The class B#;$fl of strong solutions of the boundary problem for a stochastic differential equation with given drift and diffusion coefficients implies here a class of all continuous processes having the given stochastic differential within the interval [y,, y,] and not leaving it. It is shown that the class BR.;i;] can be characterized as a class of all solutions of some stochastic integral equation. The instantly reflecting process (IRP) in A. Skorokhod's sense is proved to be the extremal (with respect to some ordering) element of this class and in the case when the drift and diffusion coefficients satisfy the Lipschitz condition it can be constructed by the method of successive approximations. The convergence of discrete approximations, which are constructed by the natural analogy of Euler's formula to the IRP, is studied.
The semimartingale stochastic approximation procedure, namely, the Robbins-Monro type SDE is introduced which naturally includes both generalized stochastic approximation algorithms with martingale noises and recursive parameter estimation procedures for statistical models associated with semimartingales. General results concerning the asymptotic behaviour of the solution are presented. In particular, the conditions ensuring the convergence, rate of convergence and asymptotic expansion are established. The results concerning the Polyak weighted averaging procedure are also presented.1991 Mathematics Subject Classification. 62L20, 60H10, 60H30. t 0 E W (s, x, u s )(µ − ν)(ds, dx) (in case 3 • ) provided the latters are well-defined.Consider the following semimartingale stochastic differential equationWe call SDE (0.1) the Robbins-Monro (PM) type SDE if the drift coefficient H t (u), t ≥ 0, u ∈ R 1 satisfies the following conditions: for all t ∈ [0, ∞) P -a.s.
(A)H t (0) = 0,The question of strong solvability of SDE (0.1) is well-investigated (see, e.g., [8], [9], [13]).
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