Some classes of increment martingales, and the corresponding localised classes, are studied. An increment martingale is indexed by $\mathbb{R}$ and its increment processes are martingales. We focus primarily on the behavior as time goes to $-\infty$ in relation to the quadratic variation or the predictable quadratic variation, and we relate the limiting behaviour to the martingale property. Finally, integration with respect to an increment martingale is studied.

Keywords: Martingales; increments; integration; compensators.

AMS Subject classification (2010): 60G44; 60G48; 60H05