Gupta Rishabh, Drzazga-Szczȩśniak Ewa A, Kais Sabre, Szczȩśniak Dominik
Department of Chemistry, Purdue University, West Lafayette, IN, 47907, United States.
Department of Physics, Faculty of Production Engineering and Materials Technology, Częstochowa University of Technology, 19 Armii Krajowej Ave., Czestochowa, 42200, Poland.
Sci Rep. 2024 Nov 17;14(1):28384. doi: 10.1038/s41598-024-79714-3.
The geometric Brownian motion (GBM) is widely used for modeling stochastic processes, particularly in finance. However, its solutions are constrained by the assumption that the underlying distribution of returns follows a log-normal distribution. This assumption limits the predictive power of GBM, especially in capturing the complexities of real-world data, where deviations from log-normality are common. In this work, we introduce entropy corrections to the GBM framework to relax the log-normality constraint and better account for the inherent structures in real data. We demonstrate that as the deterministic components within the data increase, entropy decreases, leading to refinements in GBM's predictive accuracy. Our approach shows significant improvements over conventional GBM in handling distributions that deviate from log-normal behavior, as demonstrated through both a simple dice roll experiment and real-world financial data. Beyond just financial modeling, this research also opens up new avenues for generating synthetic data that better captures real-world complexity, enhancing applications in fields like machine learning and statistical modeling.
几何布朗运动（GBM）被广泛用于对随机过程进行建模，尤其是在金融领域。然而，其解受到收益的基础分布遵循对数正态分布这一假设的限制。这一假设限制了GBM的预测能力，特别是在捕捉现实世界数据的复杂性方面，因为与对数正态性的偏差很常见。在这项工作中，我们对GBM框架引入熵修正，以放宽对数正态性约束，并更好地考虑实际数据中的固有结构。我们证明，随着数据中确定性成分的增加，熵会降低，从而提高GBM的预测准确性。通过一个简单的掷骰子实验和实际金融数据表明，我们的方法在处理偏离对数正态行为的分布方面比传统GBM有显著改进。除了金融建模之外，这项研究还为生成能更好地捕捉现实世界复杂性的合成数据开辟了新途径，增强了在机器学习和统计建模等领域的应用。
