# Clarification regarding the outcome space of a stochastic process.

In his book Stochastic Differential Equations - An Introduction with Applications, Øksendal gives the following definition of a stochastic process:

A stochastic process is a parametrized colletcion of random variables $$\{ X_t\}_{t\in T}$$ defined on a probability space $$(\Omega, \mathcal{F}, P)$$ and assuming values in $$\mathbb{R}^n$$.

He then notes that it may be useful to think of $$t$$ as time and each $$\omega \in \Omega$$ as an individual experiment, such that $$X_t(\omega)$$ would represent the result at time $$t$$ of the experiment $$\omega$$. He also notes that a path of a stochastic process is obtained by the mapping $$t \mapsto X_t(\omega)$$ for a fixed $$\omega \in \Omega$$.

This seems to indicate that the outcome space $$\Omega$$ does not vary with time, and that the set of possible outcomes for each experiment, parametrized by $$t$$, is not dependent on $$t$$. However, it is not clear to me how this view would represent such experiments in this context. Take for instance the example of a random walk. At each time $$t \in \mathbb{N}^+$$ a coin is flipped. If the outcome is $$H$$, a step is taken vertically upwards, if $$T$$ a step downwards.

If each $$X_t$$ would represent the step taken at time $$t$$, would not the outcome of the experiment (the coin toss at time $$t$$) be $$\omega \in \{ H, T, \emptyset \} = \Omega$$? But then, fixing $$\omega' \in \Omega$$, for each time $$t$$ the variable $$X_t(t)$$ would have the same outcome, so this cannot be the correct interpretation.

The question becomes:

1. What would each experiment be in this context, and is it then true that $$\Omega = \{H, T, \emptyset\}$$?

2. In this context, how would the accumulated position of the random walk at time $$t$$ be formulated?

$$\Omega$$ often takes the form of the so-called canonical space: $$\Omega=(\mathbb{R}^n)^T,$$ rather than the $$\mathbb{R}^n$$ you seem to propose. In this case, we can take a slightly different approach. A natural choice would be $$\Omega=\{1,-1\}^{\mathbb{N}^+}$$. You might then define $$X_t(\omega)=\sum_{i=1}^{\lfloor t\rfloor}\omega_i$$.
• Thanks, this was clarifying. I think what confuses me still is how one views this. I like to think each coin flip is done independently and without information about the prior and subsequent flips. But in this view, with $\Omega = \{-1, 1\}^{\mathbb{N}^+}$, we already "decided" to flip the coin countably many times. However, if we only wanted to flip a coin once, should we still take the same $\Omega$, and evaluate all the possible outcomes in $\mathcal{F}_1 = \{-1, 1, \emptyset\}$? Jul 7, 2020 at 13:26