Entropy corrected geometric Brownian motion

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.