#### Stability of stochastic differential equations
in manifolds

by Marc Arnaudon and Anton Thalmaier

We extend the so-called topology of semimartingales to continuous
semimartingales with values in a manifold and with lifetime, and prove that
if the manifold is endowed with a connection
then this topology and the
topology of compact convergence in probability coincide on the set
of continuous -martingales.
For the topology of manifold-valued semimartingales, we give results
on differentiation with respect to a parameter for second order,
Stratonovich and Itô stochastic differential equations
and identify the equation solved by the derivative processes.
In particular, we prove that both Stratonovich and Itô equations
differentiate like equations involving smooth paths
(for the Itô equation the tangent bundles must be
endowed with the complete lifts of the connections on the manifolds).

As applications, we prove that differentiation and antidevelopment
of C^{1} families of semimartingales commute,
and that a semimartingale with values in a tangent bundle is a martingale
for the complete lift of a connection
if and only if it is the derivative of a family of martingales
in the manifold.
In: J. Azema, M. Emery, M. Ledoux and M. Yor (Eds.)

* Séminaire de Probabilités * XXXII, 188-214,

Lecture Notes in Mathematics **1686**,

Springer: Berlin, 1998.

SpringerLink

The paper is available here:

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