Consider a standard Poisson process (left(N_{t} ight)_{t in mathbb{R}_{+}})with intensity (lambda>0), started at (N_{0}=0). a) Solve the

Consider a standard Poisson process \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)with intensity \(\lambda>0\), started at \(N_{0}=0\).

a) Solve the stochastic differential equation

\[d S_{t}=\eta S_{t^{-}} d N_{t}-\eta \lambda S_{t} d t=\eta S_{t^{-}}\left(d N_{t}-\lambda d t\right)\]

b) Using the first Poisson jump time \(T_{1}\), solve the stochastic differential equation

\[d S_{t}=-\lambda \eta S_{t} d t+d N_{t}, \quad t \in\left(0, T_{2}\right)\]