A note on stochastic integrals as $L^2$-curves

Abstract

In a work of van Gaans (2005a) stochastic integrals are regarded as $L^2$-curves. In Filipovi\'{c} and Tappe (2008) we have shown the connection to the usual It\^o-integral for c\`adl\`ag-integrands. The goal of this note is to complete this result and to provide the full connection to the It\^o-integral. We also sketch an application to stochastic partial differential equations.