Tsirel'son's equation in discrete time (Q1187105)

Motivated by Tsirel'son's equation in continuous time, a similar stochastic equation indexed by discrete negative time is discussed in full generality, in terms of the law of a discrete time noise. When uniqueness in law holds, the unique solution (in law) is not strong; moreover, when there exists a strong solution, there are several strong solutions. In general, for any time \(n\), the \(\sigma\)-field generated by the past of a solution up to time \(n\) is shown to be equal, up to negligible sets, to the \(\sigma\)-field generated by the 3 following components: the infinitely remote past of the solution, the past of the noise up to time \(n\), together with an adequate independent complement.