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

Consider \(\left(N_{t}\right)_{t \in \mathbb{R}_{+}}\)a standard Poisson process 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

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

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

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

b) Compute the price at time 0 of a European call option on \(S_{T}\) with strike price \(\kappa\) and maturity \(T\), using a series of multiple integrals.