This paper studies switched Riccati differential equations, whose switching is driven by a Poisson-like random signal. First we show that the expected value of the escape time of a switched Riccati differential equation satisfies an integral equation and then give a sufficient condition for the equation to admit a unique solution. Then we study a switched version of so called extended Riccati differential equations, which are obtained by extending the domain of Riccati differential equations to the Grassmannian manifold. We show that the limiting distribution of the random walk given by the switched stochastic equation converges to a unique invariant measure exponentially fast. The theory of products of random matrices is used to derive this result. We do not require Riccati differential equations to be symmetric.