Consider an asset price \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\)given by the stochastic differential equation

\[
\begin{equation*}
d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t} \tag{7.44}
\end{equation*}
\]

where \(\left(B_{t}ight)_{t \in \mathbb{R}_{+}}\)is a standard Brownian motion, with \(r \in \mathbb{R}\) and \(\sigma>0\).

a) Find the stochastic differential equation satisfied by the power \(\left(S_{t}^{p}ight)_{t \in \mathbb{R}_{+}}\)of order \(p \in \mathbb{R}\) of \(\left(S_{t}ight)_{t \in \mathbb{R}_{+}}\).

b) Using the Girsanov Theorem 7.3 and the discounting Lemma 5.13, construct a probability measure under which the discounted process \(\left(\mathrm{e}^{-r t} S_{t}^{p}ight)_{t \in \mathbb{R}_{+}}\)is a martingale.