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

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

where \(\left(Y_{k}\right)_{k \geqslant 1}\) is an i.i.d. sequence of uniformly distributed random variables on \([0,1]\).

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 the long forward contract with maturity \(T\) and payoff \(S_{T}-\kappa\).