Canonical Sample Spaces for Continuous Stochastic Processes
Summary
The document explains how to think about the sample space behind a continuous-time market model. It distinguishes an abstract probability space, where outcomes are represented by , from a canonical space of possible process paths, where a random process maps each outcome to a continuous function.
For continuous processes without jumps, it identifies Wiener space—the space of continuous paths starting at zero—as a useful canonical choice. The example of independent geometric Brownian motions for multiple stocks motivates the question, but the response does not develop the construction of the probability measure, filtration, or joint law. It points readers toward further measure-theoretic references, so it is an orientation rather than a full rigorous treatment.
Key ideas
- A probability space is an abstract foundation from which random variables are defined as measurable mappings.
- A stochastic process can be viewed as a mapping from an outcome to an entire path.
- Wiener space provides a canonical path space for continuous processes without jumps.
- The choice of canonical space does not by itself specify the probability law or dependence structure of market processes.
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# Underlying Sample Space in Continuous Market Model
# Underlying Sample Space in Continuous Market Model
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 by Richi Wa (score 3, accepted)
https://quant.stackexchange.com/a/18579
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.