Girsanov formula for $G$-Brownian motion: the degenerate case

TL;DR

This paper extends the Girsanov formula to degenerate $G$-Brownian motion by developing a perturbation approach within a nonlinear expectation framework, broadening the applicability of stochastic calculus under uncertainty.

Contribution

It proves the Girsanov formula for degenerate $G$-Brownian motion, removing the non-degenerate condition and introducing a novel perturbation method in the nonlinear setting.

Findings

Established Girsanov formula for degenerate $G$-Brownian motion.

Developed a perturbation method in the nonlinear expectation framework.

Provided estimates for exponential martingales of $G$-Brownian motion.

Abstract

In this paper, we prove the Girsanov formula for $G$-Brownian motion without the non-degenerate condition. The proof is based on the perturbation method in the nonlinear setting by constructing a product space of the $G$-expectation space and a linear space that contains a standard Brownian motion. The estimates for exponential martingale of $G$-Brownian motion are important for our arguments.