# semimartingale

Semimartingales are adapted stochastic processes^{} which can be used as integrators in the general theory of stochastic integration. Examples of semimartingales include Brownian motion^{}, all local martingales^{}, finite variation processes and Levy processes^{}.

Given a filtered probability space $(\mathrm{\Omega},\mathcal{F},{({\mathcal{F}}_{t})}_{t\in {\mathbb{R}}_{+}},\mathbb{P})$, we consider real-valued stochastic processes ${X}_{t}$ with time index $t$ ranging over the nonnegative real numbers. Then, semimartingales have historically been defined as follows.

###### Definition.

A semimartingale $X$ is a cadlag adapted process having the decomposition $X\mathrm{=}M\mathrm{+}V$ for a local martingale $M$ and a finite variation process $V$.

More recently, the following alternative definition has also become common. For simple predictable integrands $\xi $, the stochastic integral $\int \xi \mathit{d}X$ is easily defined for any process $X$. The following definition characterizes semimartingales as processes for which this integral is well behaved.

###### Definition.

A semimartingale $X$ is a cadlag adapted process such that

$$\{{\int}_{0}^{t}\xi \mathit{d}X:|\xi |\le 1\mathit{\text{is simple predictable}}\}$$ |

is bounded^{} in probability for each $t\mathrm{\in}{\mathrm{R}}_{\mathrm{+}}$.

Writing $\parallel \xi \parallel $ for the supremum norm of a process $\xi $, this definition characterizes semimartingales as processes for which

$${\int}_{0}^{t}{\xi}^{n}\mathit{d}X\to 0$$ |

in probability as $n\to \mathrm{\infty}$ for each $t>0$, where ${\xi}^{n}$ is any sequence of simple predictable processes satisfying $\parallel {\xi}^{n}\parallel \to 0$. This property is necessary and, as it turns out, sufficient for the development of a theory of stochastic integration for which results such as bounded convergence holds.

The equivalence of these two definitions of semimartingales is stated by the Bichteler-Dellacherie theorem.

A stochastic process ${X}_{t}=({X}_{t}^{1},{X}_{t}^{2},\mathrm{\dots},{X}_{t}^{n})$ taking values in ${\mathbb{R}}^{n}$ is said to be a semimartingale if ${X}_{t}^{k}$ is a semimartingale for each $k=1,2,\mathrm{\dots},n$.

Title | semimartingale |
---|---|

Canonical name | Semimartingale |

Date of creation | 2013-03-22 18:36:38 |

Last modified on | 2013-03-22 18:36:38 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 8 |

Author | gel (22282) |

Entry type | Definition |

Classification | msc 60G07 |

Classification | msc 60G48 |

Classification | msc 60H05 |