(3.4) In the Cramer-Lundberg model, let
$\theta(r) = \lambda(E(e^{rX_1} - 1)) - cr$
for those $r$-values for which $E(e^{rX_1})$ exists.
Prove that the process
$(M_r(t))_t = (e^{rU(t) - \theta(r)t})_t$
is a martingale. (Hint: Use conditioning on $S_0, S_1, \dots$).