The robbins-monro type stochastic differential equations. I. convergence of solutions

References

• Albert , A. and Gardner , L. 1967 . “ Stochastic approximation and nonlinear regression ” . Cambridge, Massachusettes : M.I.T.-Press .
   Google Scholar
• Gal'cuk , L. I. 1978 . On the uniqueness and existence of solution of stochastic equations with respect to semimartingales . Theory Probab. Appl. , 23 : 782 – 795 .
   Web of Science ®Google Scholar
• Gikhman , I. I. and Skorokhod , A. V. 1982 . “ Stochastic differential equations and its Applications ” . Kiev : Naukova Dumka . in Russian
   Google Scholar
• Gushchin , A. A. 1995 . On asymptotic optimality of parameter estimators under the LAQ condition . Probab. theory and Appl. , 40 ( 2 ) : 286 – 300 .
   Web of Science ®Google Scholar
• Hall , P. and Heide , C. C. 1980 . “ Martingale limit theory and its applications ” . New York : Academic Press .
   Google Scholar
• Jacod , J. 1979 . “ Calcul stochastique et promblemes de martingales ” . Lect. Notes in Math Berlin, , New York : Springer .
   Google Scholar
• Jacod , J. and Memin , J. 1981 . “ Weak and strong solutions of stochastic differential equations: existence and stability ” . Lect. Notes in Math Vol. 851 , 169 – 212 . Springer-Verlag .
   Google Scholar
• Kabanov , Yu. , Liptser , R. S. and Shiryayev , A. N. 1979 . Absolute continuity and singularity of locally absolutely continuous probability distributions . Math. USSR Sb. Part I , 35 : 631 – 660 . English transl
   Web of Science ®Google Scholar
• Lipster , R. S. 1980 . A strong law of large numbers for local martingales . Stochastics , 3 ( 3 ) : 217 – 228 .
   Google Scholar
• Lenglarth , E. 1977 . Sur la convergence presque sure des martingales locales . C.r. Acad. science Paris , 284 ( 17 ) : A1085 – A1088 .
   Google Scholar
• Ljung , L. , Pflug , G. and Walk , H. Stochastic approximation and optimization of random systems . DMV-seminar, Bd 17 . Basel, Boston, Berlin : Birkhäuser verlag .
   Google Scholar
• Lipster , R. S. and Shiryayev , A. N. 1986 . “ Theory of martingales ” . Moscow : Nauka . English transl. Kluwer, Dordrecht, 1989
   Google Scholar
• Lazrieva , N. L. and Toronjadze , T. A. 1987 . “ Ito-Ventzel's formula for semimartingales,asymptotic properties of MLE and recursive estimation ” . In Lect. Notes in Control and Inform. Sciences, Stochast. diff. systems , Edited by: Engelbert , H. J. and Schmidt , W. 346 – 355 . Springer .
   Google Scholar
• Melnikov , A. V. 1989 . “ Procedures of stochastic approximation for semimartingales ” . In Statistic and Control of Random Process , 147 – 156 . Moscow : Nauka .
   Google Scholar
• Melnikov , A. V. and Rodkina , A. E. 1993 . “ Martingale approach to the procedures of stochastic approximation ” . In Frontiers in Pure and Applied Probab., 1 , Edited by: Niemi. , H. 165 – 182 . TVP/VSP .
   Google Scholar
• Melnikov , A. V. , Rodkina , A. E. and Valkeila , E. 1993 . “ On a general class of stochastic approximation algorithms ” . In Frontiers in Pure and Applied Probab., 1 , Edited by: Niemi , H. 183 – 196 . TVP/VSP .
   Google Scholar
• Nevelson , M. B. and Khas'minskii , R. Z. 1972 . “ Stochastic approximation and recursive estimation ” . Moscow : Nauka . in Russian
   Google Scholar
• Robbins , H. and Monro , S. 1951 . A stochastic approximation method . Ann. Math.Stat. , 22 ( 1 ) : 400 – 407 .
   Google Scholar
• Robbins , H. and Siegmund , D. 1971 . “ A convergence theorem for nonnegative almost supermartingales ” . In Optimizing Methods in Statistics , Edited by: Rustagi , J. S. 233 – 257 . New York : Academic Press .
   Google Scholar
• Sharia , T. 1996 . “ On the recursive parameter estimation in the general discrete time statistical model ” . to appear
   Google Scholar
• Spreij , P. 1991 . Approximate maximum likelihood estimation for a class of counting process models . Journal of Multivariate Analysis , 39 ( 2 ) : 236 – 245 .
   Web of Science ®Google Scholar