Differentiable and analytic families of continuous martingales in manifolds with connection

• 成果类型: Article
• 署名作者: Arnaudon, M
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050108
• 发表日期: 1997
• 页码: 219-257
• 关键词: convexity

摘要

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 bi-continuous in t and a. We consider a filtered probability space (Omega, (F-t)(0 less than or equal to t less than or equal to 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) is an element of 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 F-1-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.