Consider a standard Poisson process (left(N_{t} ight)_{t in mathbb{R}_{+}})with intensity (lambda>0) under a probability measure (mathbb{P}). Let

Consider a standard Poisson process \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)with intensity \(\lambda>0\) under a probability measure \(\mathbb{P}\). Let \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\)be defined by the stochastic differential equation

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

where \(\eta>0\).

a) Find the value of \(\alpha \in \mathbb{R}\) such that the discounted process \(\left(\mathrm{e}^{-r t} S_{t}\right)_{t \in \mathbb{R}_{+}}\)is a martingale under \(\mathbb{P}\).

b) Compute the price at time \(t \in[0, T]\) of a power option with payoff \(\left|S_{T}\right|^{2}\) at maturity \(T\).