  The Ensemble Kalman inversion (EKI) method is a method for the estimation of
unknown parameters in the context of (Bayesian) inverse problems. The method
approximates the underlying measure by an ensemble of particles and iteratively
applies the ensemble Kalman update to evolve (the approximation of the) prior
into the posterior measure.
  For the convergence analysis of the EKI it is common practice to derive a
continuous version, replacing the iteration with a stochastic differential
equation. In this paper we validate this approach by showing that the
stochastic EKI iteration converges to paths of the continuous-time stochastic
differential equation by considering both the nonlinear and linear setting, and
we prove convergence in probability for the former, and convergence in moments
for the latter. The methods employed can also be applied to the analysis of
more general numerical schemes for stochastic differential equations in
general.
