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Dierentiable and analytic families of continuous martingales in manifolds with connection

Summary: Dierentiable and analytic families of continuous
martingales in manifolds with connection
Marc Arnaudon
Institut de Recherche Mathe matique Avance e, Universite Louis Pasteur et CNRS, 7, rue ReneÂ
Descartes, F-67084 Strasbourg Cedex, France
Received: 14 March 1996/In revised form: 12 November 1996
Summary. We prove that the derivative of a dierentiable family t of
continuous martingales in a manifold w is a martingale in the tangent space for
the complete lift of the connection in w, provided that the derivative is bi-
continuous in t and . We consider a ®ltered probability space XY Ft0 t 1Y P
such that all the real martingales have a continuous version, and a manifold w
endowed with an analytic connection and such that the complexi®cation of w
has strong convex geometry. We prove that, given an analytic family U3 v
of random variable with values in w and such that v0 x0 P w, there exists
an analytic family U3 of continuous martingales such that 1 v.
For this, we investigate the convexity of the tangent spaces n
w, and we prove
that any continuous martingale in any manifold can be uniformly approxi-
mated by a discrete martingale up to a stopping time such that P ` 1 is
arbitrarily small. We use this construction of families of martingales in complex


Source: Arnaudon, Marc - Département de mathématiques, Université de Poitiers


Collections: Mathematics