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Intrinsic Stochastic Differential Equations as jets

Research output: Contribution to journalArticle

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalRoyal Society of London. Proceedings A. Mathematical, Physical and Engineering Sciences
Volume474
Issue number2210
Early online date14 Feb 2018
DOIs
Accepted/In press17 Jan 2018
E-pub ahead of print14 Feb 2018
Published28 Feb 2018

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Abstract

We explain how Ito Stochastic Differential Equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We show how jets can be used to derive graphical representations of Ito SDEs. We show how jets
can be used to derive the differential operators associated with SDEs
in a coordinate free manner. We relate jets to vector flows, giving a
geometric interpretation of the Ito-Stratonovich transformation.
We show how percentiles can be used to give an alternative
coordinate free interpretation of the coefficients of one dimensional SDEs.
We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of ``fan diagrams''. 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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