A note on stochastic integrals as L2L2-curves

In a work of van Gaans (2005a) stochastic integrals are regarded as L2L2-curves. In the paper Filipović and Tappe (2008) we have shown the connection to the usual Itô-integral for càdlàg-integrands. The goal of this note is to complete this result and to provide the full connection to the Itô-integral. We also sketch an application to stochastic partial differential equations.