Why Brownian Motion Does Not Make Asset Prices Predictable from the Past

Summary

The document poses a probability question: if an asset price follows Brownian motion, why can’t its future values be predicted from its past values? It notes that Brownian motion has independent increments, which is the central property needed for the argument. Given observations up to the present, future increments remain random rather than being determined by the observed path.

The document contains only the question and the asker’s uncertainty; it provides no worked proof or discussion of conditional distributions. A complete argument would distinguish unpredictability from the ability to forecast a distribution: past values do not determine the next increment, but the model still specifies its distribution and expected value. The asset-price premise is also a simplified model, since real prices are often represented by transformed or drifted processes rather than standard Brownian motion directly.

Key ideas

• Brownian motion is modeled with independent increments.
• Observing the past path does not determine future increments.
• Unpredictability of a realized value does not prevent forecasting its probability distribution.
• The document raises the question but does not provide a formal solution.

Tags

# Brownian motion for modelling future asset values


# Brownian motion for modelling future asset values












> Assume that an asset price $S$ is given by a Brownian motion. Argue from the definition why it is not possible to predict future values of the asset based on the past values of $S$.

I am not sure exactly what this asks. I know that Brownian motion is a random process with independent increments (at least the way we defined it in our course). I am not sure what else I can add on or how I can formalise my argument more using the definition of Brownian motion.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.