Asymptotic Plateaus for Generalized Abel Equations with Financial Applications

TL;DR

This paper introduces a comprehensive analytical and computational approach for generalized Abel differential equations, proving a new asymptotic plateau theorem and demonstrating its application to financial models.

Contribution

It establishes a novel asymptotic plateau theorem for generalized Abel equations, providing explicit convergence rates and a degree-reduction method, with practical computational implementation.

Findings

Solutions are globally defined, monotone, and converge to a finite limit.

The framework accurately reproduces asymptotic plateaus to nine significant digits.

Application to credit risk models demonstrates economic relevance.

Abstract

We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\prime }(x)=a_n(x)\bigl(% y^n+\lambda_{n-1}(x)y^{n-1}+\dots+\lambda_0(x)\bigr)$ of arbitrary degree $% n\ge1$ on the unbounded interval $[x_0,\infty)$. Under mild structural hypotheses on the coefficients and on the existence of a stable moving equilibrium branch $E(x)$, we prove a new \emph{Asymptotic Plateau Theorem} establishing that the solution issued from $y(x_0)=0$ is globally defined, strictly monotone, trapped between zero and $E(x)$, and converges to a finite positive limit $L=\lim_{x\to\infty}E(x)$. We further obtain an explicit, computable rate of convergence and a degree-reduction principle that generalizes the classical Liouville substitution. The theory is complemented by a high-order Radau IIA implementation whose output reproduces the predicted plateaus to nine significant digits. A complete application to a generalized Merton structural credit-risk model, including the rigorous derivation of an Abel-type credit spread term structure, illustrates the economic relevance of the framework.