From Present Observation to Future Uncertainty: Memory Dynamics in the Geometric Brownian Motion

This paper presents a theoretical and empirical analysis of a new property of stochastic processes - memory - that can be used to predict future values based on past observations. Memory is analyzed with respect to the way the information contained in one observation at time t propagates through the series of GBM-derived forecasts generated for future times t + h. More specifically, this analysis seeks to quantify how much the process carries forward (i.e., retains) the information from the current observation until the time of the forecast. Ultimately, the goal is to define the persistence of predictive information as the uncertainty associated with it increases with time. The analysis is based on the closed-form solution of the Geometric Brownian Motion and provides a mathematical relationship for the change in dispersion of forecasts and the consequent change in reliability of forecasts. The results suggest that, as a result of the properties of a non-mean-reverting diffusion with a variance that increases in direct proportion to time, the influence of the initial observation on subsequent observations quickly decays. We show evidence that the decline in memory occurs according to a logistic decay model and interpret this phenomenon as a law of memory decay. We use historical data on the S&P 500 index for empirical data validation. Once we tune the model’s parameters, we create multi-horizon forecasts and develop associated prediction intervals. We then evaluate forecast accuracy through repeated validation at increasing horizon intervals of time, which allows us to quantitate the effective forecast life of the model. The findings support the theoretical memory pattern identified earlier and demonstrate the inherently short-term memory characteristic of Geometric Brownian Motion.