Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)

TL;DR

This paper investigates the quadratic variation and local times of the horizontal component of the Peano curve, revealing unique deterministic properties that distinguish it from stochastic processes like Brownian motion.

Contribution

It demonstrates that the horizontal component has a quadratic variation depending on the rational parameter p and provides a novel example of a deterministic function with local time.

Findings

Quadratic variation depends on the rational parameter p.

The function has a well-defined local time expressed as a limit of interval crossings.

Distinct from Wiener process trajectories in its deterministic nature.

Abstract

We show 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 $3^{-n}p\mathbb{Z}+3^{-n}r$, $n=1,2,\ldots$, where $p$ is a rational number, while $r$ is irrational number, but the value of such quadratic variation depends on $p$. This also yields that the horizontal component of the Peano curve is an example of a deterministic function possessing local time (density of the occupation measure) with respect to the Lebesgue measure, whose local time can be expressed as the limit of normalized numbers of interval crossings by this function but the normalization is not a smooth function of the width of the intervals. These two features distinct the horizontal component of the Peano curve from the trajectories of the Wiener process, which is widely used in financial models.