Some aspects of optimal quantization and applications to finance

Abstract : This thesis is concerned with the study of optimal quantization and its applications. We deal with theoretical, algorithmic and numerical aspects. It is composed of five chapters. In the first section, we study the link between the stratification-based variance reduction and optimal quantization. In the case where the considered random variable is a Gaussian process, a simulation scheme with a linear cost is proposed for the conditional distribution of the process in a stratum. The second chapter is devoted to the numerical approximation of the Karhunen-Loève eigensystem of a Gaussian process with the so-called Nyström method. In the third section, we propose a new approach for the quantization of the solutions of SDE, whose convergence is investigated. These results lead to a new cubature scheme for the solutions of stochastic differential equations, which is developed in the fourth chapter, and which is tested with option pricing problems. In the last chapter, we present a new tree-based fast nearest neighbor search algorithm, based on the quantization of the empirical distribution of the considered data set.
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Theses
Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2011. English


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Submitted on : Tuesday, September 27, 2011 - 1:00:39 AM
Last modification on : Friday, September 30, 2011 - 5:10:46 PM
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Sylvain Corlay. Some aspects of optimal quantization and applications to finance. Mathematics [math]. Université Pierre et Marie Curie - Paris VI, 2011. English. <tel-00626445>

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