  This paper is concerned with fully discrete mixed finite element
approximations of the time-dependent stochastic Stokes equations with
multiplicative noise. A prototypical method, which comprises of the
Euler-Maruyama scheme for time discretization and the Taylor-Hood mixed element
for spatial discretization is studied in detail. Strong convergence with rates
is established not only for the velocity approximation but also for the
pressure approximation (in a time-averaged fashion). A stochastic inf-sup
condition is established and used in a nonstandard way to obtain the error
estimate for the pressure approximation in the time-averaged fashion. Numerical
results are also provided to validate the theoretical results and to gauge the
performance of the proposed fully discrete mixed finite methods.
