  This article investigates discrete-time approximations of stochastic
integrals driven by semimartingales with jumps via weighted bounded mean
oscillation (BMO) approach. This approach enables $L_p$-estimates, $p \in (2,
\infty)$, for the approximation error depending on the weight, and it allows a
change of the underlying measure which leaves the error estimates unchanged. To
take advantage of this approach, we propose a new approximation scheme obtained
from a correction for the Riemann approximation based on tracking jumps of the
underlying semimartingale. We also discuss a way to optimize the approximation
rate by adapting the discretization times to the setting. When the small jump
activity of the semimartingale behaves like an $\alpha$-stable process with
$\alpha \in (1, 2)$, our scheme achieves under a regular regime the same
convergence rate for the error as in Rosenbaum and Tankov [\textit{Ann. Appl.
Probab.} \textbf{24} (2014) 1002--1048]. Moreover, our approach extends to the
case $\alpha \in (0, 1]$ and to the $L_p$-setting which are not treated there.
As an application, we apply the methods in the special case where the
semimartingale is an exponential L\'evy process to mean-variance hedging of
European type options.
