Martingale-type processes indexed by the real line

TL;DR

This paper investigates increment martingales indexed by the real line, analyzing their behavior at negative infinity, their quadratic variations, and how these properties relate to the martingale property, including integration aspects.

Contribution

It introduces and studies classes of increment martingales indexed by the real line, focusing on their asymptotic behavior and integration theory, which are novel contributions.

Findings

Characterization of increment martingales at negative infinity

Relation between limiting behavior and martingale property

Development of integration theory for increment martingales

Abstract

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.