Summary
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 solution. In general, for any time,n, the σ-field generated by the past of a solution up to timen is shown to be equal, up to negligible sets, to the σ-field generated by the 3 following components: the infinitely remote past of the solution, the past to the noise up to timen, together with an adequate independent complement.
Article PDF
Similar content being viewed by others
References
Dynkin, E.B.: Sufficient statistics and extreme points. Ann. Probab.6, 705–730 (1978)
Lipcer, R.S., Shyriaev, A.N.: Statistics of random processes, I. General theory. Applications of mathematics, vol. 5. Berlin Heidelberg New York: Springer 1977
Neveu, J.: Bases mathématiques du calcul des probabilités, 2nd edn. Paris: Masson 1970
Rogers, L.C.G., Williams, D.: Diffusions, Markov processes and martingales, vol. 2. Itô calculus, New York: Wiley 1987
Stroock, D.W., Yor, M.: On extremal solutions of martingale problems. Ann. Sci. Ec. Norm. Super., IV. Ser.13, 95–164 (1980)
Tsirel'son, B.S.: An example of a stochastic equation having no strong solution. Teor. Veroyatn. Primen.20(2), 427–430 (1975)
Veretennikov, A.Y.: On strong solutions and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb.39, 387–403 (1981)
Weizsäcker, H. von: Exchanging the order of taking suprema and countable intersection of sigma algebras. Ann. Inst. Henri Poincaré19, 91–100 (1983)
Williams, D.:Probability with martingales. Cambridge mathematical textbooks. Cambridge: Cambridge University Press 1991
Yamada, Y., Watanabe, S.: On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ.11, 155–167 (1971)
Yor, M.: De nouveaux résultats sur l'équation de Tsirel'son. C.R. Acad. Sci., Paris, Sér. I.309, 511–514 (1989)
Zvonkin, A.K.: A transformation of the phase space of a diffusion process that removes the drift. Math. USSR, Sb.22, 129–149 (1974)