19,642 research outputs found
Pathwise stochastic integrals for model free finance
We present two different approaches to stochastic integration in frictionless
model free financial mathematics. The first one is in the spirit of It\^o's
integral and based on a certain topology which is induced by the outer measure
corresponding to the minimal superhedging price. The second one is based on the
controlled rough path integral. We prove that every "typical price path" has a
naturally associated It\^o rough path, and justify the application of the
controlled rough path integral in finance by showing that it is the limit of
non-anticipating Riemann sums, a new result in itself. Compared to the first
approach, rough paths have the disadvantage of severely restricting the space
of integrands, but the advantage of being a Banach space theory. Both
approaches are based entirely on financial arguments and do not require any
probabilistic structure.Comment: Published at http://dx.doi.org/10.3150/15-BEJ735 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Pathwise integration with respect to paths of finite quadratic variation
We study a pathwise integral with respect to paths of finite quadratic
variation, defined as the limit of non-anticipative Riemann sums for
gradient-type integrands. We show that the integral satisfies a pathwise
isometry property, analogous to the well-known Ito isometry for stochastic
integrals. This property is then used to represent the integral as a continuous
map on an appropriately defined vector space of integrands. Finally, we obtain
a pathwise 'signal plus noise' decomposition for regular functionals of an
irregular path with non-vanishing quadratic variation, as a unique sum of a
pathwise integral and a component with zero quadratic variation.Comment: To appear in: Journal de Mathematiques Pures et Appliquee
Extending Whitney's extension theorem: nonlinear function spaces
We consider a global, nonlinear version of the Whitney extension problem for
manifold-valued smooth functions on closed domains , with non-smooth
boundary, in possibly non-compact manifolds. Assuming is a submanifold with
corners, or is compact and locally convex with rough boundary, we prove that
the restriction map from everywhere-defined functions is a submersion of
locally convex manifolds and so admits local linear splittings on charts. This
is achieved by considering the corresponding restriction map for locally convex
spaces of compactly-supported sections of vector bundles, allowing the even
more general case where only has mild restrictions on inward and outward
cusps, and proving the existence of an extension operator.Comment: 37 pages, 1 colour figure. v2 small edits, correction to Definition
A.3, which makes no impact on proofs or results. Version submitted for
publication. v3 small changes in response to referee comments, title
extended. v4 crucial gap filled, results not affected. v5 final version to
appear in Annales de l'Institut Fourie
Convergence in for Feynman path integrals
We consider a class of Schrodinger equations with time-dependent smooth
magnetic and electric potentials having a growth at infinity at most linear and
quadratic, respectively. We study the convergence in with loss of
derivatives, , of the time slicing approximations of the
corresponding Feynman path integral. The results are completely sharp and hold
for long time, where no smoothing effect is available. The techniques are based
on the decomposition and reconstruction of functions and operators with respect
to certain wave packets in phase space.Comment: 24 pages, 1 figure; in this version, some typos were corrected and
some arguments a little bit cleane
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