Consider a standard Poisson process (left(N_{t} ight)_{t in mathbb{R}_{+}})with intensity (lambda>0). a) Solve the stochastic differential equation

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

a) Solve the stochastic differential equation \(d X_{t}=\sigma X_{t^{-}} d N_{t}\) for \(\left(X_{t}\right)_{t \in \mathbb{R}_{+}}\), where \(\sigma>0\) and \(X_{0}=1\).

b) Show that the solution \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)of the stochastic differential equation

\[d S_{t}=r d t+\sigma S_{t^{-}} d N_{t},\]

is given by \(S_{t}=S_{0} X_{t}+r X_{t} \int_{0}^{t} X_{s}^{-1} d s\).

c) Compute \(\mathbb{E}\left[X_{t}\right]\) and \(\mathbb{E}\left[X_{t} / X_{s}\right], 0 \leqslant s \leqslant t\).

d) Compute \(\mathbb{E}\left[S_{t}\right], t \geqslant 0\).