$Maze of Financial Charts Leading to a Golden Key$

Unlock Financial Optimization: How New Math Breaks Barriers for Better Investments

Samir D’Costa in Business & Economy December 2025 • 4 min read.

"Ditch rigid models! Discover how relaxing differentiability assumptions revolutionizes Taylor Theorems and investment strategies for smarter financial decisions."

In the dynamic world of finance, optimization is key. But traditional mathematical models often hit a wall due to a major problem: they assume the financial world and its functions are 'smooth' and predictable. In reality, markets are volatile. The value functions that drive investment decisions often aren't as well-behaved as old models would have you believe. This has made it hard to create accurate and reliable models for investment.

Think of it this way: older methods rely on the Hamilton-Jacobi-Bellman (HJB) partial differential equation. The problem is, finding a 'smooth' solution to this equation is rare, and only exists for a few well functional forms. This is why 'weak solutions' like viscosity solutions have become popular, but a strong (smooth) solution is always preferable.

Now, researchers are changing the game. By relaxing the strict requirement of 'differentiability' (or smoothness) in the Taylor Theorem, they're making it easier to find practical solutions. This article breaks down the core ideas of this approach and explores how it could revolutionize portfolio management, consumption strategies, and more.

Rethinking Taylor's Theorem: What Does "Relaxing Differentiability" Actually Mean?

Maze of Financial Charts Leading to a Golden Key

At its core, this innovation involves adapting a foundational concept in mathematics: the Taylor Theorem. Traditionally, this theorem requires that the functions being analyzed are 'differentiable'. Differentiability implies that a function has a well-defined derivative at every point – meaning it's smooth and predictable. But financial markets rarely behave so neatly!

The new approach introduces a method to create Taylor-like expansions even when the original function isn't perfectly differentiable. This is incredibly useful across various applications, including regression analysis, optimization problems, integration, and partial differential equations (PDEs).

The Shift Parameter: Imagine a continuous but not perfectly smooth function, f(x). The method involves transforming it using a 'shift parameter,' β, so you're now working with f(x + β). Think of β as a tweak or nudge.
Differentiability Reimagined: The core insight is that even if f(x) isn't differentiable in the traditional sense, f(x + β) can be made differentiable with respect to β. Since every function can be shifted, this adjustment allows differentiability to exist.
A Simple Analogy: Graphically, this is like shifting the function horizontally. Suddenly, you can analyze it with tools that demand differentiability.

Consider the classic Taylor expansion around a point 'c': f(i) = f(c) + f'(c) (i – c) + R Where R is the error term. Now, extend this to two variables, f(x, y). The same shifting trick applies, making it differentiable with respect to a new variable 'h.' This leads to a modified Taylor expansion: f(i, h) = f(c1, c2) + f'i(c1, c2) (i – c1) + f'h(c1, c2) (h – c2) + R2

The Future of Finance: Models That Reflect Reality

By relaxing old assumptions, this approach paves the way for financial models that are better equipped to handle the complexities and uncertainties of real-world markets. This can lead to more effective investment strategies, improved risk management, and a more resilient financial system.

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.1211.5816,

Title: Relaxing The Differentiability Assumption In Taylor Theorem And Optimization: Applications To The Hjb Pde And Finance

Subject: q-fin.pm math.ap

Authors: Moawia Alghalith

Published: 25-11-2012

Everything You Need To Know

What is the core limitation of traditional mathematical models in finance that this new approach addresses?

Traditional mathematical models in finance often assume that financial functions are 'smooth' and predictable. This means they assume functions are differentiable. However, real-world markets are volatile, and value functions are not always well-behaved. The new approach addresses this by relaxing the strict requirement of differentiability, allowing for more accurate models that reflect real market conditions.

How does 'relaxing differentiability' in the Taylor Theorem help to create better financial models?

By relaxing the differentiability requirement in the Taylor Theorem, it becomes easier to find practical solutions for complex financial problems. The traditional Taylor Theorem demands that functions be differentiable, implying smoothness. The innovation introduces a method to create Taylor-like expansions even when the original function isn't perfectly differentiable. This enables the analysis and optimization of functions that were previously difficult to model, leading to more effective investment strategies and risk management.

Can you explain how the 'shift parameter' works in the context of relaxing differentiability?

The 'shift parameter,' denoted as β, is used to transform a continuous but not perfectly smooth function, f(x), into f(x + β). While f(x) may not be differentiable in the traditional sense, f(x + β) can be made differentiable with respect to β. This adjustment allows the application of mathematical tools that require differentiability. Graphically, it's like shifting the function horizontally to make it more amenable to analysis. This manipulation is crucial for creating Taylor-like expansions for non-smooth functions.

What are the potential benefits of using models based on the relaxed differentiability assumptions in real-world financial applications?

Models that relax old assumptions by using techniques to make functions differentiable can lead to more effective investment strategies because these models are better equipped to handle the complexities and uncertainties of real-world markets. This further enables improved risk management practices, a more resilient financial system, and the development of robust portfolio management strategies. The ability to work with non-smooth functions opens up new avenues for optimization and regression analysis in financial modeling.

How does the modified Taylor expansion with two variables, f(i, h), incorporate the concept of relaxed differentiability, and what is its significance?

The modified Taylor expansion for two variables, f(i, h), extends the concept of relaxed differentiability by applying the same shifting trick to make the function differentiable with respect to a new variable 'h.' This leads to an expansion: f(i, h) = f(c1, c2) + f'i(c1, c2) (i – c1) + f'h(c1, c2) (h – c2) + R2. Its significance lies in enabling the analysis of more complex, multi-dimensional financial models where differentiability assumptions may not hold. This approach is particularly useful in scenarios involving multiple factors or variables that influence financial outcomes, leading to more accurate and reliable models.