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

Let Zt be a one-dimensional symmetric stable process of order α with α∈(0,2) and consider the stochastic differential equationdXt=φ(Xt−) dZt.For β<(1/α)∧1, we show there exists a function φ that is bounded above and below by positive constants and which is Hölder continuous of order β but for which pathwise uniqueness of the stochastic differential equation does not hold. This result is sharp.