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Intrinsic stochastic differential equations as jets

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Title: Intrinsic stochastic differential equations as jets
Authors: Armstrong, J
Brigo, D
Item Type: Journal Article
Abstract: We explain how Itô 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 also show how jets can be used to derive graphical representations of Itô SDEs, and we show how jets can be used to derive the differential operators associated with SDEs in a coordinatefree manner. We relate jets to vector flows, giving a geometric interpretation of the Itô–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 an SDE solution is associated with the drift of the SDE in Stratonovich form for small times.
Issue Date: 28-Feb-2018
Date of Acceptance: 17-Jan-2018
URI: http://hdl.handle.net/10044/1/57251
DOI: 10.1098/rspa.2017.0559
ISSN: 1364-5021
Publisher: Royal Society, The
Journal / Book Title: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume: 474
Issue: 2210
Copyright Statement: © 2018 The Author(s) Published by the Royal Society. All rights reserved.
Keywords: Science & Technology
Multidisciplinary Sciences
Science & Technology - Other Topics
stochastic differential equations
stochastic differential geometry
stochastic differential equations on manifolds
Ito calculus
Stratonovich calculus
jet bundle
MANIFOLD
01 Mathematical Sciences
02 Physical Sciences
09 Engineering
Notes: The journal requires that access to the full text must be embargoed for 12 months from publication. This means that the full text version can be made available in March 2019.
Publication Status: Published
Article Number: 20170559
Online Publication Date: 2018-02-14
Appears in Collections:Financial Mathematics
Mathematics
Faculty of Natural Sciences