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Stochastic differential equations driven by stable processes for which pathwise uniqueness fails. (English) Zbl 1111.60038
The paper considers stochastic differential equations (SDE) $$dX_t=\varphi (X_{t-})dZ_t$$, where $$Z_t$$ is a one-dimensional symmetric stable process of order $$\alpha$$ with $$0<\alpha <2$$ and proves that pathwise uniqueness can fail even when $$\varphi$$ is a Hölder continuous function of order $$\beta<\min(1/\alpha,1)$$ bounded above and below by strictly positive finite constants. The result is sharp in what concerns $$\beta$$. This is an extension of a similar result by M. T. Barlow [J. Lond. Math. Soc., II. Ser. 26, 335–347 (1982; Zbl 0456.60062)] for SDE $$dX_t=\varphi(X_t)dB_t$$ driven by a one-dimensional Brownian motion $$B_t$$.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G52 Stable stochastic processes
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##### References:
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