$\sigma$-localization and $\sigma$-martingales

This paper introduces the concept of $\sigma$-localization, which is a generalization of localization in the general theory of stochastic processes. The $\sigma$-localized class derived from the set of martingales is the class of $\sigma$-martingales, which plays an important role in mathematical finance. These processes and the corresponding $\sigma$-martingale measures are considered in detail. By extending the stochastic integral with respect to compensated random measures, a canonical representation of $\sigma$-martingales as for local martingales is derived.