Authors/Creators UNSPECIFIED (2007) Sequentials Monte Carlo methods in stochastic control. PhD thesis CMAP, CMAP, EP/X p.272.

Alternative Locations: http://www.imprimerie.polytechnique.fr/Theses/Files/Labart.pdf

Abstract

My thesis deals with two different themes of numerical probabilities and their financial applications: the first one is the approximation and the simulation of backward stochastic differential equations (BSDE). The second one concerns the American options and tackle their pricing using domain optimization and boundary perturbations.
The first part of my thesis analyzes the convergence of n, the time discretization of markovian BSDE (Y,Z). We establish a Taylor expansion for the error on (Y,Z): it strongly depends on the error on X. Had we been able to perfectly simulate X, we would have obtained an error on (Y,Z) of order 1/n.
The second part of my thesis is devoted to solving BSDE using Picard's procedure and a sequential Monte Carlo method. We prove that our algorithm converges geometrically fast. Moreover, the accuracy is independent (at the first order) of the number of Monte Carlo simulations.
The last part of my thesis presents basic results on the pricing of American options using an optimization of the exercise region. The keystone of such an approach is the ability of computing a gradient w.r.t the boundary. In continuous time, this work has been done by Costantini et al (2006). This thesis deals with the discrete time.

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