It\^{o}--F\"{o}llmer Calculus in Banach Spaces II: Transformations of Quadratic Variations

TL;DR

This paper extends Itô-Föllmer calculus in Banach spaces by establishing a transformation formula for quadratic variations and exploring the relationship between tensor and scalar quadratic variations.

Contribution

It introduces a $C^1$-type transformation formula for quadratic variations in Banach spaces and analyzes tensor versus scalar quadratic variations.

Findings

Proved a $C^1$-type transformation formula for quadratic variations.

Explored relations between tensor and scalar quadratic variations.

Enhanced understanding of quadratic variations in infinite-dimensional settings.

Abstract

In this paper, we study properties of quadratic variations of c\`{a}dl\`{a}g paths within the framework of the It\^{o}--F\"{o}llmer calculus in Banach spaces. We prove a $C^1$-type transformation formula for quadratic variations. We also investigate relations between tensor and scalar quadratic variations.