# adapted process

Let $\{{X}_{t}\mid t\in T\}$ be a stochastic process^{} defined on a probability space^{} $(\mathrm{\Omega},\mathcal{F},P)$ and $\{{\mathcal{F}}_{t}\mid t\in T\}$ a filtration^{} (an increasing sequence of sigma subalgebras of $\mathcal{F}$), where $T$ is a linearly ordered subset of $\mathbb{R}$ with a minimum ${t}_{0}$. Then the process $\{{X}_{t}\}$ is said to be *adapted to* the filtration $\{{\mathcal{F}}_{t}\}$ if for each $t\ge {t}_{0}$, ${X}_{t}$ is ${\mathcal{F}}_{t}$-measurable (http://planetmath.org/MathcalFMeasurableFunction):

$${X}_{t}^{-1}(B)\in {\mathcal{F}}_{t}\text{for each Borel set}B\in \mathbb{R}.$$ |

A stochastic process is an *adapted process* if it is adapted to some filtration.

Title | adapted process |
---|---|

Canonical name | AdaptedProcess |

Date of creation | 2013-03-22 16:16:43 |

Last modified on | 2013-03-22 16:16:43 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 19 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 60A99 |

Classification | msc 60G07 |

Synonym | adapted |