This paper focuses on the modeling of time-varying covariance matrices using the state covariance of linear systems. Following concepts from optimal mass transport, we investigate and compare three types of covariance paths, which are solutions to different optimal control problems. One of the covariance paths solves the Schrodinger bridge problem. The other two types of covariance paths are based on generalizations of the Fisher-Rao metric in information geometry, which are the major contributions of this paper. The general framework is an extension of the approach proposed in the paper "Smooth interpolation of covariance matrices and brain network estimation" (IEEE Trans. Autom. Control), which focuses on linear systems without stochastic input The performances of the three covariance paths are compared using synthetic data and a real-data example on the estimation of dynamic brain networks using functional magnetic resonance imaging.