  Some classes of increment martingales, and the corresponding localized
classes, are studied. An increment martingale is indexed by the real line and
its increment processes are martingales. We focus primarily on the behavior as
time goes to minus infinity in relation to the quadratic variation or the
predictable quadratic variation, and we relate the limiting behavior to the
martingale property. Finally, integration with respect to an increment
martingale is studied.

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