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Differentiability of SDEs with drifts of super-linear growth

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https://arxiv.org/abs/1803.06947
Original languageEnglish
Article number3
Number of pages43
JournalElectronic journal of probability
Volume24
DOIs
Publication statusPublished - 8 Feb 2019

Abstract

We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth (and random coefficients), namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Ray Absolute Continuity and Stochastic G\^ateaux Differentiability. This method enables one to take limits in probability rather than mean square which bypasses the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology from Nualart's 2006 work, Lemma 1.2.3 for this setting. Several examples illustrating the range and scope of our results are presented. We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.

    Research areas

  • math.PR, Primary: 60H07. Secondary: 60H10, 60H30

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