filtered probability space

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

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

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

Given any filtered probability space, it can always be enlarged by passing to the completion of the probability space, adding zero probability sets to $\mathcal{F}_{t}$, and by replacing $\mathcal{F}_{t}$ by $\mathcal{F}_{t+}$. 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 $\mathcal{F}_{t}$ 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.

Title filtered probability space FilteredProbabilitySpace 2013-03-22 18:36:51 2013-03-22 18:36:51 gel (22282) gel (22282) 5 gel (22282) Definition msc 60G05 FiltrationOfSigmaAlgebras stochastic basis usual conditions usual hypotheses