Girsanov's formula for G-Brownian motion

TL;DR

This paper extends Girsanov's formula to G-Brownian motion, enabling measure change techniques in the framework of sublinear expectations and multi-dimensional processes, broadening the scope of stochastic analysis under uncertainty.

Contribution

It establishes Girsanov's formula for multi-dimensional G-Brownian motion using a novel approach based on the representation of G-expectation and martingale measure enlargement.

Findings

Girsanov's formula is valid for G-Brownian motion.

The method applies to multi-dimensional G-Brownian motion.

It broadens stochastic calculus under model uncertainty.

Abstract

In this paper, we establish Girsanov's formula for $G$-Brownian motion. Peng (2007, 2008) constructed $G$-Brownian motion on the space of continuous paths under a sublinear expectation called $G$-expectation; as obtained by Denis et al. (2011), $G$-expectation is represented as the supremum of linear expectations with respect to martingale measures of a certain class. Our argument is based on this representation with an enlargement of the associated class of martingale measures, and on Girsanov's formula for martingales in the classical stochastic analysis. The methodology differs from that of Xu et al. (2011), and applies to the multi-dimensional $G$-Brownian motion.