= Exponential martingale from a jump generator
{title2=$Z_t=g(X_t)\exp\left(-\int_0^t\frac{Lg(X_{s-})}{g(X_{s-})}\,ds\right)$}
For a positive function $g$ in the domain of a <Markov jump-process generator>, the generator martingale for $g(X_t)$ and <semimartingale integration by parts> give the displayed <local martingale>. Suitable finite-horizon deterministic bounds on $g$ and $Lg/g$ make it a true <martingale>. For a <single marked exponential jump process>, take $g(x)=e^{\theta x}$. Then the exponent's compensating process is $r_\theta(t\wedge T)$, where $r_\theta=(e^\theta-1)/\theta-1$ for $\theta\ne0$ and $r_0=0$; it is continuous adapted and hence <predictable>.
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