Stochastic differential equations driven by stable processes for which pathwise uniqueness fails

Let Zt be a one-dimensional symmetric stable process of order [alpha] with [alpha][set membership, variant](0,2) and consider the stochastic differential equationdXt=[phi](Xt-) dZt.For [beta]<(1/[alpha])[logical and]1, we show there exists a function [phi] that is bounded above and below by positive constants and which is Hölder continuous of order [beta] but for which pathwise uniqueness of the stochastic differential equation does not hold. This result is sharp.