  We consider the time-inhomogeneous Markovian jump process introduced by John
S. Bell [Phys.Rep. 137, 49] for a lattice quantum field theory, which runs on
the associated configuration space. Its jump rates, tailored to give the
process the quantum distribution $|\Psi_t|^2$ at all times $t$, typically
exhibit singularities. We establish the existence of a unique such process for
all times, under suitable assumptions on the Hamiltonian or the initial state
vector $\Psi_0$. The proof of non-explosion takes advantage of the special role
of the $|\Psi_t|^2$ distribution.
