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 \(d X_{t}=\alpha X_{t} d t+\sigma d N_{t}\) over the time intervals \(\left[0, T_{1}\right),\left[T_{1}, T_{2}\right),\left[T_{2}, T_{3}\right),\left[T_{3}, T_{4}\right)\), where \(X_{0}=1\).

b) Write a differential equation for \(f(t):=\mathbb{E}\left[X_{t}\right]\), and solve it for \(t \in \mathbb{R}_{+}\).