INSTABILITY OF CERTAIN NONLINEAR STOCHAST IC DIFFERENTIAL-EQUATIONS

We prove existence and uniqueness of the solution X(t)(epsilon) of the SDE, X(t)(epsilon) = epsilon B-t+ integral(o)(') u(epsilon)(q-1) (s, X(t)(epsilon)) ds, where X(t)(epsilon) is a one-dimensional process an d u(epsilon)(t, x) the density of X(t)(epsilon) (epsilon > 0, q > 1): We show that the closure of (X(t)(epsilon); 0 less than or equal to t less than or equal to 1) with respect to Holder norm, when epsilon goe s to 0, is a.s. equal to an explicit family of continuous functions. W e obtain similar results, considering SDE's where the drift coefficien t is equal to +/- sgn(x) u(t,x). (C) 1995 Academic Press, Inc.