Underlying Sample Space in Continuous Market Model

Question

E.g., a model for $N$ stocks might have each follow a GBM $dS_i = \mu_i S_i dt + \sigma_i S_i dW_i$, where each $W_i$ is independent of the others. Letting $(\Omega, \mathcal{F}, P)$ be the underlying probability space, what should I be thinking of for $\Omega$?

Perhaps it's easier with just one stochastic process? Some candidate spaces I've heard of are $\Omega = \{$infinite sequences of coin tosses$\}$ and $\Omega = \{$continuous functions on $[0,T]$ starting at $0\}$, but I can't really get a good handle on these. Is there a a good explanation for these, or a better example of the possible underlying space? I would prefer a rigorous (measure-theoretic) explanation, if possible.

Answer 1

I would say the following:

• the tripple $(\Omega,\mathcal{F},P)$ is an abstract probability space with all the properties that I assume that you know.
• then we can define random variables as mappings from this probability space to the real numbers $$ X: \omega \mapsto X(\omega) \in \mathbb{R}. $$ But we want to study processes $$ (X_t)_{t \ge 0}: \omega \mapsto (X_t(\omega))_{t \ge 0} \in W, $$ where the canonical space, $W$, for these continuous (no jumps) stochastic processes is Wiener space - the space of continuous functions on the real (half-) line.

If you search the internet for these keywords (Wiener space, stochastic process) then you find the mathematical details. You can start e.g. here.