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Kusuoka-Stroock gradient bounds for the solution of the filtering equation

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Publication details

JournalJournal of Functional Analysis
DateAccepted/In press - 13 Dec 2014
DateE-pub ahead of print - 12 Jan 2015
DatePublished (current) - 1 Apr 2015
Issue number7
Volume268
Number of pages44
Pages (from-to)1928-1971
Early online date12/01/15
Original languageEnglish

Abstract

We obtain sharp gradient bounds for perturbed diffusion semigroups. In contrast with existing results, the perturbation is here random and the bounds obtained are pathwise. Our approach builds on the classical work of Kusuoka and Stroock [13,14,16,17], and extends their program developed for the heat semi-group to solutions of stochastic partial differential equations. The work is motivated by and applied to nonlinear filtering. The analysis allows us to derive pathwise gradient bounds for the un-normalised conditional distribution of a partially observed signal. It uses a pathwise representation of the perturbed semigroup following Ocone [22]. The estimates we derive have sharp small time asymptotics.

    Research areas

  • Filtering, Gradient bounds, Randomly perturbed semigroup, Stochastic partial differential equation

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