Under the Girsanov theorem, if we are given a Brownian motion $W_t$, then multiplying the probability distribution by the Girsanov term
\begin{align*}
G_t = e^{-\gamma W_t \; – \; \frac{\gamma^2}{2} t}
\end{align*}we can obtain a new Brownian motion $W_t$ with probability distribution Q. These two processes are related through
\begin{align*}
d\widehat{W}_t = dW_t + \gamma dt
\end{align*}or alternatively,\begin{align*}
dW_t = d\widehat{W}_t \; – \; \gamma dt
\end{align*}Under the Girsanov theorem, we have a new measure under which the Brownian motion with drift becomes driftless under new measure $\mathbb{Q}$.
When we compute the discounted dynamics $d\widetilde{S}_t = d( e^{-rt} S_t)$, using product rule, we substitute the SHARPE RATIO for $\gamma$ to make it driftless. The Sharpe Ratio is defined as
$$\gamma = \dfrac{\mu – r}{\sigma}$$
This is an important consideration because by the First Fundamental Theorem of Asset Pricing discounted asset prices are Q-martingales.
Additionally, by Girsanov’s theorem, we can go from the risk-neutral measure to the forward neutral measure.