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

Approximation of stochastic integrals with jumps via weighted BMO approach