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Pathwise vs. path-by-path uniqueness
Alexander Shaposhnikov, Lukas Wresch · 2020-01-09 · via math.PR updates on arXiv.org

We construct a series of stochastic differential equations of the form $dX_t = b(t, X_t) dt + dB_t$ which exhibit nonuniqueness in the path-by-path sense while having a unique adapted solution in the sense of stochastic processes, i.e. pathwise uniqueness holds.

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