A theory of Hilbert-space-valued traces and multiple integration is developed for kernels in L(2)([0, 1](p) x Omega). The multiple Ogawa and the multiple Stratonovich integrals for such kernels are introduced and sufficient conditions for their existence are obtained. The derivation of the Hu-Meyer formula connecting the multiple Ogawa and the multiple Stratonovich integrals requires the introduction of traces of random kernels, Such a derivation is obtained under appropriate conditions.