Stochastic Fubini theorem ($\int du\int H(s,u)d{W}_s=\int (\int H(s,u)du)d{W}_s$)

A theorem interchanging a parameter integral with an Itô integral, under appropriate measurability and integrability hypotheses. For a bounded deterministic integrand on a finite parameter-time rectangle, the square-integrability conditions hold; triangular domains can be handled by an indicator of the domain.