In a first course on probability one typically works with a sequence of random variables X1, X2, … For stochastic processes, instead of indexing the random variables by the positive integers, we index them by t ∈ [0, ∞) and we think of Xt as being the value at time t. The random variable could be the location of a particle on the real line, the strength of a signal, the price of a stock, and many other possibilities as well.

We will also work with increasing families of σ-fields {ℱt}, known as filtrations. The σ-field ℱt is supposed to represent what we know up to time t.

Processes and σ-fields

Let (Ω, ℱ, ℙ) be a probability space. A real-valued stochastic process (or simply a process) is a map X from [0,∞) × Ω to the reals. We write Xt = Xt(ω) = X (t, ω). We will impose stronger measurability conditions shortly, but for now we require that the random variables Xt be measurable with respect to ℱ for each t ≥ 0.

A collection of σ-fields ℱt such that ℱt ⊂ ℱ for each t and ℱs ⊂ ℱt if s ≤ t is called a filtration. Define ℱt+ = ∩∈ > 0ℱt+∈. A filtration is right continuous if ℱt+ = ℱt for all t ≥ 0. The σ-field ℱt+ is supposed to represent what one knows if one looks ahead an infinitesimal amount. Most of the filtrations we will come across will be right continuous, but see Exercise 1.1.

A null set N is one that has outer probability 0.