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Stability of traveling wave solutions in a credit rating migration Free Boundary Problem

Claude-Michel Brauner, Yuchao Dong, Jin Liang, Luca Lorenzi

TL;DR

This work analyzes the stability of traveling-wave solutions arising from a credit-rating migration free-boundary problem. The authors transform the free-boundary system into a fully nonlinear parabolic problem on a fixed domain, then establish nonlinear stability by spectral analysis in exponentially weighted spaces and a careful dispersion-relation study, showing absence of unstable spectrum under financially admissible parameters. They prove that the traveling wave profile $K$ is nonlinearly stable and that the discounted bond value $\phi(t,x)$ converges to the attenuated traveling wave $e^{-rt}K(x+ct)$ with explicit exponential rates, thereby validating the pricing framework under credit risk. The results provide a robust mathematical justification of the model (LWH16) and demonstrate that the wave-based pricing rule remains stable to perturbations in model coefficients. The methodological contributions—analytic semigroup theory in weighted spaces, a detailed dispersion analysis, and a nonlinear Stefan-type condition—may inform stability analyses for other free-boundary problems in finance and physics.

Abstract

In this paper, we study the stability of traveling wave solutions arising from a credit rating migration problem with a free boundary, After some transformations, we turn the Free Boundary Problem into a fully nonlinear parabolic problem on a fixed domain and establish a rigorous stability analysis of the equilibrium in an exponentially weighted function space. It implies the convergence of the discounted value of bonds that stands as an attenuated traveling wave solution.

Stability of traveling wave solutions in a credit rating migration Free Boundary Problem

TL;DR

This work analyzes the stability of traveling-wave solutions arising from a credit-rating migration free-boundary problem. The authors transform the free-boundary system into a fully nonlinear parabolic problem on a fixed domain, then establish nonlinear stability by spectral analysis in exponentially weighted spaces and a careful dispersion-relation study, showing absence of unstable spectrum under financially admissible parameters. They prove that the traveling wave profile is nonlinearly stable and that the discounted bond value converges to the attenuated traveling wave with explicit exponential rates, thereby validating the pricing framework under credit risk. The results provide a robust mathematical justification of the model (LWH16) and demonstrate that the wave-based pricing rule remains stable to perturbations in model coefficients. The methodological contributions—analytic semigroup theory in weighted spaces, a detailed dispersion analysis, and a nonlinear Stefan-type condition—may inform stability analyses for other free-boundary problems in finance and physics.

Abstract

In this paper, we study the stability of traveling wave solutions arising from a credit rating migration problem with a free boundary, After some transformations, we turn the Free Boundary Problem into a fully nonlinear parabolic problem on a fixed domain and establish a rigorous stability analysis of the equilibrium in an exponentially weighted function space. It implies the convergence of the discounted value of bonds that stands as an attenuated traveling wave solution.
Paper Structure (13 sections, 9 theorems, 217 equations)

This paper contains 13 sections, 9 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Financial motivation
  3. The model
  4. Notation.
  5. The linearized model
  6. The system with fixed interface
  7. Ansatz
  8. The fully nonlinear problem
  9. Analysis of the linear problem
  10. The generation result
  11. A thorough analysis of the dispersion relation
  12. Nonlinear stability analysis
  13. Convergence to the attenuated traveling wave

Key Result

Lemma 3.3

The space ${\mathscr X}_2$ can be equivalently characterized as the set of all pairs $\bm{w}=(w_1,w_2)$ such that the functions $\dfrac{w_1}{q_L}$ and $\dfrac{w_2}{q_H}$ belong to $C^2_b((-\infty,\eta^*];{\mathbb{C}})$ and $C^2_b([\eta^*,+\infty);{\mathbb{C}})$, respectively. Moreover, the norm of $

Theorems & Definitions (26)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 16 more