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 \begin{equation*} [X,Y]_t=X_tY_t-X_0Y_0-\int_0^tX_{s-}\,dY_s-\int_0^tY_{s-}\,dX_s. \end{equation*}Furthermore, suppose that $P_n$ is a sequence of partitions of $\mathbb{R}_+$ , \begin{equation*} P_n=\left\{0=\tau^n_0\le\tau^n_1\le\cdots\uparrow\infty\right\} \end{equation*}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 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 \begin{equation*} \Delta[X]=(\Delta X)^2,\ \Delta[X,Y]=\Delta X\Delta Y \end{equation*}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 \begin{equation*} \sum_{s\le t}(\Delta X_s)^2 \le [X]_t <\infty. \end{equation*} 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, So, the continuous part of the covariation \begin{equation*} [X,Y]^c_t\equiv [X,Y]_t-\sum_{s\le t}\Delta X_s\Delta Y_s \end{equation*}is well defined and continuous.