Stochastic Process

Topics: Stochastic Process

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Empirically, we have a stochastic process whenever we’re studying the behaviour of a random variable across time. We’ll be working with stochastic processes when trying to model the future behaviour of a given process.

(Almost Formal) Definition §

(definition)

A stochastic process is a sequence of random variables $\left\{X_n\mid n\in \mathbb{N}\right\}$, where the sub-index indicates the corresponding time or space.

This idea can be generalised so that the time instants are continuous; that is, $\left\{X_t\mid t\in \mathbb{R}\right\}$.

(Formal) Definition §

(definition)

Given the probability space $(\Omega ,\alpha ,\mathbb{P})$ such that for every fixed $t\in T\subset \mathbb{R}$, we have:

$$X_t:(\Omega ,\alpha ,\mathbb{P})\to X_t(\omega )\in \mathbb{R}$$

That is, $X_t$ is a random variable and for every fixed $\omega \in \Omega$, $X_t(\omega )$ is a time function.

Generally, we’ll think of the sub-index $t$ as the time indicator and of $X_t$ as the state/position of the stochastic process at the instant $t$.