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Conditionnement de processus markoviens

Abstract : The aim of this work is to describe the conditional law of a multidimensional Markov process knowing linear combinations of its coordinates at given times. We are looking for a process of the same kind, whose law is equivalent to the targeted one.The diffusion processes represent the most studied process class in this thesis. We first use techniques of enlargement of filtrations (Jacod 1985) in order to determine the parameters of the conditional stochastic differential equation (SDE). This theoretical result does not allow direct simulation of conditional paths because of its drift. Indeed, this one depends on the transition density functions of the initial diffusion, and those functions are generally unknown. That is why, we provide an alternative, inspired by a Delyon & Hu(2006), consisting in proposing a SDE, whose law is equivalent to the targeted conditional distribution. Moreover, this SDE possesses explicit coefficents, and is easy to simulate thanks to an Euler scheme. Same kind of results are also established in the case of realpoint processes.An application in collaboration with Anne Cuzol and Etienne Mémin from the INRIA is also presented. It consists in applying the precedent result to a model, whose construction is based on 2D-Navier-Stokes equations.
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Submitted on : Tuesday, September 18, 2012 - 1:18:06 PM
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  • HAL Id : tel-00733301, version 1

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Jean-Louis Marchand. Conditionnement de processus markoviens. Probabilités [math.PR]. Université Rennes 1, 2012. Français. ⟨NNT : 2012REN1S048⟩. ⟨tel-00733301⟩

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