On the Continuity of Natural Filtrations of Processes with Independent Increments

Abstract

We study the continuity of the natural filtration of an arbitrary random process with a convex parametric set with values in an arbitrary separable metric space. It is proved that the natural filtration is left-continuous if the random process is left-continuous. The extended natural filtration is left-continuous if the random process is stochastically left-continuous. If a stochastically right-continuous random process takes values in the finite-dimensional Euclidean space and has independent increments, then its extended natural filtration is right-continuous. These statements generalize the well-known fact that the extended natural filtration of any Lévy process defined on the complete probability space is right-continuous.