Fixed point analysis of expectation convergence in financial learning in perturbed metric spaces

Şeyma Bilazeroğlu; Şeyma Bilazeroğlu

doi:10.3934/math.20261013

AIMS Mathematics

2026, Volume 11, Issue 8: 25192-25228. doi: 10.3934/math.20261013

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Research article

Fixed point analysis of expectation convergence in financial learning in perturbed metric spaces

Şeyma Bilazeroğlu ^,

Department of Computer Science, Çankaya University, 06815 Etimesgut, Ankara, Türkiye

Received: 08 June 2026 Revised: 20 July 2026 Accepted: 28 July 2026 Published: 14 August 2026
MSC : 47H10, 54H25, 91B42, 91G80

Financial markets rarely operate under perfect rationality. Price expectations are formed and revised in environments shaped by limited information, behavioral biases, and institutional frictions. In such settings, investors' perceptions of differences between forecasts may deviate from purely rational benchmarks. In this paper, we develop a mathematical framework for expectation updating that captures these perceptual distortions in a perturbed-metric setting. We introduce a new contractive-type learning condition that simultaneously incorporates individual inertia and institutional adjustment frictions. Unlike classical contraction structures, the proposed condition reflects directional interactions and layered perception effects in the updating process. Under a natural stability requirement linking individual and institutional adjustment intensities, we establish the existence and uniqueness of an equilibrium price expectation and prove the convergence of the learning dynamics. Linear and nonlinear specifications illustrate how behavioral and institutional mechanisms jointly determine not only stability but also persistence and the speed of convergence.
- fixed point theory,
- perturbed metric space,
- generalized contraction,
- financial learning dynamics,
- expectation equilibrium
Citation: Şeyma Bilazeroğlu. Fixed point analysis of expectation convergence in financial learning in perturbed metric spaces[J]. AIMS Mathematics, 2026, 11(8): 25192-25228. doi: 10.3934/math.20261013

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Abstract

Financial markets rarely operate under perfect rationality. Price expectations are formed and revised in environments shaped by limited information, behavioral biases, and institutional frictions. In such settings, investors' perceptions of differences between forecasts may deviate from purely rational benchmarks. In this paper, we develop a mathematical framework for expectation updating that captures these perceptual distortions in a perturbed-metric setting. We introduce a new contractive-type learning condition that simultaneously incorporates individual inertia and institutional adjustment frictions. Unlike classical contraction structures, the proposed condition reflects directional interactions and layered perception effects in the updating process. Under a natural stability requirement linking individual and institutional adjustment intensities, we establish the existence and uniqueness of an equilibrium price expectation and prove the convergence of the learning dynamics. Linear and nonlinear specifications illustrate how behavioral and institutional mechanisms jointly determine not only stability but also persistence and the speed of convergence.

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