Topics: Stochastic Process

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


A stochastic process is a sequence of random variables , where the sub-index indicates the corresponding time or space.

This idea can be generalised so that the time instants are continuous; that is, .

(Formal) Definition


Given the probability space such that for every fixed , we have:

That is, is a random variable and for every fixed , is a time function.

Generally, we’ll think of the sub-index as the time indicator and of as the state/position of the stochastic process at the instant .