In Jacod and Shiryaev's book "Limit Theorems for Stochastic Processes", the optional quadratic variation [X,Y] for two semimartingales is introduced after having defined the stochastic integral $H \cdot X$ for $H$ a locally bounded predictable process and $X$ a semimartingale. For two semimartingales $X$ and $Y$, they define the optional quadratic variation (or quadratic co-variation) as the process given by $$ [X,Y] := XY - X_0 Y_0 - X_- \cdot Y - Y_- \cdot X, $$ but I do not understand why the integrals $X_- \cdot Y$ and $Y_- \cdot X$ make sense. So the question I have can be stated more generally as:
If $H$ is only left-continuous and adapted, is it also locally bounded?
Thanks a lot in advance!