  Model order reduction (MOR) techniques are often used to reduce the order of
spatially-discretized (stochastic) partial differential equations and hence
reduce computational complexity. A particular class of MOR techniques is
balancing related methods which rely on simultaneously diagonalizing the system
Gramians. This has been extensively studied for deterministic linear systems.
The balancing procedure has already been extended to bilinear equations [1], an
important subclass of nonlinear systems. The choice of Gramians in [1] is
referred to be the standard approach. In [18], a balancing related MOR scheme
for bilinear systems called singular perturbation approximation (SPA) has been
described that relies on the standard choice of Gramians. However, no error
bound for this method could be proved. In this paper, we extend the setting
used in [18] by considering a stochastic system with bilinear drift and linear
diffusion term. Moreover, we propose a modified reduced order model and choose
a different reachability Gramian. Based on this new approach, an $L^2$-error
bound is proved for SPA which is the main result of this paper. This bound is
new even for deterministic bilinear systems.
