A variational representation for G-Brownian functionals

TL;DR

This paper establishes a variational formula for functionals of G-Brownian motion under G-expectation, providing a new proof for large deviation principles related to G-Brownian motion.

Contribution

It introduces a variational representation for G-Brownian functionals, extending classical results to the G-expectation framework and offering an alternative proof for large deviations.

Findings

Derived a variational formula for G-Brownian functionals

Provided a new proof for large deviation results of G-Brownian motion

Extended classical stochastic analysis tools to the G-expectation setting

Abstract

The purpose of this paper is to establish a variational representation \log \E [e^{f(B)}] = \sup_h \E [f(B + \int_0^{\cdot} d<B>_s h_s) - 1/2 \int_0^1 h_s \cdot (d<B>_s h_s)] for functionals of the d-dimensional G-Brownian motion B. Here \E is a sublinear expectation called G-expectation, f is any bounded function in the domain of \E mapping C([0,1];\R^d) to \R, the integrals are taken with respect to the quadratic variation of B, and the supremum runs over all h's for which these integrals are well-defined. As an application, we give another proof of the results obtained by Gao-Jiang (2010), large deviations for G-Brownian motion.