 
Summary: To appear in: S’ eminaire de Probabilit’ es, XXXVI
Lecture Notes in Mathematics
HORIZONTAL MARTINGALES IN VECTOR BUNDLES
MARC ARNAUDON AND ANTON THALMAIER
Abstract. Canonical prolongations of manifoldvalued martingales to vector bun
dles 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 r is a connection on M (i.e. a covariant derivative on TM ) and r 0
the lifted connection on E (i.e. a covariant derivative on TE), we consider semi
martingales J in E, living above X and linked to X via d r 0
J = h J (d r X) where
h J : TXM ! T J E is the horizontal lift; d r X and d r 0
J denote the It “
o differentials
with respect to the given connection. Such semimartingales J in E will be called hori
zontal semimartingales, resp. horizontal martingales in case when X is a rmartingale.
There are numerous ways of lifting r to r 0 . We mainly deal with horizontal and com
plete lifts. Horizontal lifts give rise to the notion of covariant It “
o differentials. For
covariant It “
