# What are the underlying events that the random variables map to the real line in the derivation of the Black-Scholes PDE?

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?

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...
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: