Asymptotic Plateaus for Generalized Abel Equations with Financial Applications

Abstract

We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\prime }(x)=a_n(x)($ of arbitrary degree 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 detailed application to a generalized Merton structural credit-risk model derives an Abel-type equation for the long-maturity state profile of the credit spread and illustrates the economic relevance of the framework.

28 pages, 9 figures