Quadratic variation and quadratic roughness

Rama Cont (1, 2) , Purba Das (2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 MI - Mathematical Institute [Oxford]

We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We define the quadratic roughness of a path along a partition sequence and show that, for Hölder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Paths of Brownian motion are shown to satisfy this quadratic roughness property almost-surely. Using these results we derive a formulation of Föllmer's pathwise integration along paths with finite quadratic variation which is invariant with respect to the partition sequence.

Mots clés: Quadratic variation; Pathwise integral; Itô calculus; Local time

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

Rama Cont, Purba Das. Quadratic variation and quadratic roughness. 2019. ⟨hal-02176236v1⟩