Consider a standard Poisson process (left(N_{t} ight)_{t in[0, T]}) with intensity (lambda>0) and a standard Brownian motion

Consider a standard Poisson process \(\left(N_{t}\right)_{t \in[0, T]}\) with intensity \(\lambda>0\) and a standard Brownian motion \(\left(B_{t}\right)_{t \in[0, T]}\) independent of \(\left(N_{t}\right)_{t \in[0, T]}\) under the probability measure \(\mathbb{P}_{\lambda}\). Let also \(\left(Y_{t}\right)_{t \in[0, T]}\) be a compound Poisson process with i.i.d. jump sizes \(\left(Z_{k}\right)_{k \geqslant 1}\) of distribution \(u(d x)\) under \(\mathbb{P}_{\lambda}\), and consider the jump process \(\left(S_{t}\right)_{t \in[0, T]}\) solution of

\[d S_{t}=r S_{t} d t+\sigma S_{t} d B_{t}+\eta S_{t^{-}}\left(d Y_{t}-\tilde{\lambda} \mathbb{E}\left[Z_{1}\right] d t\right)\]

with \(r, \sigma, \eta, \lambda, \widetilde{\lambda}>0\).

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martingale | a) Assume that X =A. Under the probability measure PA, the discounted price process (ert St) te [0,T] is a: submartingale | supermartingale | b) Assume > A. Under the probability measure PA, the discounted price process (e-rt St) te [0.T] is a: submartingale | martingale | supermartingale | c) Assume < A. Under the probability measure PA, the discounted price process rt (e "St)te [0,T] is a: submartingale martingale | supermartingale | d) Consider the probability measure Px defined by its Radon-Nikodym density dP dPx := e-(-A)T (A) NT