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Martingales sur les variétés de valeur terminale donnée

Abstract : Defined several decades ago, martingales in manifolds are very canonical objects. About these objects very simple questions are still unresolved. For instance, given a random variable with values in a complete manifold and a continuous filtration (one with respect to which all real-valued martingales admit a continuous version), does there exist a continuous martingale in the manifold with terminal value given by this random variable ? What about semimartingales with prescribed drift and terminal value ? The main aim of this thesis is to provide answers to these questions. Under convex geometry assumption, answers are given in the articles of Kendall (1990), Picard (1991), Picard (1994), Darling (1995) or Arnaudon (1997). The case of semimartingales was widely treated by Blache (2004). The martingales in the manifolds make it possible to define the barycenters associated to a filtration, which are sometimes simpler to compute than the usual barycenters or averages, and which have an associative property. They are strongly related to control theory, stochastic optimization, and backward stochastic differential equations (BSDEs). Solving the problem with geometric arguments also gives tools for solving multidimensional quadratic EDSRs.During the thesis, two methods have been used for studying the problem of existence of a martingale with prescribed terminal value. The first one is based on a stochastic algorithm. The random variable that we try to reach will be deformed into a C¹-family ξ(a), and we deal with the following newer problem: does there exist a martingale X(a) with terminal value ξ(a)? A shooting method, using the same kind of principle as the deterministic geodesic shooting, will be used with respect to a parameter a towards ξ(a). The second one is the resolution of a multidimensional quadratic BSDE. The aim of this part will be to adapt to the multidimensional framework a recent strategy developed by Briand and Elie (2013) to treat multidimensional quadratic BSDEs. This new approach makes it possible to rediscover the results obtained by different methods. Beyond the unification, this new approach paves the way for potential future works.
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Submitted on : Friday, October 19, 2018 - 10:01:07 AM
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Jonathan Harter. Martingales sur les variétés de valeur terminale donnée. Probabilités [math.PR]. Université de Bordeaux, 2018. Français. ⟨NNT : 2018BORD0074⟩. ⟨tel-01898942⟩



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