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Fundamentals usually do not enter the parameters of GBM. But it depends on the purpose: if you want to price options, then the drift is the risk-less rate and volatility is implied from other traded derivatives. if you want to use GBM for risk management then you usually apply statistical methods for $\sigma^2$.

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It turns out that GBM with constant drift and constant volatility is not really used in real life. It is well known that volatility as well as drift may vary over time. Hence, if you want to use a model with time-varying parameters, you need to come up with a model to define $\mu_t$ and $\sigma_t$. There are classic models that use some mean-reverting ...

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It does not necessarily have to be using historical data (you could use implied volatilities for example), but indeed fundamental analysis is not taken into account in geometric Brownian motions: you just assume returns are normally distributed with some mean and volatility and it does not change in time. So if you want to "incorporate" fundamental analysis ...

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