Extend formula (8.2.5) to functions \(f\) defined on \(\mathbb{R}^{+} \times \mathbb{N}\) that are \(C^{1}\) with respect to the first variable, and prove that if \(\beta\) is a constant with \(\beta>-1\) and \(L_{t}=\exp \left(\log (1+\beta) N_{t}-\lambda \beta t\right)\), then \(d L_{t}=L_{t^{-}} \beta d M_{t}\).

More generally, let \(L_{t}=(1+a)^{N_{t}} e^{-\lambda a t}\) for \(a \in \mathbb{R}\). Prove that \(L\) satisfies \(d L_{t}=L_{t}-a d M_{t}\), i.e.,

\[L_{t}=1+\int_{0}^{t} L_{s^{-}} a d M_{s}=1+a \int_{0}^{t} L_{s^{-}} d N_{s}-\lambda a \int_{0}^{t} L_{s^{-}} d s .\]

Note that, for \(a<-1, L_{t}\) takes values in \(\mathbb{R}\). The process \(L\) is the Doléans-Dade exponential of the martingale \(a M\).

\[\begin{equation*}
f\left(N_{t}\right)-f(0)=\int_{0}^{t}\left(f\left(N_{s-}+1\right)-f\left(N_{s-}\right)\right) d N_{s}, \tag{8.2.5}
\end{equation*}\]