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:
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What would each experiment be in this context, and is it then true that $\Omega = \{H, T, \emptyset\}$?
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In this context, how would the accumulated position of the random walk at time $t$ be formulated?
Best Answer
$\Omega$ 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 $X_t(\omega)=\sum_{i=1}^{\lfloor t\rfloor}\omega_i$.