Stochastic Extinction with Relaxed Boundedness Conditions

TL;DR

This paper develops a streamlined approach to stochastic extinction in Markov processes, relaxing previous boundedness conditions and providing new criteria applicable in broader settings.

Contribution

It introduces a simplified method that relaxes boundedness assumptions, extending extinction criteria to cases with unbounded quadratic variation.

Findings

Established extinction criteria with relaxed boundedness conditions.

Provided examples illustrating broader applicability of the new criteria.

Improved upon previous results in certain stochastic extinction models.

Abstract

We study stochastic extinction for a class of Markov processes motivated by models in ecology and epidemiology. Extinction is often characterized by a boundedness condition and a condition on boundary Lyapunov exponents (invasion rates). While the latter is typically sharp, the former is often restrictive and can be improved. Building on the ideas initiated in \cite{benaim2018stochastic}, we develop a streamlined approach that relaxes this boundedness condition and yields concise and accessible criteria for extinction. In particular, we establish extinction criteria in two settings: with and without a linearly bounded quadratic variation condition. In the first case, our result is comparable to, and slightly improves upon, the main results in \cite{foldes2024stochastic}. In the second case, where the quadratic variation is not linearly bounded, we obtain new extinction results that fall outside the scope of existing frameworks. Several examples are provided to illustrate the applicability of our results and to highlight situations where previous conditions are not practically verifiable.