#### Horizontal Martingales in Vector Bundles

by Marc Arnaudon and Anton Thalmaier

**
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*,
if
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
where
is the horizontal lift;
and
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
-martingale.
There are numerous ways of lifting
to .
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.
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:

Marc Arnaudon
<arnaudon@math.univ-poitiers.fr>
Anton Thalmaier
<anton.thalmaier@uni.lu>
Back to Homepage