  The binary systems consisting of a Be star and a white dwarf (BeWDs) are very
interesting.They can originate from the binaries composed of a Be star and a
subdwarf O or B star (BesdOBs), and they can merge into red giants via luminous
red nova or can evolve into double WD potentially detected by $LISA$ mission.
Using the method of population synthesis, we investigate the formation and the
destiny of BeWDs,and discuss the effects of the metallicity ($Z$) and the
common envelope evolution parameters. We find that BesdOBs are significant
progenitors of BeWDs. About 30\% ($Z=0.0001$)-50\% ($Z=0.02$) of BeWDs come
from BesdOBs. About 60\% ($Z=0.0001$) -70\% ($Z=0.02$) of BeWDs turn into red
giants via a merger between a WD and a non-degenerated star. About 30\%
($Z=0.0001$) -40\% ($Z=0.02$) of BeWDs evolve into double WDs which are
potential gravitational waves of $LISA$ mission at a frequency band between
about $3\times10^{-3}$ and $3\times10^{-2}$ Hz. The common envelope evolution
parameter introduces an uncertainty with a factor of about 1.3 on BeWD
populations in our simulations.
