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

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Abstract

Let Zt be a one-dimensional symmetric stable process of order α with α∈(0,2) and consider the stochastic differential equation dXt=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. © 2004 Elsevier B.V. All rights reserved.

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Bass, R. F., Burdzy, K., & Chen, Z. Q. (2004). Stochastic differential equations driven by stable processes for which pathwise uniqueness fails. Stochastic Processes and Their Applications, 111(1), 1–15. https://doi.org/10.1016/j.spa.2004.01.010

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