= Exponential Laplace martingale of a compound Poisson process
{title2=$Z_t=\exp[-qY_t+\lambda t(1-\phi(q))]$}
Let $Y_t=\sum_{j=1}^{N_t}X_j$ for a rate-$\lambda$ <Poisson process> and <independent and identically distributed> nonnegative marks independent of that process, with $\phi(q)=\mathbb E e^{-qX_1}$. The <Laplace transform of a nonnegative random variable> for the increment over $[s,t]$ is $\exp[\lambda(t-s)(\phi(q)-1)]$. That increment is independent of the past of $Y$, so the displayed nonnegative process has mean one and satisfies $\mathbb E[Z_t\mid\sigma(Y_u:u\leq s)]=Z_s$. No finite first moment of the marks is needed for this <martingale> when $q\geq0$.
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