# stopped process

A stochastic process^{} ${({X}_{t})}_{t\in \mathbb{T}}$ defined on a measurable space^{} $(\mathrm{\Omega},\mathcal{F})$ can be stopped at a random time $\tau :\mathrm{\Omega}\to \mathbb{T}\cup \{\mathrm{\infty}\}$. The resulting stopped process is denoted by ${X}^{\tau}$,

$${X}_{t}^{\tau}\equiv {X}_{\mathrm{min}(t,\tau )}.$$ |

The random time $\tau $ used is typically a stopping time.

If the process ${X}_{t}$ has left limits (http://planetmath.org/CadlagProcess) for every $t\in \mathbb{T}$, then it can alternatively be stopped just before the time $\tau $, resulting in the pre-stopped process

$$ |

Stopping is often used to enforce boundedness or integrability constraints on a process.
For example, if $B$ is a Brownian motion^{} and $\tau $ is the first time at which $|{B}_{\tau}|$ hits some given positive value, then the stopped process ${B}^{\tau}$ will be a continuous^{} and bounded^{} martingale^{}.
It can be shown that many properties of stochastic processes, such as the martingale property, are stable under stopping at any stopping time $\tau $. On the other hand, a pre-stopped martingale need not be a martingale.

For continuous processes, stopping and pre-stopping are equivalent^{} procedures.
If $\tau $ is the first time at which $|{X}_{\tau}|\ge K$, for any given real number $K$, then the pre-stopped process ${X}^{\tau -}$ will be uniformly bounded.
However, for some noncontinuous processes it is not possible to find a stopping time $\tau >0$ making ${X}^{\tau}$ into a uniformly bounded process. For example, this is the case for any Levy process (http://planetmath.org/LevyProcess) with unbounded^{} jump distribution^{}.

Stopping is used to generalize properties of stochastic processes to obtain the related localized property. See, for example, local martingales^{}.

Title | stopped process |
---|---|

Canonical name | StoppedProcess |

Date of creation | 2013-03-22 18:37:38 |

Last modified on | 2013-03-22 18:37:38 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 5 |

Author | gel (22282) |

Entry type | Definition |

Classification | msc 60G40 |

Classification | msc 60G05 |

Synonym | optional stopping |

Defines | pre-stopped process |

Defines | prestopped process |