Horizontal Martingales in Vector Bundles
Canonical prolongations of manifold-valued martingales to vector bundles
over a manifold are considered. Such prolongations require a lift of
the connection from the manifold to the corresponding bundle.
Given a continuous semimartingale X in M,
is a connection on M (i.e. a covariant derivative on TM) and
the lifted connection on E
(i.e. a covariant derivative on TE), we consider semimartingales J
in E, living above X and linked to X via
is the horizontal lift;
denote the Itô differentials with respect
to the given connection.
Such semimartingales J in E will be called horizontal
semimartingales, resp. horizontal martingales in case when X is a
There are numerous ways of lifting
We mainly deal with horizontal and complete lifts.
Horizontal lifts give rise to the notion of covariant Itô differentials.
For covariant Itô differentials a commutation formula with ordinary
covariant differentials is established.
As an application, covariant variations of stochastic parallel transport
and their relation to Yang-Mills connections are investigated.
On the other side, within the framework of complete lifts of connections,
the martingale property is preserved under taking derivatives
(or exterior derivatives) of families of martingales and is inherited
to diffusions on the exterior cotangent bundle
with the de Rham-Hodge Laplacian as generator. Moreover,
in a natural way, derivatives of harmonic maps are harmonic as well.
by Marc Arnaudon and Anton Thalmaier
In: J. Azema, M. Emery, M. Ledoux and M. Yor (Eds.)
Séminaire de Probabilités XXXVI, 419-456,
Lecture Notes in Mathematics 1801,
Springer: Berlin, 2003.
The paper is available here:
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