Dual Pairs of Generalized Lyapunov Inequalities and Balanced Truncation of Stochastic Linear Systems

We consider two approaches to balanced truncation of stochastic linear systems, which follow from different generalizations of the reachability Gramian of deterministic systems. Both preserve mean-square asymptotic stability, but only the second leads to a stochastic H
^∞
-type bound for the approximation error of the truncated system.