  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.
