quadratic variation of a semimartingale
Given any semimartingale , its quadratic variation exists and, for any two semimartingales , their quadratic covariation exists. This is a consequence of the existence of the stochastic integral, and the covariation can be expressed by the integration by parts formula
where, can, in general, be stopping times. Suppose that the mesh tends to zero in probability as , for each time . Then, the approximations to the quadratic covariation converge ucp (http://planetmath.org/UcpConvergence) to 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, is continuous whenever or 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 ,, the polarization identity 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|
|Date of creation||2013-03-22 18:41:21|
|Last modified on||2013-03-22 18:41:21|
|Last modified by||gel (22282)|