stochastic process

A stochastic process is a collection of random variables $X_t$ defined on a common probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Equivalently, it is a joint function $X(t, \omega)$ of an index $t$ and a possible world $\omega \in \Omega$.

If the 'index set' $t\in I$ admits a total order, then we can break down the probability space as a filtration $(\mathcal{F}_t)_{t\in I}$, where a sequence of random variables (such as $X_t$) is adapted to the filtration if $X_t$ is a measurable function under $\mathcal{F}_t$. Informally, this means that we can compute probabilities of outcomes $X_t$ using only the information available from $\mathcal{F}_t$.