# quadratic variation of a semimartingale

Given any semimartingale $X$, its quadratic variation $[X]$ exists and, for any two semimartingales $X,Y$, their quadratic covariation $[X,Y]$ exists. This is a consequence of the existence of the stochastic integral, and the covariation can be expressed by the integration by parts formula

$${[X,Y]}_{t}={X}_{t}{Y}_{t}-{X}_{0}{Y}_{0}-{\int}_{0}^{t}{X}_{s-}\mathit{d}{Y}_{s}-{\int}_{0}^{t}{Y}_{s-}\mathit{d}{X}_{s}.$$ |

Furthermore, suppose that ${P}_{n}$ is a sequence of partitions^{} (http://planetmath.org/Partition3) of ${\mathbb{R}}_{+}$,

$${P}_{n}=\{0={\tau}_{0}^{n}\le {\tau}_{1}^{n}\le \mathrm{\cdots}\uparrow \mathrm{\infty}\}$$ |

where, ${\tau}_{k}^{n}$ can, in general, be stopping times. Suppose that the mesh $|{P}_{n}^{t}|={sup}_{k}({\tau}_{k}^{n}\wedge t-{\tau}_{k-1}^{n}\wedge t)$ tends to zero in probability as $n\to \mathrm{\infty}$, for each time $t>0$. Then, the approximations ${[X,Y]}^{{P}_{n}}$ to the quadratic covariation converge ucp (http://planetmath.org/UcpConvergence) to $[X,Y]$ and, convergence also holds in the semimartingale topology.

A consequence of ucp convergence is that the jumps of the quadratic variation and covariation satisfy

$$\mathrm{\Delta}[X]={(\mathrm{\Delta}X)}^{2},\mathrm{\Delta}[X,Y]=\mathrm{\Delta}X\mathrm{\Delta}Y$$ |

at all times.
In particular, $[X,Y]$ is continuous^{} whenever $X$ or $Y$ is continuous.
As quadratic variations are increasing processes, this shows that the sum of the squares of the jumps of a semimartingale is finite over any bounded interval

$$ |

Given any two semimartingales $X$,$Y$, the polarization identity^{} $[X,Y]=([X+Y]-[X-Y])/4$ expresses the covariation as a difference of increasing processes and, therefore is of finite variation (http://planetmath.org/FiniteVariationProcess), So, the continuous part of the covariation

$${[X,Y]}_{t}^{c}\equiv {[X,Y]}_{t}-\sum _{s\le t}\mathrm{\Delta}{X}_{s}\mathrm{\Delta}{Y}_{s}$$ |

is well defined and continuous.

Title | quadratic variation of a semimartingale |
---|---|

Canonical name | QuadraticVariationOfASemimartingale |

Date of creation | 2013-03-22 18:41:21 |

Last modified on | 2013-03-22 18:41:21 |

Owner | gel (22282) |

Last modified by | gel (22282) |

Numerical id | 4 |

Author | gel (22282) |

Entry type | Theorem |

Classification | msc 60G07 |

Classification | msc 60G48 |

Classification | msc 60H05 |

Related topic | QuadraticVariation |