  In the first part of the paper we develop the sensitivity analysis for the
nonlinear McKean-Vlasov diffusions stressing precise estimates of growth of
solutions and their derivatives with respect to the initial data, under rather
general assumptions on the coefficients. The exact estimates become
particularly important when treating the extension of these equations having
random coefficient, since the noise is usually assumed to be unbounded.
  The second part contains our main results dealing with the sensitivity of
stochastic McKean-Vlasov diffusions. By using the method of stochastic
characteristics, we transfer these equations to the non-stochastic equations
with random coefficients thus making it possible to use the estimates obtained
in the first part. The motivation for studying sensitivity of McKean-Vlasov
SPDEs arises naturally from the analysis of the mean-field games with common
noise.
