Let \(N\) be a Poisson process with intensity \(\lambda\). Prove that, if \(S_{t}=S_{0} e^{\mu t+\sigma N_{t}}\), then

\[d S_{t}=S_{t^{-}}\left(\mu d t+\left(e^{\sigma}-1\right) d N_{t}\right)\]

and that \(S\) is a martingale iff \(\mu=-\lambda\left(e^{\sigma}-1\right)\). Prove that, for \(a+1>0\), the process \(\left(L_{t}=\exp \left(N_{t} \ln (1+a)-\lambda a t\right), t \geq 0\right)\) is a martingale and that, if \(\left.\mathbb{Q}\right|_{\mathcal{F}_{t}}=\left.L_{t} \mathbb{P}\right|_{\mathcal{F}_{t}}\), the process \(N\) is a \(\mathbb{Q}\)-Poisson process with intensity \(\lambda(1+a)\).