Sur quelques problèmes concernant le calcul de Malliavin et son application à la théorie du filtrage non linéaire

Abstract : The aim of this work is to some problems of Malliavin calculus and its application to nonlinear filtering theory. It is divided in five chapters and one appendix. In chapter I we present some generalities on the Malliavin calculus and the theory of nonlinear filtering. We give the main ideas of these two theories and we recall the classical results that we need in the following chapters. The aim of chapters II, III and V is to study diffusions with time dependent coefficients under different conditions. In chapter II we suppose that on a sub-manifold of codimension 1 we have an exponential degenerencence condition, weaker than the general Hörmander condition. In chapter III we study diffusions generated by an infinity of Brownian motions and in chapter V we consider diffusions on riemannian manifolds. In the latter chapters, we establish a Malliavin calculus permitting to prove the existence of a smooth density for these diffusions. Furthermore, the results of Malliavin calculus are used to show that the unnormalized filter associated with a nonlinear filtering system, which admits the diffusion studied in the previous chapter as signal process, has a smooth density. In chapter IV we consider a nonlinear filtering system with an observation process depending on a white noise of order E. We define a suboptimal filter of finite dimension and we show that the error commited by replacing the filter by the suboptimal filter is of order E. In the appendix we derive the filtering equations associated with a nonlinear filering system with an infinite dimensional signal process and we give a robust form of the Zakai equation in the case of non correlated noises
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Jean Schiltz. Sur quelques problèmes concernant le calcul de Malliavin et son application à la théorie du filtrage non linéaire. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 1997. Français. ⟨NNT : 1997METZ026S⟩. ⟨tel-01777211⟩

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