On pathwise quadratic variation for càdlàg functions

Rama Cont (1) , Henry Chiu (2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 Department of Mathematics [Imperial College London]

We revisit Föllmer's concept of quadratic variation of a càdlàg 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 càdlàg processes , one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. The definition then simplifies and one obtains the Lebesgue decomposition of the pathwise quadratic variation as a result, rather than requiring it as an extra condition.

Mots clés: Quadratic variation; semimartingale; pathwise calculus; Ito formula; pathwise integration; cadlag functions; Skorokhod topology

Domaines: Probabilités [math.PR]; Analyse classique [math.CA]

Rama Cont, Henry Chiu. On pathwise quadratic variation for càdlàg functions. 2018. ⟨hal-01818930v1⟩