(untitled) (Contemporary Mathematics, 2025)
This paper proposes a unified mathematical framework for the formalization and forecasting of Support and Resistance (S/R) zones in financial time series. Empirical evaluation shows that our method improves Precision and Recall by 10-16 percentage points compared to classical extremum-based approaches. In contrast to traditional heuristic approaches based on local extrema, the method relies on a volume-weighted potential function, stochastic differential equations, and absorbing Markov processes to rigorously describe zone persistence and breakout probabilities. This formulation ensures reproducibility, theoretical grounding, and interpretability, bridging the gap between technical analysis and stochastic modeling. To address irregularly sampled high-frequency data, we employ cubic spline interpolation and continuous-time stochastic models, while incorporating microstructural features such as Volume-Weighted Average Price (VWAP), bid-ask spreads, and realized volatility. Empirical evaluation on a multi-asset high-frequency dataset (1-second and 1-minute grids) demonstrates consistent improvements over classical extremum-based methods: Precision and Recall increase by 10-16 percentage points, while false breakout rates decline by 12-15%. High-volume S/R zones exhibit significantly longer lifetimes, confirming the central role of liquidity clustering in price dynamics. Beyond improving detection accuracy, the framework also generates theoretically grounded labels that enhance machine learning models by reducing overfitting and increasing predictive interpretability. The results establish S/R zones as dynamic, volume-dependent structures rather than static heuristic levels, providing a reproducible foundation for quantitative finance. The proposed approach advances both the theoretical understanding of market boundaries and their practical application in forecasting, algorithmic trading, and risk management. Future research may extend the framework toward reinforcement learning architectures and cross-asset generalization, further expanding its relevance for modern financial markets.