Explicit Predictable Compensators for Single Jump Processes with Initial Information

We study the predictable compensators of stochastic processes in a single jump filtration augmented with initial information represented by a sub-$\sigma$-algebra $\mathcal{H}$. We consider adapted càdlàg processes of finite variation and give an explicit construction of their predictable compensators. The main difficulty arises when the jump size has a heavy tail and lacks integrability, so that the classical Doob-Meyer decomposition does not apply. To overcome this, we use the theory of $\sigma$-martingales. We establish necessary and sufficient conditions for a process to be a $\sigma$-martingale and explicitly compute the compensator of a suitably weighted process. This yields an explicit relation between the continuous drift and the expected jump component of the original process.