The anisotropic norm of discrete-time linear stochastic systems with state dependent noise is considered. Using first principles analysis applying completing the square arguments, it is proved that the anisotropic norm of such systems is upper bounded by a given positive real scalar, if a specific Riccati equation has a stabilizing positive semidefmite solution satisfying two additional conditions. It is shown that these conditions are sufficient and necessary for the boundedness of the anisotropic norm. Numerical algorithms to determine the stabilizing solution of this Riccati equation allowing thus to compute the anisotropic norm of stochastic systems with multiplicative noise are also presented. The theoretical results are illustrated by numerical examples. (C) 2017 Elsevier Ltd. All rights reserved.