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

Recent results concerning cross Grammians for both SISO and symmetric MIMO systems are extended for 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 symmtry properties similar to those for the cross Grammian are introduced, along with connections to balanced stochastic realizations.