 
Summary: Dierentiable and analytic families of continuous
martingales in manifolds with connection
Marc Arnaudon
Institut de Recherche MatheÂ matique AvanceÂ e, UniversiteÂ Louis Pasteur et CNRS, 7, rue ReneÂ
Descartes, F67084 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
