# 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

 $[X,Y]_{t}=X_{t}Y_{t}-X_{0}Y_{0}-\int_{0}^{t}X_{s-}\,dY_{s}-\int_{0}^{t}Y_{s-}% \,dX_{s}.$

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

 $P_{n}=\left\{0=\tau^{n}_{0}\leq\tau^{n}_{1}\leq\cdots\uparrow\infty\right\}$

where, $\tau^{n}_{k}$ can, in general, be stopping times. Suppose that the mesh $|P_{n}^{t}|=\sup_{k}(\tau^{n}_{k}\wedge t-\tau^{n}_{k-1}\wedge t)$ tends to zero in probability as $n\rightarrow\infty$, 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

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

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

 $\sum_{s\leq t}(\Delta X_{s})^{2}\leq[X]_{t}<\infty.$

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

 $[X,Y]^{c}_{t}\equiv[X,Y]_{t}-\sum_{s\leq t}\Delta X_{s}\Delta Y_{s}$

is well defined and continuous.

Title quadratic variation of a semimartingale QuadraticVariationOfASemimartingale 2013-03-22 18:41:21 2013-03-22 18:41:21 gel (22282) gel (22282) 4 gel (22282) Theorem msc 60G07 msc 60G48 msc 60H05 QuadraticVariation