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

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

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

b) Compute \(\mathbb{E}\left[X_{t}\right]\) for \(t \in \mathbb{R}_{+}\).