$Surreal maze illustrating the challenges of non-smooth obstacle problems.$

Non-Smooth Obstacles? How New Math Solves Real-World Puzzles

Samir D’Costa in Tech & Innovation December 2025 • 3 min read.

"Breakthrough in obstacle problem solutions could reshape economics, finance, and more."

Imagine trying to navigate a maze where the walls shift and change unpredictably. That's the kind of challenge mathematicians face when dealing with 'obstacle problems,' especially when those obstacles aren't smooth and well-behaved. These problems pop up everywhere, from figuring out the best time to sell a stock to understanding how a tumor grows, or the best time to stop in information acquisition models.

Traditional math often struggles with these 'non-smooth' scenarios. Think of a perfectly round ball (smooth) versus a jagged rock (non-smooth). The sharp edges of the rock create complications. But recently, researchers Théo Durandard and Bruno Strulovici have made a significant leap forward. They've developed new mathematical techniques that can handle these tricky non-smooth obstacle problems, opening doors to more realistic and accurate solutions in economics, finance, and other fields.

Their work focuses on 'parabolic obstacle problems,' which involve situations that evolve over time. Unlike previous methods that require everything to be nice and smooth, this new approach can handle obstacles with kinks and irregularities. This is a game-changer because the real world is full of those kinds of complexities.

What Are Obstacle Problems and Why Should You Care?

Surreal maze illustrating the challenges of non-smooth obstacle problems.

At its core, an obstacle problem involves finding the 'smallest' function that satisfies two conditions: it has to stay above a certain obstacle, and it has to behave in a specific way according to a given equation. Imagine a trampoline (the function) and a ball placed underneath (the obstacle). The trampoline has to stay above the ball, but it also has to maintain its shape and tension.

These problems show up in diverse fields:

Physics: Modeling phase transitions (like ice melting into water).
Biology: Studying tumor growth.
Economics: Making learning and investment decisions.
Finance: Pricing American options (contracts that can be exercised at any time before expiration).

In economics and finance, these obstacles often represent constraints or limitations. For example, an investor might face an obstacle in the form of transaction costs or regulatory hurdles. Understanding how to overcome these obstacles is crucial for making optimal decisions.

The Future of Problem-Solving: From Theory to Application

The work of Durandard and Strulovici provides a more robust and realistic framework for tackling complex problems. By relaxing the smoothness requirements, their methods can be applied to a wider range of scenarios, leading to more accurate predictions and better decision-making tools. This could revolutionize how we approach problems in economics, finance, and beyond. This not only helps us understand the world better, but also gives us the tools to navigate it more effectively.

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: https://doi.org/10.48550/arXiv.2404.01498,

Title: Existence, Uniqueness, And Regularity Of Solutions To Nonlinear And Non-Smooth Parabolic Obstacle Problems

Subject: math.ap econ.th math.oc

Authors: Durandard Théo, Strulovici Bruno

Published: 01-04-2024

Everything You Need To Know

What is a 'non-smooth obstacle' in the context of the new research, and why is it significant?

In the context, a 'non-smooth obstacle' refers to a complex element within mathematical problems that possess irregularities, such as sharp edges or unpredictable changes. Traditional methods struggle with these because they often assume smoothness. Théo Durandard and Bruno Strulovici's new techniques can handle these 'non-smooth obstacles,' enabling more realistic solutions in fields where real-world complexities are common, such as economics and finance. The significance lies in the ability to model and solve problems more accurately by accommodating the nuances of reality.

How do 'parabolic obstacle problems' relate to the advancements made by Durandard and Strulovici?

Durandard and Strulovici's work specifically focuses on 'parabolic obstacle problems'. These problems involve situations that evolve over time. Their innovative approach allows for the handling of obstacles with kinks and irregularities, unlike prior methods that relied on smoothness. This advancement is crucial for applying these methods to a wider array of scenarios, leading to more precise predictions and improved decision-making tools within areas like economics and finance. This new method allows a better reflection of the changing and unpredictable nature of real-world challenges.

Can you explain the core concept of an 'obstacle problem' and provide examples of where it applies?

At its core, an 'obstacle problem' seeks to find the 'smallest' function, meeting two conditions: it must stay above a predefined obstacle and behave according to a specific equation. This is like a trampoline staying above a ball, maintaining its shape. These problems arise in various fields: in physics, modeling phase transitions; in biology, studying tumor growth; in economics, making learning and investment decisions; and in finance, pricing American options. The common thread is the presence of constraints or limitations that the solution must navigate, making the understanding of these problems critical for optimal decision-making.

What are the practical implications of these new mathematical techniques in economics and finance?

In economics and finance, the new mathematical techniques, developed by Théo Durandard and Bruno Strulovici, have the potential to reshape how complex problems are approached. These techniques provide a more robust and realistic framework for tackling problems by relaxing smoothness requirements. This can lead to more accurate predictions and the development of improved decision-making tools. In these fields, obstacles can represent things like transaction costs or regulatory hurdles, and the ability to understand and overcome these obstacles is crucial for optimizing investment strategies, financial modeling, and economic planning.

What's the key innovation in the new research by Durandard and Strulovici, and why does it matter for real-world problem-solving?

The key innovation is the development of new mathematical techniques capable of handling 'non-smooth obstacles' within 'parabolic obstacle problems'. This approach allows for a broader application of these methods to real-world scenarios, which often involve complexities and irregularities. The significance of this research extends to economics, finance, and other fields, resulting in more precise predictions and superior decision-making capabilities. These enhancements are essential in understanding and navigating a world filled with dynamic and unpredictable elements.