A variational representation for G-Brownian functionals

Abstract

The purpose of this paper is to establish a variational representationHere 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.

Girsanov’s formula forG-Brownian motion

“…Among them is the derivation of a variational representation for functionals of Brownian motion due to Boué-Dupuis [2], where they also showed the usefulness of the representation by applying it to prove Laplace principles for families of functionals of Brownian motion. Using the main result of the present paper, we establish in [5] a variational representation for functionals of G-Brownian motion and show that a similar application is possible under the framework of G-expectation space. Independently of our work [5], Gao [3] also obtains the representation by using our Girsanov's formula, and discusses an application to a large deviation for stochastic flows driven by G-Brownian motion.…”

“…Using the main result of the present paper, we establish in [5] a variational representation for functionals of G-Brownian motion and show that a similar application is possible under the framework of G-expectation space. Independently of our work [5], Gao [3] also obtains the representation by using our Girsanov's formula, and discusses an application to a large deviation for stochastic flows driven by G-Brownian motion.…”