  In this paper, we study the local-in-time existence of solutions to
randomized forms of the Gross-Pitaevskii hierarchy on periodic domains. In
particular, we study the independently randomized Gross-Pitaevskii hierarchy
and the dependently randomized Gross-Pitaevskii hierarchy, which were first
introduced in the author's joint work with Staffilani \cite{SoSt}. For these
hierarchies, we construct local-in-time low-regularity solutions in spaces
which contain a random component. The constructed density matrices will solve
the full randomized hierarchies, thus extending the results from \cite{SoSt},
where solutions solving arbitrarily long subhierarchies were given.
  Our analysis will be based on the truncation argument which was first used in
the deterministic setting in the work of T. Chen and Pavlovi\'{c} \cite{CP4}.
The presence of randomization in the problem adds additional difficulties, most
notably to estimating the Duhamel expansions that are crucial in the truncation
argument. These difficulties are overcome by a detailed analysis of the Duhamel
expansions. In the independently randomized case, we need to keep track of
which randomization parameters appear in the Duhamel terms, whereas in the
dependently randomized case, we express the Duhamel terms directly in terms of
the initial data. In both cases, we can obtain stronger results with respect to
the time variable if we assume additional regularity on the initial data.
