We generalize existing limit theory for stochastic integrals driven by semimartingales and with left-continuous integrands. Joint Skorohod convergence is replaced with joint finite dimensional convergence plus an assumption excluding the case when oscillations of the integrand appear immediately before oscillations of the integrator. Integrands may converge in a very weak topology. It is also proved that convergence of integrators implies convergence of stochastic integrals with respect to the same topology.
"Convergence in various topologies for stochastic integrals driven by semimartingales." Ann. Probab. 24 (4) 2141 - 2153, October 1996. https://doi.org/10.1214/aop/1041903222