I've found the following statement (but it is rather an example) in a book that states if $X$ is a local martingale and its quadratic variation has the form
$$\left[X\right]=t+cX$$
where $c$ is a constant, then $X$ is a compensated Poisson process. The example wanted to emphasize the similarity between this formula and the Levy characterization of a Brownian motion.
However this formula just doesn't seem right. Let $\pi$ denotes a Poisson process with $\lambda$ intensity parameter, so $X\dot{=}\pi-\lambda t$ is a compensated Poisson process. This way
$$\left[X\right]=\left[\pi-\lambda t\right]=\left[\pi\right]=\pi\neq t+c\left(\pi-\lambda t\right).$$
Can we somehow correct this formula or do I just miss something trivial here? Is there any similar formula for compensated Poisson processes?