  We perform a non-perturbative study of the scale-dependent renormalization
factors of a multiplicatively renormalizable basis of $\Delta{B}=2$ parity-odd
four-fermion operators in quenched lattice QCD. Heavy quarks are treated in the
static approximation with various lattice discretizations of the static action.
Light quarks are described by non-perturbatively ${\rm O}(a)$ improved
Wilson-type fermions. The renormalization group running is computed for a
family of Schroedinger functional (SF) schemes through finite volume techniques
in the continuum limit. We compute non-perturbatively the relation between the
renormalization group invariant operators and their counterparts renormalized
in the SF at a low energy scale. Furthermore, we provide non-perturbative
estimates for the matching between the lattice regularized theory and all the
SF schemes considered.
