Pathwise Uniqueness of Non-uniformly Elliptic SDEs with Rough Coefficients
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In this paper, we review and improve pathwise uniqueness results for some types of one-dimensional stochastic differential equations (SDE) involving the local time of the unknown process. The diffusion coefficient of the SDEs we consider is allowed to vanish on a set of positive measure and is not assumed to be smooth. As opposed to various existing results, our arguments are mainly based on the comparison theorem for local time and the occupation time formula. We apply our pathwise uniqueness results to derive strong existence and other properties of solutions for SDEs with rough coefficients.
KeywordsStochastic differential equations Pathwise uniqueness Comparison theorem for local times Local time of the unknown
Mathematics Subject Classification60H10 60H60 60J55
Financial support from the Alexander von Humboldt Foundation under the programme financed by the German Federal Ministry of Education and Research entitled German Research Chair No. 01DG15010 is gratefully acknowledged.
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