Quadratic variation and stochastic calculus for deterministic paths and model-free price paths with jumps

Abstract

The construction of the Itô integral by means of the pathwise approach developed by Hans Föllmer is a delicate task, in a sense that it depends on the choice of the sequences of partitions from which the quadratic variation is defined; different sequences of partitions may lead to different quadratic variations, depriving us of the uniqueness of the Itô integral. Due to this, researchers have considered using modified schemes in order to achieve the uniqueness of the Itô formula. One of the latest of such schemes is known as the Lebesgue partitions and it was used to prove the existence of quadratic variation for continuous typical price paths. Other researchers have considered partition-independent approaches to quadratic variation, as an example they proved the existence of quadratic variation in terms of what is known as the normalized truncated variation. This study investigates the existence of quadratic variations defined in terms of Lebesgue partitions and modified truncated variation for deterministic paths with jumps. The study considers a generalization of Lebesgue partitions into what is known as shifted Lebesgue partitions and provides detailed stability analysis of the quadratic variations along shifted partitions for the horizontal component of the Peano curve. In particular the study shows that the horizontal component of the Peano curve has quadratic variation equal the limit of quadratic variations along the Lebesgue partitions for grids of the form 3−npZ + 3−nr, n = 1, 2, · · · , where p is rational number and r is irrational.