# progressively measurable process

A stochastic process^{} ${({X}_{t})}_{t\in {\mathbb{Z}}_{+}}$ is said to be adapted to a filtration^{} (http://planetmath.org/FiltrationOfSigmaAlgebras) $({\mathcal{F}}_{t})$ on the measurable space^{} $(\mathrm{\Omega},\mathcal{F})$ if ${X}_{t}$ is an ${\mathcal{F}}_{t}$-measurable random variable^{} for each $t=0,1,\mathrm{\dots}$. However, for continuous-time processes, where the time $t$ ranges over an arbitrary index set^{} $\mathbb{T}\subseteq \mathbb{R}$, the property of being adapted is too weak to be helpful in many situations. Instead, considering the process as a map

$$X:\mathbb{T}\times \mathrm{\Omega}\to \mathbb{R},(t,\omega )\mapsto {X}_{t}(\omega )$$ |

it is useful to consider the measurability of $X$.

The process $X$ is *progressive* or *progressively measurable* if, for every $s\in \mathbb{T}$, the stopped process ${X}_{t}^{s}\equiv {X}_{\mathrm{min}(s,t)}$ is $\mathcal{B}(\mathbb{T})\otimes {\mathcal{F}}_{s}$-measurable. In particular, every progressively measurable process will be adapted and jointly measurable. In discrete time, when $\mathbb{T}$ is countable^{}, the converse^{} holds and every adapted process is progressive.

A set $S\subseteq \mathbb{T}\times \mathrm{\Omega}$ is said to be progressive if its characteristic function^{} ${1}_{S}$ is progressive. Equivalently,

$$S\cap \left((-\mathrm{\infty},s]\times \mathrm{\Omega}\right)\in \mathcal{B}(\mathbb{T})\otimes {\mathcal{F}}_{s}$$ |

for every $s\in \mathbb{T}$. The progressively measurable sets form a $\sigma $-algebra, and a stochastic process is progressive if and only if it is measurable with respect to this $\sigma $-algebra.

Title | progressively measurable process |
---|---|

Canonical name | ProgressivelyMeasurableProcess |

Date of creation | 2013-03-22 18:37:31 |

Last modified on | 2013-03-22 18:37:31 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Definition |

Classification | msc 60G05 |

Synonym | progressive process |

Related topic | PredictableProcess |

Related topic | OptionalProcess |

Defines | progressive |

Defines | progressively measurable |