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.     «

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...     »