Lithuanian Mathematical Journal, Vol. 32, No. 2, 1992 ON APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH COEFFICIENTS DEPENDING ON THE PAST V. MackeviSius 1. INTRODUCTION Let us consider a multidimensional stochastic differential equation dX (t) = y(x(t)) o dZ (t), x(o) = (1) (in matrix notation), or, equivalently (using a convention of summation over repeating indices), dXi(t)= fij(X(t))odZj(t), Xi(O)=xi, i=l,2,...,d, (2) where Z = (Z1, Z2,... ,Zr) is an r-dimensional continuous semimartingale, f(x) = (fq(x), i = 1,..., d, j = 1;..., r), z E I$ d, is a sufficiently smooth matrix function (for example, fij E C~ (I~d)), the stochastic differential (denoted by the sign o) is understood in the Stratonovich sense. Let Z 8, 6 > O, be a family of r-dimensional continuous semimartingales, and let X 6, 6 > O, be the family of solutions of (1) with Z replaced by Z 8, that is, dX~(t)-- fij(X'(t))odZ](t), X~(O)=xl, i= l,2,...,d. (2') When does the uniform convergence of Z ~ to Z (as 6 --* 0 ) on some time interval 11) imply the uniform convergence ofX z to X? It is known (see, e.g., Mackevigius [11, 12], Pieard [13] and the bibliography therein) that for the positive answer the approximation (Z ~) of Z has to be symmetric in a certain sense. One way to define the symmetry is by the following condition: z', o Z] -~ Z, o Z~, ~-~ o, i,j = 1,2,...,~, where X o Y denotes the Stratonovich integral X o Y(t) = fo X(s) o dY(s). If not all of the limits Rq = lim~_~0(Z~ o Z~ - Zi o Zj) are equal to 0, and R~j are FV (finite variation) processes, then one has to add "correction terms" either to the limit equation or to the approximating ones. What happens if the coefficient f depends on paths of solutions, and we consider equations dXi(t) = fq (X)(t) o dZj (t), axr : Si~(X')(t) o dZ~(t), Xi(O)=xi, i=1,2,...,d, (3) X~(O)=xi, i=1,2,...,d, (3 ~) X)By the uniform convergence we mean the convergence suptet IZ6(t) - Z(t)I --, 0, 6 ~ 0, in some sense (in probability, in L P etc.). Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 32, No. 2, pp. 285-298, April-June, 1992. Original artide submitted 30 August, 1991. 0363-1672/92/3202-0227512.50 (~) 1993 Plenum Publishing Corporation 227