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


Furthermore, suppose that Pn is a sequence of partitionsPlanetmathPlanetmath (http://planetmath.org/Partition3) of +,


where, τkn can, in general, be stopping times. Suppose that the mesh |Pnt|=supk(τknt-τk-1nt) tends to zero in probability as n, for each time t>0. Then, the approximations [X,Y]Pn 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


at all times. In particular, [X,Y] is continuousMathworldPlanetmath 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 identityPlanetmathPlanetmath [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


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