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Tsirel’son’s equation in discrete time. (English) Zbl 0744.60033
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.
Reviewer: M.Yor (Paris)

MSC:
 60G07 General theory of stochastic processes 60H05 Stochastic integrals
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References:
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