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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    $\begingroup$ From the distribution of the increments, you can argue that asset prices are i.i.d. which means that their realization does not depend on any of the previous prices, nor will it affect future prices. So, you cannot predict the future. In discrete time, you can see this by comparing the Brownian motion to the random walk $\endgroup$ – Waie Apr 17 at 12:00

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