I'm reading Chapter 2 of Protter's Stochastic Integration and Differential Equations, Version 2.1. The definition of quadratic variation of a semi-martingale boils down to \begin{equation*} [X,X]_{t} = X_{0}^{2} + \lim_{n \to \infty} \sum_{i = 0}^{2^{n} - 1} (X_{(i + 1) t2^{-n}} - X_{it2^{-n}})^{2}, \end{equation*} where the limit is in probability. The key point is the $X_{0}^{2}$ term. If $X$ and $Y$ are semi-martingales, $[X,Y]$ is defined similarly (or by polarization).
Later, Theorem 29 effectively states that \begin{equation} [H \cdot X, Y]_{t} = \int_{(0,t]} H_{s} d[X,Y]_{s}, \end{equation} provided $H$ is caglad and adapted. (Here $H \cdot X$ is the stochastic integral of $H$ with respect to $X$, and both $X$ and $Y$ are semi-martingales.)
My question is: where has the $H_{0} X_{0} Y_{0}$ term gone? I suspect this is a typo and it should read \begin{equation*} [H\cdot X,Y]_{t} = \int_{(0,t]} H_{s} d[X,Y]_{s} + H_{0} X_{0} Y_{0}. \end{equation*} Otherwise, taking $H \equiv 1$ in Theorem 29, one would obtain \begin{equation*} [X,Y]_{t} = \int_{(0,t]} d[X,Y]_{s} = [X,Y]_{t} - [X,Y]_{0} = [X,Y]_{t} - X_{0} Y_{0}. \end{equation*}
One answer could be "caglad implies zero at $t = 0$" (which is philosophically consistent with the book's convention that $H_{0^{-}} = 0$). But when Protter defines "simple predictable processes," he allows their value at $0$ to be a random variable.
Am I missing something? This is the corrected second edition of the book so I'm willing to believe it.
It's not $H_0X_0Y_0$ that you would expect, but $(H\cdot X)_0Y_0$, which is $0$ as $(H\cdot X)_0=0$.