A Mathematical Framework for Modeling Financial Resilience Through Regime Persistence: Change-Point Detection and Explainable Machine Learning

Financial markets exhibit complex nonlinear dynamics driven by interactions between firm-specific characteristics and macroeconomic conditions, requiring robust mathematical models capable of capturing structural changes and temporal heterogeneity. This study proposes an explainable machine learning framework that combines regime-switching analysis and predictive modeling to investigate the long-term resilience of firms listed in the BIST 100 Index. Structural breaks in monthly return and volatility series are first detected using the Pruned Exact Linear Time (PELT) algorithm, enabling the classification of firm trajectories into resilient and fragile market regimes. A Regime Persistence Score is then introduced to quantify the proportion of time each firm remains in the resilient state. This score serves as the response variable in a Random Forest model that evaluates the influence of firm-specific financial indicators and macroeconomic variables, including exchange rates, producer price inflation, and commercial loan interest rates. In addition, a forward-looking classification model estimates the probability that firms will transition into a fragile regime within the subsequent six months. The proposed framework achieves an AUC of 0.706 and demonstrates stable predictive performance under alternative regime definitions, penalty parameters, cost functions, and cross-validation strategies. Explainability analysis derived from Shapley Additive Explanations (SHAP) data shows book-to-market ratio and sensitivities to inflation and interest rate changes as the most important components of enduring resilience. The proposed methodology provides an interpretable mathematical framework for regime detection, nonlinear time-series modeling, and decision support in sustainable financial systems, offering practical value for risk assessment and resilience-oriented portfolio management.