= Exponential surplus martingale
{title2=$Z_t=\exp(R(\sum_{j=1}^{N_t}X_j-ct))$}
In a <classical risk model>, an <adjustment coefficient> $R$ makes $\lambda(M_X(R)-1)-cR=0$. The <random-sum transform identity> and <independent increments> show that $\mathbb E[Z_t\mid\mathcal F_s]=Z_s$. This nonnegative <martingale> starts at one. At the ruin <stopping time> $\tau$, $Z_\tau>e^{Ru}$. Applying the <optional stopping theorem> only to $\tau\wedge t$ yields $\mathbb P(\tau\le t)\le e^{-Ru}$; passage to increasing horizons proves the <Lundberg inequality> without an unjustified expectation identity at an unbounded stopping time.
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