= Exponential martingale of a pure birth process
{title2=$M_t=\exp\!\left(\theta X_t-(e^\theta-1)\int_0^t\lambda(X_s)ds\right)$}
For a <pure birth process> with fixed initial state and finite statewise rates, the jump-count compensator gives $dM=(e^\theta-1)M_{-}[dN-\lambda(X_{-})dt]$, so $M$ is a <local martingale> up to explosion. If $\lambda\le C$, a rate-$C$ Poisson candidate clock with accepted jumps of probability $\lambda(X_{-})/C$ prevents explosion and dominates the jump count. Its finite exponential moments dominate every stopped $M$ on each finite horizon, upgrading the local martingale to a true martingale. A random initial state requires $\mathbb E e^{\theta X_0}<\infty$; uniformly bounded rates do not supply this missing initial integrability.
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