  We use the linear scalar SDE as a test problem to show that it is possible to
construct almost sure stable first-order weak balanced schemes based on the
addition of stabilizing functions to the drift terms. Then, we design balanced
schemes for multidimensional bilinear SDEs achieving the first order of weak
convergence, which do not involve multiple stochastic integrals. To this end,
we follow two methodologies to find appropriate stabilizing weights; through an
optimization procedure or based on a closed heuristic formula. Numerical
experiments show a promising performance of the new numerical schemes.
