Instability of Certain Nonlinear Stochastic Differential Equations

We prove existence and uniqueness of the solution Xϵt of the SDE, Xϵt = ϵBt + ∫t0uq −1 ϵ(s, Xϵt) ds, where Xϵt is a one-dimensional process and uϵ(t, x) the density of Xϵt (ϵ > 0, q > 1). We show that the closure of (Xϵt; 0 ≤ t ≤ 1) with respect to Hölder norm, when ϵ goes to 0, is a.s. equal to an explicit family of continuous functions. We obtain similar results, considering SDE′s where the drift coefficient is equal to ± sgn(x) u(t, x).