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Itô stochastic differential equations as 2-jets

Research output: Chapter in Book/Report/Conference proceedingOther chapter contribution

John Armstrong, Damiano Brigo

Original languageEnglish
Title of host publicationGeometric Science of Information - 3rd International Conference, GSI 2017, Proceedings
PublisherSpringer Verlag
Pages543-551
Number of pages9
Volume10589 LNCS
ISBN (Electronic)978-3-319-68445-1
ISBN (Print)9783319684444
DOIs
Publication statusE-pub ahead of print - 24 Oct 2017
Event3rd International Conference on Geometric Science of Information, GSI 2017 - Paris, France
Duration: 7 Nov 20179 Nov 2017

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume10589 LNCS
ISSN (Print)03029743
ISSN (Electronic)16113349

Conference

Conference3rd International Conference on Geometric Science of Information, GSI 2017
CountryFrance
CityParis
Period7/11/20179/11/2017

King's Authors

Abstract

We explain how Itô Stochastic Differential Equations on manifolds may be defined as 2-jets of curves and show how this relationship can be interpreted in terms of a convergent numerical scheme. We use jets as a natural language to express geometric properties of SDEs. We explain that the mainstream choice of Fisk-Stratonovich-McShane calculus for stochastic differential geometry is not necessary. We give a new geometric interpretation of the Itô–Stratonovich transformation in terms of the 2-jets of curves induced by consecutive vector flows. We discuss the forward Kolmogorov equation and the backward diffusion operator in geometric terms. In the one-dimensional case we consider percentiles of the solutions of the SDE and their properties. In particular the median of a SDE solution is associated to the drift of the SDE in Stratonovich form for small times.

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