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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)

60G07 General theory of stochastic processes
60H05 Stochastic integrals
Full Text: DOI
[1] Dynkin, E.B.: Sufficient statistics and extreme points. Ann. Probab.6, 705-730 (1978) · Zbl 0403.62009
[2] Lipcer, R.S., Shyriaev, A.N.: Statistics of random processes, I. General theory. Applications of mathematics, vol. 5. Berlin Heidelberg New York: Springer 1977
[3] Neveu, J.: Bases mathématiques du calcul des probabilités, 2nd edn. Paris: Masson 1970 · Zbl 0203.49901
[4] Rogers, L.C.G., Williams, D.: Diffusions, Markov processes and martingales, vol. 2. Itô calculus, New York: Wiley 1987 · Zbl 0627.60001
[5] Stroock, D.W., Yor, M.: On extremal solutions of martingale problems. Ann. Sci. Ec. Norm. Super., IV. Ser.13, 95-164 (1980) · Zbl 0447.60034
[6] Tsirel’son, B.S.: An example of a stochastic equation having no strong solution. Teor. Veroyatn. Primen.20(2), 427-430 (1975)
[7] Veretennikov, A.Y.: On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb.39, 387-403 (1981) · Zbl 0462.60063
[8] Weizsäcker, H. von: Exchanging the order of taking suprema and countable intersection of sigma algebras. Ann. Inst. Henri Poincaré19, 91-100 (1983) · Zbl 0509.60002
[9] Williams, D.:Probability with martingales. Cambridge mathematical textbooks. Cambridge: Cambridge University Press 1991
[10] Yamada, Y., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ.11, 155-167 (1971) · Zbl 0236.60037
[11] Yor, M.: De nouveaux résultats sur l’équation de Tsirel’son. C.R. Acad. Sci., Paris, Sér. I.309, 511-514 (1989) · Zbl 0697.60062
[12] Zvonkin, A.K.: A transformation of the phase space of a diffusion process that removes the drift. Math. USSR, Sb.22, 129-149 (1974) · Zbl 0306.60049
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