On pathwise quadratic variation for cadlag functions

Abstract

We revisit H. Foellmer's concept of quadratic variation of a cadlag function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of cadlag processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition of quadratic variation which implies the Lebesgue decomposition as a result, rather than requiring it as an extra condition.