Tempered positive Linnik processes and their representations

TL;DR

This paper explores the properties and representations of tempered positive Linnik processes, including their subordinated forms, self-similarity, and applications in multivariate models for finance and fraud detection.

Contribution

It introduces new subordinated representations, explicit compound Poisson forms with Mittag-Leffler innovations, and a multivariate TPL framework using negative binomial mixing.

Findings

Established stochastic self-similarity with negative binomial subordination.

Derived explicit compound Poisson representations with Mittag-Leffler innovations.

Developed a multivariate TPL Lévy framework for applications in finance and anti-fraud.

Abstract

This paper analyzes various classes of processes associated with the tempered positive Linnik (TPL) distribution. We provide several subordinated representations of TPL L\'evy processes and in particular establish a stochastic self-similarity property with respect to negative binomial subordination. In finite activity regimes we show that the explicit compound Poisson representations gives rise to innovations following Mittag-Leffler type laws which are apparently new. We characterize two time-inhomogeneous TPL processes, namely the Ornstein-Uhlenbeck (OU) L\'evy-driven processes with stationary distribution and the additive process determined by a TPL law. We finally illustrate how the properties studied come together in a multivariate TPL L\'evy framework based on a novel negative binomial mixing methodology. Some potential applications are outlined in the contexts of statistical anti-fraud and financial modelling.