Consider an exponential compound Poisson process of the form

\[S_{t}=S_{0} \mathrm{e}^{\mu t+\sigma B_{t}+Y_{t}}, \quad t \geqslant 0\]

where \(\left(Y_{t}\right)_{t \in \mathbb{R}_{+}}\)is a compound Poisson process of the form (20.8).

a) Derive the stochastic differential equation with jumps satisfied by \(\left(S_{t}\right)_{t \in \mathbb{R}_{+}}\).

b) Let \(r>0\). Find a family \(\left(\widetilde{\mathbb{P}}_{u, \tilde{\lambda}, \tilde{u}}\right)\) of probability measures under which the discounted asset price \(\mathrm{e}^{-r t} S_{t}\) is a martingale.