In a paper of me of a few years ago, Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients, Theorem 3.6 gives sufficient conditions. The proof is indeed by iteration, but of the continuous time Lyapunov equation, which I found gave better results in the case I was interested in.

These sufficient conditions are, in terms of your notation: $\sum_{i=1}^p ||F_i||^2 < 1$, which looks very similar to the condition you state but is perhaps more restrictive. Unfortunately at the moment I am not too fluent in Kronecker products. Hope this helps.

In a paper of me of a few years ago, Estimate on the Pathwise Lyapunov Exponent of Linear Stochastic Differential Equations with Constant Coefficients, Theorem 3.6 gives sufficient conditions. The proof is indeed by iteration, but of the continuous time Lyapunov equation, which I found gave better results in the case I was interested in.

These sufficient conditions are, in terms of your notation: $\sum_{i=1}^p ||F_i||^2 < 1$, which looks very similar to the condition you state but is perhaps more restrictive. Unfortunately at the moment I am not too fluent in Kronecker products, so I find it difficult to compare.