Existence, Asymptotic Behavior, and Numerical Analysis of a Generalized Abel Differential Equation with Applications in Financial Modeling
For mathematicians and financial modelers, this work offers the first systematic treatment of generalized Abel ODEs, but the contribution is primarily theoretical and incremental in the context of existing ODE theory.
This paper provides a rigorous analysis of generalized Abel ODEs with arbitrary polynomial nonlinearities, proving existence, uniqueness, and asymptotic plateaus, and validates the theory with high-order numerical methods. The results are applied to financial models such as Merton credit risk and stochastic control.
We present a comprehensive investigation into a generalized class of Abel ordinary differential equations (ODEs), extending the classical cubic form to arbitrary polynomial nonlinearities of degree $n \geq 1$. This work provides a rigorous treatment of the existence and uniqueness of regular solutions on both bounded and unbounded domains. Utilizing a unified barrier-based approach, we derive sharp growth rates and prove the existence of exact asymptotic plateaus, establishing the first systematic treatment of such generalizations in the literature. The regularity of solutions is rigorously justified through the lens of Sobolev spaces and classical ODE theory. To complement our analytical findings, we implement a high-order numerical framework based on Radau IIA implicit Runge--Kutta schemes, providing detailed stability arguments and error analysis. The numerical results demonstrate exceptional consistency with our theoretical predictions, particularly in capturing the asymptotic behavior. Finally, we discuss the implications of our results for real-world models, including Merton-type credit risk analysis and Hamilton--Jacobi--Bellman stochastic control problems, bridging the gap between abstract nonlinear dynamics and applied science.