Entropy corrected geometric Brownian motion.

Area of Science:

• Quantitative Finance
• Statistical Modeling
• Stochastic Processes

Background:

• Geometric Brownian Motion (GBM) is a standard model for stochastic processes, especially in finance.
• GBM's reliance on log-normal distribution assumptions limits its accuracy with real-world data exhibiting deviations.
• Capturing complex data structures requires models that relax strict distributional constraints.

Purpose of the Study:

• To introduce entropy corrections into the GBM framework to overcome log-normality limitations.
• To enhance the predictive accuracy of GBM for non-log-normal data distributions.
• To explore applications beyond finance, including synthetic data generation.

Main Methods:

• Developed an entropy-corrected geometric Brownian motion (GBM) model.
• Introduced entropy as a measure to quantify deviations from log-normality.
• Validated the enhanced GBM model using a dice roll experiment and real financial datasets.

Main Results:

• Entropy corrections significantly improve GBM's predictive accuracy for non-log-normal distributions.
• A decrease in data entropy correlates with an increase in deterministic components and improved GBM refinement.
• The enhanced model demonstrates superior performance compared to conventional GBM on complex datasets.

Conclusions:

• Entropy-corrected GBM provides a more robust framework for stochastic modeling, particularly for financial data.
• This approach enhances the ability to model and generate complex synthetic data for machine learning and statistical applications.
• The findings suggest broader applicability of entropy-based corrections in various scientific modeling domains.