# local martingale

Let $(\mathrm{\Xi \copyright},\mathrm{\beta \x84\pm},{({\mathrm{\beta \x84\pm}}_{t})}_{t\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}},\mathrm{\beta \x84\x99})$ be a filtered probability space, where the time index set^{} $\mathrm{\pi \x9d\x95\x8b}\beta \x8a\x86\mathrm{\beta \x84\x9d}$ has minimal element ${t}_{0}$. The most common cases are discrete-time, with $\mathrm{\pi \x9d\x95\x8b}={\mathrm{\beta \x84\u20ac}}_{+}$, and continuous^{} time where $\mathrm{\pi \x9d\x95\x8b}={\mathrm{\beta \x84\x9d}}_{+}$, in which case ${t}_{0}=0$.

A process $X$ is said to be a *local martingale ^{}* if it is locally (http://planetmath.org/LocalPropertiesOfProcesses) a right-continuous martingale

^{}. That is, if there is a sequence of stopping times ${\mathrm{{\rm O}\x84}}_{n}$ almost surely increasing to infinity

^{}and such that the stopped processes ${1}_{\{{\mathrm{{\rm O}\x84}}_{n}>{t}_{0}\}}\beta \x81\u2019{X}^{{\mathrm{{\rm O}\x84}}_{n}}$ are martingales. Equivalently, ${1}_{\{{\mathrm{{\rm O}\x84}}_{n}>{t}_{0}\}}\beta \x81\u2019{X}_{{\mathrm{{\rm O}\x84}}_{n}\beta \x88\S t}$ is integrable and

$${1}_{\{{\mathrm{{\rm O}\x84}}_{n}>{t}_{0}\}}{X}_{{\mathrm{{\rm O}\x84}}_{n}\beta \x88\S s}=\mathrm{\pi \x9d\x94\u038c}[{1}_{\{{\mathrm{{\rm O}\x84}}_{n}>{t}_{0}\}}{X}_{{\mathrm{{\rm O}\x84}}_{n}\beta \x88\S t}\beta \x88\pounds {\mathrm{\beta \x84\pm}}_{s}]$$ |

for all $$. In the discrete-time case where $\mathrm{\pi \x9d\x95\x8b}={\mathrm{\beta \x84\u20ac}}_{+}$ then it can be shown that a local martingale $X$ is a martingale if and only if $$ for every $t\beta \x88\x88{\mathrm{\beta \x84\u20ac}}_{+}$. More generally, in continuous-time where $\mathrm{\pi \x9d\x95\x8b}$ is an interval of the real numbers, then the stronger property that

$$\{{X}_{\mathrm{{\rm O}\x84}}:\mathrm{{\rm O}\x84}\beta \x89\u20act\beta \x81\u2019\text{\Beta is a stopping time}\}$$ |

is uniformly integrable for every $t\beta \x88\x88\mathrm{\pi \x9d\x95\x8b}$ gives a necessary and sufficient condition for a local martingale to be a martingale.

Local martingales form a very important class of processes in the theory of stochastic calculus. This is because the local martingale property is preserved by the stochastic integral^{}, but the martingale property is not.
Examples of local martingales which are not proper martingales are given by solutions to the stochastic differential equation

$$d\beta \x81\u2019X={X}^{\mathrm{\Xi \pm}}\beta \x81\u2019d\beta \x81\u2019W$$ |

where $X$ is a nonnegative process, $W$ is a Brownian motion^{} and $\mathrm{\Xi \pm}>1$ is a fixed real number.

An alternative definition of local martingales which is sometimes used requires ${X}^{{\mathrm{{\rm O}\x84}}_{n}}$ to be a martingale for each $n$. This definition is slightly more restrictive, and is equivalent^{} to the definition given above together with the condition that ${X}_{{t}_{0}}$ must be integrable.

Title | local martingale |
---|---|

Canonical name | LocalMartingale |

Date of creation | 2013-03-22 15:12:43 |

Last modified on | 2013-03-22 15:12:43 |

Owner | skubeedooo (5401) |

Last modified by | skubeedooo (5401) |

Numerical id | 8 |

Author | skubeedooo (5401) |

Entry type | Definition |

Classification | msc 60G07 |

Classification | msc 60G48 |

Related topic | Martingale |

Related topic | LocalPropertiesOfProcesses |