filtered probability space

A filtered probability space, or stochastic basis, $(\mathrm{\Omega },ℱ,{({ℱ}_t)}_{t\in T},\mathbb{P})$ consists of a probability space $(\mathrm{\Omega },ℱ,\mathbb{P})$ and a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) ${({ℱ}_t)}_{t\in T}$ contained in $ℱ$. Here, $T$ is the time index set, and is an ordered set — usually a subset of the real numbers — such that ${ℱ}_s\subseteq {ℱ}_t$ for all in $T$.

Filtered probability spaces form the setting for defining and studying stochastic processes. A process $X_t$ with time index $t$ ranging over $T$ is said to be adapted if $X_t$ is an ${ℱ}_t$-measurable random variable for every $t$.

When the index set $T$ is an interval (http://planetmath.org/Interval) of the real numbers (i.e., continuous-time), it is often convenient to impose further conditions. In this case, the filtered probability space is said to satisfy the usual conditions or usual hypotheses if the following conditions are met.

• The probability space $(\mathrm{\Omega },ℱ,\mathbb{P})$ is complete (http://planetmath.org/CompleteMeasure).

• The $\sigma$-algebras ${ℱ}_t$ contain all the sets in $ℱ$ of zero probability.

• The filtration ${ℱ}_t$ is right-continuous. That is, for every non-maximal $t\in T$, the $\sigma$-algebra ${ℱ}_{t+}\equiv {\bigcap }_{s>t}{ℱ}_s$ is equal to ${ℱ}_t$.

Given any filtered probability space, it can always be enlarged by passing to the completion of the probability space, adding zero probability sets to ${ℱ}_t$, and by replacing ${ℱ}_t$ by ${ℱ}_{t+}$. This will then satisfy the usual conditions.In fact, for many types of processes defined on a complete probability space, their natural filtration will already be right-continuous and the usual conditions met.However, the process of completing the probability space depends on the specific probability measure $\mathbb{P}$ and in many situations, such as the study of Markov processes, it is necessary to study many different measures on the same space. A much weaker condition which can be used is that the $\sigma$-algebras ${ℱ}_t$ are universally complete, which is still strong enough to apply much of the ‘heavy machinery’ of stochastic processes, such as the Doob-Meyer decomposition, section theorems, etc.