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Stochastic algebraic Riccati equations, also known as rational algebraicRiccati equations, arising in linear-quadratic optimal control for stochasticlinear time-invariant systems, were considered to be not easy to solve.The-state-of-art numerical methods most rely on differentiability orcontinuity, such as Newton-type method, LMI method, or homotopy method. In thispaper, we will build a novel theoretical framework and reveal the intrinsicalgebraic structure appearing in this kind of algebraic Riccati equations. Thisstructure guarantees that to solve them is almost as easy as to solvedeterministic/classical ones, which will shed light on the theoretical analysisand numerical algorithm design for this topic.

Stochastic algebraic Riccati equations are almost as easy as deterministic ones theoretically