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Miscellany

The Robbins–Monro type stochastic differential equations. II. Asymptotic behaviour of solutions

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Pages 153-180 | Received 30 Nov 2000, Accepted 17 Dec 2002, Published online: 17 Oct 2011

References

  • Albert A. Gardner L. 1967 Stochastic Approximation and Nonlinear Regression M.I.T. Press Cambridge, MA
  • Chen H.F. 1993 Asymptotic efficient stochastic approximation Stochastics 45 1 6
  • Fabian V. 1978 On asymptotically efficient recursive estimation Ann. Stat. 6 854 867
  • Gal'čuk L.I. 1978 On the uniqueness and existence of solution of stochastic equations with respect to semimartingales Theory Probab. Appl. 23 782 795
  • Gikhman I.I. Skorokhod A.V. 1982 Stochastic Differential Equations and its Applications Naukova Dumka Kiev In Russian
  • Hall P. Heide C.C. 1980 Martingale Limit Theory and its Applications Academic Press New York
  • Jacod J. 1979 Calcul Stochastique et Promblémes de Martingales Lect. Notes in Math. 714 Springer Berlin/Heidelberg/New York
  • Jacod J. Memin J. 1981 Weak and Strong Solutions of Stochastic Differential Equations: Existence and Stability Lect. Notes in Math., Springer Berlin 851 169 212
  • Jacod J. Shiryayev A.N. 1987 Limit Theorems for Stochastic Processes Springer Berlin
  • Kushner H.J. Yin G. 1997 Stochastic Approximation Algorithms and Applications Springer Berlin
  • Lazrieva, N.L., Sharia, T. and Toronjadze, T.A. “The Robbins–Monro type stochastic differential equations. I. Convergence of solutions”, Stochastics and Stochastic Reports 61
  • Lazrieva N.L. Toronjadze T.A. 1987 Engelbert H.W. Schmidt W. Ito–Ventzel's Formula For Semimartingales, Asymptotic Properties of MLE and Recursive Estimation Lect. Notes in Control and Inform. Sciences, 96, Stochast. Diff. Systems 346 355
  • Liptser R.S. Shiryayev A.N. 1986 Theory of Martingales Nauka Moscow English translation, Kluwer, Dordrecht, 1989
  • Ljung L. Pflug G. Walk H. 1992 Stochastic Approximation and Optimization of Random Systems Birkhäuser Verlag Basel/Boston/Berlin DMV-seminar, Bd 17
  • Ljung L. 2001 Recursive least-squares and accelerated convergence in stochastic approximation schemes Int. J. Adapt. Control Signal Process. 15 2 169 178
  • Melnikov A.V. 1989 Procedures of stochastic approximation for semimartingales Statistics and Control of Random Processes Nauka Moscow 147 156
  • Melnikov A.V. Rodkina A.E. 1993 Martingale approach to the procedures of stochastic approximation Niemi H. et al. Frontiers in Pure and Applied Probab. 1 TVP/VSP 165 182
  • Melnikov A.V. Rodkina A.E. Valkeila E. 1993 On a general class of stochastic approximation algorithms Niemi H. et al. Frontiers in Pure and Applied Probab. 1 TVP/VSP 183 196
  • Melnikov A.V. Valkeila E. 1999 Martingale models of stochastic approximation and their convergence Theory Probab. Appl. 44 2 333 360
  • Nevelson M.B. Khas'minskii R.Z. 1972 Stochastic Approximation and Recursive Estimation Nauka Moscow In Russian
  • Prakasa Rao B.L.S. 1999 Semimartingales and their Statistical Inference Chapman & Hall/CRC Press New York/Boca Raton, FL
  • Robbins H. Monro S. 1951 A stochastic approximation method Ann. Math. Stat. 22 1 400 407
  • Sharia T. 1998 On the recursive parameter estimation for the general discrete time statistical model Stochastic Processes Appl. 73 2 151 172
  • Spreij P. 1991 Approximate maximum likelihood estimation for a class of counting process models J. Multivariate Anal. 39 2 236 245
  • Yin G. Gupta I. 1993 On a continuous time stochastic approximation problem Acta Aplicandae Math. 3 20
  • Yin G. 2001 Stochastic approximation: theory and applications Kannan D. et al. Handbook of Stochastic Analysis and Applications Stat., Textb. Monogr. 163 Marcel Dekker New York, NY 577 624

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