  In this paper, we bring together the worlds of model order reduction for
stochastic linear systems and $\mathcal H_2$-optimal model order reduction for
deterministic systems. In particular, we supplement and complete the theory of
error bounds for model order reduction of stochastic differential equations.
With these error bounds, we establish a link between the output error for
stochastic systems (with additive and multiplicative noise) and modified
versions of the $\mathcal H_2$-norm for both linear and bilinear deterministic
systems. When deriving the respective optimality conditions for minimizing the
error bounds, we see that model order reduction techniques related to iterative
rational Krylov algorithms (IRKA) are very natural and effective methods for
reducing the dimension of large-scale stochastic systems with additive and/or
multiplicative noise. We apply modified versions of (linear and bilinear) IRKA
to stochastic linear systems and show their efficiency in numerical
experiments.
