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We are concerned with efficient numerical methods for stochasticcontinuous-time algebraic Riccati equations (SCARE). Such equations frequentlyarise from the state-dependent Riccati equation approach which is perhaps theonly systematic way today to study nonlinear control problems. Often involvedRiccati-type equations are of small scale, but have to be solved repeatedly inreal time. Important applications include the 3D missile/target engagement, theF16 aircraft flight control, and the quadrotor optimal control, to name a few.A new inner-outer iterative method that combines the fixed-point strategy andthe structure-preserving doubling algorithm (SDA) is proposed. It is provedthat the method is monotonically convergent, and in particular, taking the zeromatrix as initial, the method converges to the desired stabilizing solution.Previously, Newton's method has been called to solve SCARE, but it was mostlyinvestigated from its theoretic aspect than numerical aspect in terms of robustand efficient numerical implementation. For that reason, we revisit Newton'smethod for SCARE, focusing on how to calculate each Newton iterative stepefficiently so that Newton's method for SCARE can become practical. It isproposed to use our new inner-outer iterative method, which is provablyconvergent, to provide critical initial starting points for Newton's method toensure its convergence. Finally several numerical experiments are conducted tovalidate the new method and robust implementation of Newton's method.

Numerical Solutions for Stochastic Continuous-time Algebraic Riccati Equations