A note on the cross Riccatian and related properties for symmetric stochastic realizations

Recent results concerning cross Grammians for both SISO (single-input, single-output) and symmetric MIMO (multi-input, multioutput) systems are extended for time-invariant stochastic systems. The extension is based on a cross Riccatian matrix and several results from stochastic realization theory. It is shown that the cross Riccatian can be obtained from the solution to a cross-Riccati equation carrying properties from both a forward and a backwards innovations representation. Other symmetry properties similar to those of the cross Grammian are introduced, along with their connection to balanced stochastic realizations