  We examine the possibility that supernovae type Ia (SN Ia) are produced by
white dwarfs accreting from Roche-lobe filling evolved companions, under the
assumption that a strong optically thick stellar wind from accretor is able to
stabilize the mass transfer. We show that if a mass transfer phase on a thermal
timescale precedes a nuclear burning driven phase, then such systems (of which
the supersoft X-ray sources are a subgroup) can account for about 10% of the
inferred SN Ia rate.
  In addition, we examine the cosmic history of the supernova rate, and we show
that the ratio of the rate of SN Ia to the rate of supernovae produced by
massive stars (supernovae of types II, Ib, Ic) should increase from about z = 1
towards lower redshifts.
