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When we first try and set up a model for the evolution of S, the value of the underlying stock, I have seen in a lot of textbooks that they model the evolution by the formula $$\frac{dS_t}{S_t}=\mu dt+\sigma dB_t$$ where $\mu$ is the mean average growth of S, $\sigma$ is the volatility of the stock and $dB_t$ is an increment of a Brownian motion.

My question is that if we view $dB\sim N(0,\sqrt{dt})$, what is the underlying event $\omega \in \Omega$ that $dB$ maps to the real line?

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Think of it this way, $\Omega$ elements are the states of the world. This means the set of {macroeconomic data, geopolitic situation, weather, fundamentals of the firms, sentiment of market participants, etc.} and anything else that might have any influence on the stocks prices...

As Ezy explained, what makes probability theory useful is that it allows us to study seemingly random phenomena, and get to interesting conclusions with simply the probability law that seems to govern these phenomena, without having to concern ourselves with the underlying causes that drive them.

Please note that these phenomena are not necessarily really random, but might seem so because of our lack of knowledge of the causes that drive them, or because these causes are too complex for us to be able to get to any conclusion in a reasonable amount of time.

You can take as an example, stocks prices, but also weather forecasts, the waiting time in a bus line, etc.

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In probability theory $\Omega$ is called the « sample space » of possible outcomes. It does not have an actual representation and it does not matter much since the only thing that really matters is the probability measure $P$. More details here:

https://en.m.wikipedia.org/wiki/Probability_space

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