Surreal illustration of falling dominoes symbolizing simultaneous events and risk management.

Decoding the Domino Effect: How Understanding 'Simultaneous Stopping Times' Can Help Predict Catastrophes

Reese Marlowe in Science & Nature September 2025 • 4 min read.

"From COVID-19 Spread to Infrastructure Failure, Unveiling the Mathematics That Could Revolutionize Risk Assessment"

In our increasingly interconnected world, predicting when random events will occur is more vital than ever. From modeling customer arrivals at a restaurant to forecasting the spread of a pandemic or the collapse of a bridge, probability models play a crucial role in modern society. At the heart of many of these models lies the concept of 'stopping times' – random moments when specific processes halt. But what happens when multiple events stop simultaneously? A new area of mathematical exploration, 'simultaneous stopping times,' is offering fresh perspectives on complex risk scenarios.

Traditional approaches often assume that stopping times are conditionally independent, meaning they occur without influencing one another. While this assumption simplifies calculations, it doesn't always reflect reality. In many real-world situations, the timing of one event can significantly affect the likelihood of another occurring at the same time. Think about the confluence of factors that might lead to a financial crisis, or the simultaneous failure of multiple components in a complex engineering system.

This article dives into the innovative research that challenges the conventional assumption of independent stopping times. We will explore how modifying existing mathematical frameworks can better capture the dependencies between events, allowing for a more accurate assessment of risk in various domains, from epidemiology and civil engineering to credit risk and beyond. Join us as we unpack the mathematics behind these models and explore their potential to revolutionize how we understand and prepare for the unexpected.

What Are Simultaneous Stopping Times and Why Do They Matter?

Surreal illustration of falling dominoes symbolizing simultaneous events and risk management.

At its core, a 'stopping time' refers to a random moment in time when a specific process or event ceases to continue. In mathematical terms, it's defined with respect to a 'filtration,' which represents the information available up to a certain point in time. The standard approach assumes that multiple stopping times are conditionally independent, given an underlying filtration. This means that knowing when one event stops doesn't tell you anything about when another event will stop.

However, the real world is rarely so simple. Events often influence each other, creating dependencies that traditional models fail to capture. For example, consider a manufacturing plant with multiple machines. If one machine breaks down, it might put extra strain on the others, increasing the likelihood of a simultaneous breakdown. Or, in the context of financial markets, a sudden economic downturn could trigger a cascade of defaults across multiple sectors.

COVID-19 Contagion: Modeling the simultaneous spread of different variants.
Civil Engineering: Assessing the simultaneous failure of multiple structural components in a bridge or building.
Credit Risk: Predicting the simultaneous default of multiple borrowers in a portfolio.

The new research introduces a modified Cox construction, combined with the bivariate exponential distribution developed by Marshall and Olkin, to create a family of stopping times that are not necessarily conditionally independent. This allows for a positive probability that the stopping times will be equal, reflecting the real-world possibility of simultaneous events. The models also explore scenarios with negative dependencies, where the occurrence of one event makes another less likely.

The Future of Risk Prediction: From Theory to Practice

While the research on simultaneous stopping times is still evolving, its potential impact is far-reaching. By moving beyond the limitations of independent models, these new frameworks offer a more realistic and nuanced approach to risk assessment. As our world becomes increasingly complex and interconnected, the ability to accurately predict and prevent simultaneous failures will be crucial for safeguarding our infrastructure, economies, and public health. Continued research and development in this area promise to equip us with the tools we need to navigate an uncertain future and build a more resilient world.

About this Article -

This article was crafted using a human-AI hybrid and collaborative approach. AI assisted our team with initial drafting, research insights, identifying key questions, and image generation. Our human editors guided topic selection, defined the angle, structured the content, ensured factual accuracy and relevance, refined the tone, and conducted thorough editing to deliver helpful, high-quality information.See our About page for more information.

This article is based on research published under:

DOI-LINK: 10.1051/ps/2024001,

Title: Stopping Times Occurring Simultaneously

Subject: math.pr q-fin.mf

Authors: Philip Protter, Alejandra Quintos

Published: 17-11-2021

Everything You Need To Know

What are 'stopping times' in the context of probability models, and why are they important?

In probability models, 'stopping times' refer to random moments when a specific process or event ceases. They are crucial because they allow us to model and predict when events will halt, which is vital in various applications, such as modeling customer arrivals, forecasting pandemics like COVID-19, and predicting structural failures. Traditional approaches often assume these stopping times are conditionally independent, but this doesn't always reflect real-world scenarios where events influence each other. 'Simultaneous stopping times' extend this concept to address scenarios where multiple events stop at the same time, offering a more accurate assessment of risk.

What are 'simultaneous stopping times,' and how do they differ from traditional approaches to modeling stopping times?

'Simultaneous stopping times' represent a mathematical concept that addresses the scenario where multiple events stop at the same time. Traditional approaches often assume that stopping times are conditionally independent, meaning that the timing of one event doesn't influence the timing of another. However, in many real-world situations, events are interconnected. 'Simultaneous stopping times' offer a way to model these dependencies, allowing for a more accurate assessment of risk in domains such as epidemiology, civil engineering, and credit risk. This approach can use constructions such as a modified Cox construction, combined with the bivariate exponential distribution developed by Marshall and Olkin.

How can the concept of 'simultaneous stopping times' be applied to predict and prevent disasters?

The concept of 'simultaneous stopping times' allows for a more realistic and nuanced approach to risk assessment by capturing the dependencies between events. For example, in civil engineering, it can be used to assess the simultaneous failure of multiple structural components in a bridge or building. In finance, it can help predict the simultaneous default of multiple borrowers in a portfolio. In epidemiology, it can model the simultaneous spread of different variants of a disease like COVID-19. By moving beyond the limitations of independent models, 'simultaneous stopping times' equip us with tools to better understand and prepare for unexpected events, safeguarding our infrastructure, economies, and public health.

What is the modified Cox construction mentioned, and what role does the bivariate exponential distribution by Marshall and Olkin play in modeling 'simultaneous stopping times'?

The modified Cox construction, combined with the bivariate exponential distribution developed by Marshall and Olkin, is used to create a family of stopping times that are not necessarily conditionally independent. This is important because it allows for modeling scenarios where the occurrence of one event affects the likelihood of another event happening simultaneously. The bivariate exponential distribution by Marshall and Olkin enables the creation of models that allow for a positive probability that the stopping times will be equal, reflecting the real-world possibility of simultaneous events. These models can also explore scenarios with negative dependencies, where the occurrence of one event makes another less likely.

What are the limitations of assuming conditionally independent stopping times, and what are some real-world examples where this assumption fails?

Assuming conditionally independent stopping times simplifies calculations but often fails to capture the complex dependencies between events in the real world. For example, in a manufacturing plant, the breakdown of one machine can put extra strain on others, increasing the likelihood of a simultaneous breakdown. In financial markets, an economic downturn can trigger a cascade of defaults across multiple sectors. Similarly, in civil engineering, the failure of one component in a bridge can weaken other components, leading to a simultaneous failure. These scenarios highlight the need for models that account for the interdependencies between events, such as those based on 'simultaneous stopping times,' to provide a more accurate assessment of risk.