Asymptotic Plateaus for Generalized Abel Equations with Financial Applications
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911693650526208 |
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| author | Covei, Dragos-Patru |
| author_facet | Covei, Dragos-Patru |
| contents | We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\prime }(x)=a_n(x)\bigl(% y^n+λ_{n-1}(x)y^{n-1}+\dots+λ_0(x)\bigr)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02831 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Plateaus for Generalized Abel Equations with Financial Applications Covei, Dragos-Patru Numerical Analysis Analysis of PDEs We develop a unified analytical and computational framework for the generalized Abel ordinary differential equation $y^{\prime }(x)=a_n(x)\bigl(% y^n+λ_{n-1}(x)y^{n-1}+\dots+λ_0(x)\bigr)$ 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. |
| title | Asymptotic Plateaus for Generalized Abel Equations with Financial Applications |
| topic | Numerical Analysis Analysis of PDEs |
| url | https://arxiv.org/abs/2605.02831 |