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Piecewise Recursive Sequences with Adaptive Thresholds : Boundary Convergence and Applications

Slimane Alaoui Soulimani Valenti

TL;DR

The paper develops a framework for analyzing discrete-time systems whose switching threshold adapts endogenously, by transforming the coupled update into a triangular, fixed-boundary map and deriving one-sided stability criteria from branch Jacobians. A key result is the Common-Limit Lemma: if an orbit switches infinitely often and converges, its limit must lie on the boundary where the state and threshold coincide. The authors apply the theory to an asset-pricing model with a trailing-stop sentiment threshold and extend it numerically to a network of coupled markets to illustrate how adaptive thresholds influence stability, bubbles, and contagion. The work provides a tractable, general toolkit for assessing convergence, synchronization, and stability in nonsmooth, adaptive-threshold dynamics with potential implications for financial systems and other decision-rule frameworks.

Abstract

We study discrete-time dynamical systems that switch between different evolution rules based on thresholds that themselves adapt over time. Specifically, we analyze the coupled recursion $a_{n+1} = f(a_n)$ if $a_n \leq c_n$ and $a_{n+1} = g(a_n)$ if $a_n > c_n$, where the threshold evolves according to $c_{n+1} = h(a_n, c_n)$. By transforming the system into triangular coordinates, we map the problem to a piecewise-smooth system with a fixed switching boundary. We derive explicit local stability criteria based on the lower-triangular structure of the associated Jacobians and establish a ``common-limit constraint'': we prove that any convergent orbit that switches regimes infinitely often must converge to a limit where the state and threshold coincide. To demonstrate the framework's utility, we develop an asset-pricing model where investor sentiment follows a ``trailing-stop'' rule. We characterize the parameter regions for market stability versus bubble formation and numerically extend the analysis to coupled markets, illustrating how adaptive thresholds can facilitate the propagation of instability and contagion across financial networks.

Piecewise Recursive Sequences with Adaptive Thresholds : Boundary Convergence and Applications

TL;DR

The paper develops a framework for analyzing discrete-time systems whose switching threshold adapts endogenously, by transforming the coupled update into a triangular, fixed-boundary map and deriving one-sided stability criteria from branch Jacobians. A key result is the Common-Limit Lemma: if an orbit switches infinitely often and converges, its limit must lie on the boundary where the state and threshold coincide. The authors apply the theory to an asset-pricing model with a trailing-stop sentiment threshold and extend it numerically to a network of coupled markets to illustrate how adaptive thresholds influence stability, bubbles, and contagion. The work provides a tractable, general toolkit for assessing convergence, synchronization, and stability in nonsmooth, adaptive-threshold dynamics with potential implications for financial systems and other decision-rule frameworks.

Abstract

We study discrete-time dynamical systems that switch between different evolution rules based on thresholds that themselves adapt over time. Specifically, we analyze the coupled recursion if and if , where the threshold evolves according to . By transforming the system into triangular coordinates, we map the problem to a piecewise-smooth system with a fixed switching boundary. We derive explicit local stability criteria based on the lower-triangular structure of the associated Jacobians and establish a ``common-limit constraint'': we prove that any convergent orbit that switches regimes infinitely often must converge to a limit where the state and threshold coincide. To demonstrate the framework's utility, we develop an asset-pricing model where investor sentiment follows a ``trailing-stop'' rule. We characterize the parameter regions for market stability versus bubble formation and numerically extend the analysis to coupled markets, illustrating how adaptive thresholds can facilitate the propagation of instability and contagion across financial networks.

Paper Structure

This paper contains 21 sections, 4 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Standing assumptions
  4. Notation
  5. Orbits, switching times, and limit sets
  6. Fixed points and local terminology
  7. Back-translation to the original variables
  8. Fixed Points and Local Stability
  9. Characterization of fixed points
  10. Jacobian structure and hyperbolicity
  11. Local stability criteria (branch-wise)
  12. A Common-Limit Lemma for Infinite Switching
  13. Economic Application: Dynamic Sentiment Thresholds in Financial Markets
  14. Single Market with Trailing-Stop Sentiment Thresholds
  15. Mispricing and Adaptive Threshold
  16. ...and 6 more sections

Key Result

Proposition 3.3

Let $z^\ast\in\operatorname{int}(X^\pm)$ satisfy eq:fp-interior and (A2). Then the eigenvalues of $DT^\pm(z^\ast)$ are In particular:

Theorems & Definitions (9)

  • Definition 3.1: Interior and boundary fixed points
  • Definition 3.2: Hyperbolicity
  • Proposition 3.3: Interior, single-branch
  • Proposition 3.4: Boundary, one-sided
  • Remark 3.5: Back-translation
  • Definition 4.1: Infinite switching
  • Lemma 4.1: Common-limit constraint
  • Corollary 4.2: Common limit in triangular coordinates
  • Remark 4.3: Interpretation and relation to the literature