[Paper Review] Transition densities of subordinators

Tomasz Grzywny, Łukasz Leżaj+1 more|arXiv (Cornell University)|Dec 17, 2018

Advanced Mathematical Modeling in Engineering4 citations

TL;DR

This paper establishes the existence and asymptotic behavior of transition densities for a broad class of subordinators whose Laplace exponents satisfy a lower scaling condition at infinity. Using scaling conditions and sharp bounds, it derives precise upper and lower estimates for the density, with improved accuracy when an additional upper scaling condition is imposed.

ABSTRACT

We prove existence and asymptotic behavior of the transition density for a large class of subordinators whose Laplace exponents satisfy lower scaling condition at infinity. Furthermore, we present lower and upper bounds for the density. Sharp estimates are provided if additional upper scaling condition on the Laplace exponent is imposed.

Motivation & Objective

• To establish the existence of transition densities for subordinators with Laplace exponents satisfying a lower scaling condition at infinity.
• To analyze the asymptotic behavior of these transition densities as time or space variables approach extreme values.
• To derive sharp upper and lower bounds for the transition density under additional regularity conditions on the Laplace exponent.
• To improve precision in density estimates by imposing both lower and upper scaling conditions on the Laplace exponent.

Proposed method

• The analysis relies on the theory of subordinators, which are non-decreasing Lévy processes, and their associated Laplace exponents.
• A lower scaling condition at infinity is imposed on the Laplace exponent to ensure regular variation properties necessary for density existence.
• The method uses pathwise estimates and asymptotic analysis to derive the behavior of the transition density in the limit.
• Upper and lower bounds for the density are constructed using scaling techniques and properties of the Laplace exponent.
• Sharp estimates are derived when an additional upper scaling condition is assumed, refining the bounds.
• The approach combines probabilistic methods with analytic techniques from regular variation theory.

Experimental results

Research questions

• RQ1 Under what conditions on the Laplace exponent does the transition density of a subordinator exist?
• RQ2 How does the transition density behave asymptotically as the spatial or temporal variable tends to infinity?
• RQ3 What are the tightest possible upper and lower bounds for the transition density under minimal assumptions?
• RQ4 How do additional upper scaling conditions on the Laplace exponent improve the sharpness of density estimates?

Key findings

• The transition density exists for all subordinators whose Laplace exponent satisfies a lower scaling condition at infinity.
• The asymptotic behavior of the density is characterized in terms of the scaling properties of the Laplace exponent.
• Sharp upper and lower bounds for the transition density are established when both lower and upper scaling conditions are satisfied.
• The bounds are shown to be optimal in the sense of matching the exact asymptotic decay rate of the density.
• The results extend known estimates for stable subordinators and apply to a broader class of sample paths with heavy-tailed increments.