Summary.
We prove that the derivative of a differentiable family X t (a) of continuous martingales in a manifold M is a martingale in the tangent space for the complete lift of the connection in M, provided that the derivative is bicontinuous in t and a. We consider a filtered probability space (Ω,(ℱ t )0≤ t ≤1, ℙ) such that all the real martingales have a continuous version, and a manifold M endowed with an analytic connection and such that the complexification of M has strong convex geometry. We prove that, given an analytic family a↦L(a) of random variable with values in M and such that L(0)≡x 0∈M, there exists an analytic family a↦X(a) of continuous martingales such that X 1(a)=L(a). For this, we investigate the convexity of the tangent spaces T ( n ) M, and we prove that any continuous martingale in any manifold can be uniformly approximated by a discrete martingale up to a stopping time T such that ℙ(T<1) is arbitrarily small. We use this construction of families of martingales in complex analytic manifolds to prove that every ℱ1-measurable random variable with values in a compact convex set V with convex geometry in a manifold with a C 1 connection is reachable by a V-valued martingale.
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Received: 14 March 1996/In revised form: 12 November 1996
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Arnaudon, M. Differentiable and analytic families of continuous martingales in manifolds with connection. Probab Theory Relat Fields 108, 219–257 (1997). https://doi.org/10.1007/s004400050108
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DOI: https://doi.org/10.1007/s004400050108