  An enormous number of compact binary systems, spanning from stellar to
supermassive levels, emit substantial gravitational waves during their final
evolutionary stages, thereby creating a stochastic gravitational wave
background (SGWB). We calculate the merger rates of stellar compact binaries
and massive black hole binaries using a semi-analytic galaxy formation model --
Galaxy Assembly with Binary Evolution (GABE) in a unified and self-consistent
approach, followed by an estimation of the multi-band SGWB contributed by those
systems. We find that the amplitudes of the principal peaks of the SGWB energy
density are within one order of magnitude $\Omega_{GW} \sim 10^{-9}- 10^{-8}$.
This SGWB could easily be detected by the Square Kilometre Array (SKA), as well
as planned interferometric detectors, such as the Einstein Telescope (ET) and
the Laser Interferometer Space Antenna (LISA). The energy density of this
background varies as $\Omega_{GW} \propto f^{2/3}$ in SKA band. The shape of
the SGWB spectrum in the frequency range $\sim[10^{-4}$,$1]$Hz could allow the
LISA to distinguish the black hole seed models. The amplitude of the SGWB from
merging stellar binary black holes (BBHs) at $\sim 100$ Hz is approximately 10
and 100 times greater than those from merging binary neutron stars (BNSs) and
neutron-star-black-hole (NSBH) mergers, respectively. Note that, since the
cosmic star formation rate density predicted by GABE is somewhat lower than
observational results by $\sim 0.2$ dex at z < $\sim 2$, the amplitude of the
SGWB in the frequency range $\sim[1$, $10^{4}]$ Hz may be underestimated by a
similar factor at most.
