The Sample-Space Meaning of Omega in Stochastic Differential Equations

Summary

The note clarifies the role of the sample point, often written as omega, in a stochastic differential equation. It describes a probability space through its sample space, event sigma algebra, and probability measure. In the Brownian-motion setting, sample outcomes can be represented as continuous paths, and the probability measure is chosen so that evaluating a path at time t yields Brownian motion.

This interpretation means omega selects a possible outcome or path in the underlying probability space. It does not make the displayed equation an ordinary deterministic equation merely by fixing omega, since the stochastic integral and solution are defined within the probabilistic framework. The example that differentiable paths have probability zero illustrates how the measure assigns likelihood to path properties. The discussion is conceptual and does not provide a procedure for solving or calibrating an SDE.

Key ideas

• An SDE is defined relative to a probability space consisting of outcomes, measurable events, and a probability measure.
• Omega commonly denotes the sample space or an individual outcome in that space.
• For Brownian-driven models, outcomes can be viewed as continuous paths and the measure determines their probabilistic behavior.
• Brownian paths are almost surely nondifferentiable, illustrating that path properties depend on the probability measure.

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# Meaning of w in SDE


# Meaning of w in SDE












I'm missing meaning of $w$ in typical SDE like $dX_t(w) = f_t(X_t(w)) + \sigma(X_t(w))dW_t$, in context of $w \in F_{xxx}$. Does it mean that both

- $w$ is one of events that could happen before moment $xxx$

- If we fix $w$ then this SDE becomes deterministic

?

## Answer by Sergio Almada (score 3)

https://quant.stackexchange.com/a/14582

In general SDE's are defined on a probability space which consists of a triplet $(\Omega, P, B)$: the space $\Omega$, a probability measure $P$, and a sigma algebra $B$. In short, the sigma algebra consist on the set of all events that we can assign probability to.

For SDE's driven by Brownian Motion this probability space is the so called Wiener space, which consist of $\Omega = C([0,T])$, the space of continuous functions equipped with the uniform topology. The sigma algebra is the one generated by the open sets under this topology, and $P$ is the so called Weiner measure, which essentially says that the projection $$\pi( \omega ) = \omega(t), \quad \omega \in \Omega,$$ is a Brownian Motion. So for example, under this measure, $$ P \{ \omega \in \Omega: \omega \text{ is differentiable } \} = 0. $$

This is the $\omega$ sometimes you see written in the math books.

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