3
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ thanks for the response. I've asked an unrelated question regarding that note set you linked about joint probabilities I've been confused about lately. I wouldn't mind hearing your thoughts: quant.stackexchange.com/questions/18588/… $\endgroup$ – bcf Jun 30 '15 at 14:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.