This paper is concerned with the identification of discrete-time, time invariant, state affine state space models driven by an independent identically distributed (IID) random input, and in the presence of process and measurement noise. The identification problem is treated using a cumulant based approach. It is shown that the input-output and input-state crosscumulant equations in the time domain have the form of a linear autonomous system. An algorithmic procedure is then developed, for the computation of the unknown system matrices, based on a standard deterministic linear subspace identification algorithm, provided the input signal has some persistent excitation properties. The special case of Gaussian IID input is also examined. The proposed method is computationally very efficient and its accuracy is illustrated by simulations.