The Stochastic Evolution of Financial Asset Prices

This paper examines the relationship between dependence and independence alternatives in general stochastic processes and explores the duality between the true (yet unknown) stochastic process and the functional representation that fits the observed data. We demonstrate that the solution depends on its historic realizations, challenging existing theoretical frameworks that assume independence between the solution and the history of the true process. Under orthogonality conditions, we investigate parameter spaces within data-generating processes and establish conditions under which data exhibit mean-reverting, random, cyclical, history-dependent, or explosive behaviors. We validate our theoretical framework through empirical analysis of an extensive dataset comprising daily prices from the S&P500, 10-year US Treasury bonds, the EUR/USD exchange rate, Brent oil, and Bitcoin from 1 January 2002 to 1 February 2024. Our out-of-sample predictions, covering the period from 17 February 2019 to 1 February 2024, demonstrate the model’s exceptional forecasting capability, yielding correct predictions with between 73% and 92% accuracy, significantly outperforming naïve and moving average models, which only achieved 47% to 54% accuracy.