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

Abstract

Let Z [subscript] t be a one-dimensional symmetric stable process of order [alpha] with [alpha is an element of the set] (0, 2) and consider the stochastic differential equation

dX [subscript] t = [omega] (X [subscript] t−)dZ [subscript]t.

For [beta] < 1 [divided by alpha] ^ 1, we show there exists a function that is bounded above and below by positive constants and which is Holder continuous of order [beta] but for which pathwise uniqueness of the stochastic differential equation does not hold. This result is sharp.