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 ${ℝ}_{+}$,

$$P_n=\{0={\tau }_0^n\le {\tau }_1^n\le \mathrm{\cdots }\uparrow \mathrm{\infty }\}$$

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.