In this paper we deal with a perturbed algebraic Riccati equation in an infinite dimensional Banach space. Besides the interest in its own right, this class of equations appears, for instance, in the optimal control problem for infinite Markov jump linear systems (from now on iMJLS). Infinite or finite here, has to do with the state space of the Markov chain being infinite countable or finite (see, e.g., ). By using a certain concept of stochastic stability (a sort of L2-stability), we prove existence (and uniqueness) of maximal solution for this class of equation. When we recast the problem in the finite setting (finite state space of the Markov chain), we recover the result of  set to the Markovian jump scenario, now free from an inconvenient technical hypothesis used there, originally introduced in .