed fBm-driven RDEs.

Proved Hilbert-Schmidt property of the Hessian matrix in this context.

Constructed a more precise exponential scale asymptotic approximation.

Abstract

This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path with as . Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the It\^o map restricted on the Cameron-Martin space of the mixed fBm. To this end, we imbed the Cameron-Martin space into a larger Hilbert space, then the Hessian is computable. Subsequently, the probability representation for the Hessian is shown. Finally, the Laplace approximation is constructed, which asserts the more precise asymptotics in the exponential scale.