Martingale Transformations of Jump Processes

We provide a large class of functions and their respective parameters to transform a jump-process into a martingale w.r.t. its natural filtration. The proofs are based on a discrete Doob-decomposition and a limiting procedure to continuous time, in turn resulting in a time-continuous Doob-Meyer decomposition. Martingale transformations are then determined by solving the Doob-Meyer decomposition for functions that elim inate the compensator. We discuss several related results and single jump filtrations. The results are provided for single-jump processes and are systematically generalized to the multi-jump case, highlighting the necessity of dependencies between current jumps and the processes paths. Eventually we apply the result to branching random walks as an instructive example.