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Etude des taux d'interet long terme Analyse stochastique des processus ponctuels determinantaux

Abstract : The first part of this thesis concerns a nancial point of view of the study of long term interest rates. We seek an alternative to classical interest rates models for longer maturities (15 years and more). Our work is inspired by the work of economists, but takes into account the existence of a (complete) financial market. We show that classical expected utility maximization techniques lead to the Ramsey Rule, linking the yield curve and marginal utility from consumption. We extend the Ramsey Rule to the case of an incomplete financial market and examine how the yield curve is modied. It is then possible to consider the case where there is incertainty on a parameter of the model, then to extend these results to the case of dynamic utility functions, where the yield curve depends on level of wealth in the economy. The other main result we present is a new way of considering the consumption, as a quantity of supplies that the investor puts aside and uses in case of a default event. Then the expected utility maximization from consumption and terminal wealth can be interpreted as a problem of maximization of expected utility from terminal wealth with a random horizon. The topic of the second part of this thesis is the stochastic analysis of determinantal point processes. Determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent. Their atoms exhibit mutual attraction or repulsion, thus these processes are very far from the uncorrelated situation encountered in Poisson models. We establish a quasi-invariance result : we show that if atoms locations are perturbed along a vector eld, the resulting process is still a determinantal (respectively permanental) process, the law of which is absolutely continuous with respect to the original distribution. Based on this formula, following Bismut approach of Malliavin calculus, we then give an integration by parts formula.
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Submitted on : Thursday, March 3, 2011 - 9:37:22 PM
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Camilier Isabelle. Etude des taux d'interet long terme Analyse stochastique des processus ponctuels determinantaux. Probabilités [math.PR]. Ecole Polytechnique X, 2010. Français. ⟨pastel-00573437⟩

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