Paper: arXiv 2403.06253

Abstract

The geometric Brownian motion (GBM) is widely employed for modeling stochastic processes, yet its solutions are characterized by the log-normal distribution. This comprises predictive capabilities of GBM mainly in terms of forecasting applications. Here, entropy corrections to GBM are proposed to go beyond log-normality restrictions and better account for intricacies of real systems. It is shown that GBM solutions can be effectively refined by arguing that entropy is reduced when deterministic content of considered data increases. Notable improvements over conventional GBM are observed for several cases of non-log-normal distributions, ranging from a dice roll experiment to real world data.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a method that requires deriving entropy metrics and simulating Monte Carlo trajectories, indicating significant mathematical density. However, the empirical validation relies on conceptual examples (dice rolls) and qualitative visualizations rather than statistical metrics, backtesting, or robust datasets, limiting its backtest-readiness.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Model stochastic processes<br>beyond log-normality"] --> B["Methodology:<br>Entropy Correction to GBM"]
  B --> C["Inputs:<br>Dice Rolls &<br>Equities Data"]
  C --> D["Process:<br>Refine GBM by reducing<br>entropy with increased data"]
  D --> E["Outcome:<br>Effective predictions for<br>non-log-normal distributions"]

Cited References (10)

  1. Ergodic observables in non-ergodic systems: The example of the harmonic chain (2023) — link
  2. Entropy of Financial Time Series Due to the Shock of War — → 2303.16155
  3. Exact distributions of the maximum and range of random diffusivity processes (2020) — link
  4. Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing (2020) — link
  5. On The Validity of The Geometric Brownian Motion Assumption (2005) — link
  6. A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options (1993) — link
  7. The Pricing of Options and Corporate Liabilities (1973) — link
  8. Multiple Wiener Integral (1951) — link
and 2 more
  1. On Information and Sufficiency (1951) — link
  2. A Mathematical Theory of Communication (2006)

1 of 10 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.