Average-tempered stable subordinators with applications

TL;DR

This paper introduces a new class of infinite-activity subordinators derived from the running average of tempered stable processes, providing explicit formulas, asymptotic estimates, and applications in structural and financial modeling.

Contribution

It explores a novel family of subordinators induced by running averages, deriving their properties and demonstrating their practical applications in various fields.

Findings

Derived explicit formulas for distribution functions, cumulants, and moments.

Provided asymptotic behavior estimates for the new subordinators.

Showcased applications in structural degradation and financial derivatives pricing.

Abstract

In this paper the running average of a subordinator with a tempered stable distribution is considered. We investigate a family of previously unexplored infinite-activity subordinators induced by the probability distribution of the running average process and determine their jump intensity measures. Special cases including gamma processes and inverse Gaussian processes are discussed. Then we derive easily implementable formulas for the distribution functions, cumulants, and moments, as well as provide explicit estimates for their asymptotic behaviors. Numerical experiments are conducted for illustrating the applicability and efficiency of the proposed formulas. Two important extensions of the running average process and its equi-distributed subordinator are examined with concrete applications to structural degradation modeling and financial derivatives pricing, where their advantages relative to several existing models are highlighted together with the mention of Euler discretization and compound Poisson approximation techniques.