Arithmetic Brownian motion subordinated by tempered stable and inverse tempered stable processes

TL;DR

This paper compares two types of subordinated arithmetic Brownian motions driven by tempered stable and inverse tempered stable processes, analyzing their properties, estimation methods, and real-world calibration for indoor air quality data.

Contribution

It introduces a detailed comparison of these two subordinated processes, including their properties, parameter estimation, and application to real indoor air quality data.

Findings

The probability density functions of both processes are characterized.

Parameter estimation procedures are proposed for practical calibration.

The models are successfully calibrated to real indoor air quality data.

Abstract

In the last decade the subordinated processes have become popular and found many practical applications. Therefore in this paper we examine two processes related to time-changed (subordinated) classical Brownian motion with drift (called arithmetic Brownian motion). The first one, so called normal tempered stable, is related to the tempered stable subordinator, while the second one - to the inverse tempered stable process. We compare the main properties (such as probability density functions, Laplace transforms, ensemble averaged mean squared displacements) of such two subordinated processes and propose the parameters' estimation procedures. Moreover we calibrate the analyzed systems to real data related to indoor air quality.