On pathwise quadratic variation for càdlàg functions

Henry Chiu,Rama Cont

doi:10.1214/18-ecp186

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.

Highlights

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
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 the quadratic variation as a limit, in Skorokhod topology, of discrete approximations defined along the partition

Summary

Pathwise quadratic variation for cadlag functions

Observe that the pointwise limits of (sn) and (qn) coincide i.e. converges to 0 by the right-continuity of x, where t(in) := max {πn ∩ [0, t]} and that x ∈ Qπ0 , Prop. Denote Qπ1 the set of x ∈ D such that (sn) defined in (2.3) has a pointwise limit s with Lebesgue decomposition given by (2.4). Denote Qπ2 the set of x ∈ D such that the quadratic sums (qn) defined in (2.7) have a pointwise limit q with Lebesgue decomposition (2.8). Qπ2 the set of x ∈ D such that the quadratic sums (qn) defined by (2.7) have a pointwise limit q with Lebesgue decomposition (2.8). If one is interested in applying this definition to a process, say a semimartingale, in general there may exist no sequence of partitions satisfying this condition almost surely across sample paths. Our definition does not require such a condition and carries over to stochastic processes without requiring the use of random partitions (see Theorem 4.1)

Quadratic variation for multidimensional functions

Some applications