Given the tilting of a normal R.V., it is intuitive that the exponential tilting of X_t, a Brownian motion with drift \mu and variance \sigma^2, is a Brownian motion with drift \mu + \theta\sigma^2 and variance \sigma^2. Thus, any Brownian motion with drift under \mathbb{P} can be thought of as a Brownian motion without drift under \mathbb{P}_{\theta^*}. To observe this, consider the process X_t = B_t + \mu_t. f(X_t) = f_{\theta^*}(X_t)\frac{d\mathbb{P}}{d\mathbb{P}_{\theta^*}} = f(B_t)\exp\{\mu B_T - \frac{1}{2}\mu^2T\}. The likelihood ratio term, \exp\{\mu B_{T} - \frac{1}{2}\mu^2 T\}, is a martingale and commonly denoted M_T. Thus, a Brownian motion with drift process (as well as many other continuous processes adapted to the Brownian filtration) is a \mathbb{P}_{\theta^*}-martingale.
