  In this paper, the problem of full state approximation by model reduction is
studied for stochastic and bilinear systems. Our proposed approach relies on
identifying the dominant subspaces based on the reachability Gramian of a
system. Once the desired subspace is computed, the reduced order model is then
obtained by a Galerkin projection. We prove that, in the stochastic case, this
approach either preserves mean square asymptotic stability or leads to reduced
models whose minimal realization is mean square asymptotically stable. This
stability preservation guarantees the existence of the reduced system
reachability Gramian which is the basis for the full state error bounds that we
derive. This error bound depends on the neglected eigenvalues of the
reachability Gramian and hence shows that these values are a good indicator for
the expected error in the dimension reduction procedure. Subsequently, we
establish the stability preservation result and the error bound for a full
state approximation to bilinear systems in a similar manner. These latter
results are based on a recently proved link between stochastic and bilinear
systems. We conclude the paper by numerical experiments using a benchmark
problem. We compare this approach with balanced truncation and show that it
performs well in reproducing the full state of the system. \end{abstract}
