# optional process

Suppose we are given a filtration^{} (http://planetmath.org/FiltrationOfSigmaAlgebras) ${(\mathrm{\beta \x84\pm})}_{t\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}}$ on a measurable space^{} $(\mathrm{\Xi \copyright},\mathrm{\beta \x84\pm})$. A stochastic process^{} is said to be adapted if ${X}_{t}$ is ${\mathrm{\beta \x84\pm}}_{t}$-measurable for every time $t$ in the index set^{} $\mathrm{\pi \x9d\x95\x8b}$. For an arbitrary, uncountable, index set $\mathrm{\pi \x9d\x95\x8b}\beta \x8a\x86\mathrm{\beta \x84\x9d}$, this property is too restrictive to be useful. Instead, we can impose measurability conditions on $X$ considered as a map from $\mathrm{\pi \x9d\x95\x8b}\Gamma \x97\mathrm{\Xi \copyright}$ to $\mathrm{\beta \x84\x9d}$.
For instance, we could require $X$ to be progressively measurable, but that is still too weak a condition for many purposes. A stronger condition is for $X$ to be *optional*. The index set $\mathrm{\pi \x9d\x95\x8b}$ is assumed to be a closed subset of $\mathrm{\beta \x84\x9d}$ in the following definition.

The class of optional processes forms the smallest set containing all adapted and right-continuous processes, and which is closed under taking limits of sequences of processes.

The $\mathrm{{\rm O}\x83}$-algebra, $\mathrm{\pi \x9d\x92\u037a}$, on $\mathrm{\pi \x9d\x95\x8b}\Gamma \x97\mathrm{\Xi \copyright}$ generated by the right-continuous and adapted processes is called the *optional* $\mathrm{{\rm O}\x83}$-algebra. Then, a process is optional if and only if it is $\mathrm{\pi \x9d\x92\u037a}$-measurable.

Alternatively, the optional $\mathrm{{\rm O}\x83}$-algebra may be defined as

$$\mathrm{\pi \x9d\x92\u037a}=\mathrm{{\rm O}\x83}\beta \x81\u2019\left(\{[T,\mathrm{\beta \x88\x9e}):T\beta \x81\u2019\text{\Beta is a stopping time}\}\right).$$ |

Here, $[T,\mathrm{\beta \x88\x9e})$ is a stochastic interval, consisting of the pairs $(t,\mathrm{{\rm O}\x89})\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}\Gamma \x97\mathrm{\Xi \copyright}$ such that $T\beta \x81\u2019(\mathrm{{\rm O}\x89})\beta \x89\u20act$. In continuous-time, the equivalence of these two definitions for $\mathrm{\pi \x9d\x92\u037a}$ does require mild conditions on the filtration β it is enough for ${\mathrm{\beta \x84\pm}}_{t}$ to be universally complete.

In the discrete-time case where the index set $\mathrm{\pi \x9d\x95\x8b}$ countable^{}, then the definitions above imply that a process ${X}_{t}$ is optional if and only if it is adapted.

Title | optional process |
---|---|

Canonical name | OptionalProcess |

Date of creation | 2013-03-22 18:37:34 |

Last modified on | 2013-03-22 18:37:34 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 5 |

Author | gel (22282) |

Entry type | Definition |

Classification | msc 60G07 |

Related topic | ProgressivelyMeasurableProcess |

Related topic | PredictableProcess |

Defines | optional |