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APPROXIMATION DE PROCESSUS DE DIFFUSION À COEFFICIENTS DISCONTINUS EN DIMENSION UN
ET APPLICATIONS À LA SIMULATION

Pierre Etore 1, 2
2 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : In this thesis numerical schemes for processes /X/ generated by
operators with discontinuous coefficients are studied.
A first scheme for the one-dimensional case uses Differential
Stochastic Equations with Local Time. Indeed, in dimension one, the
processes /X/ are solutions of such equations. We construct a grid on
the real line, that is transformed by a proper bijection in a uniform
grid of step /h/. This bijection also transforms /X/ in some process
/Y/, that behaves locally like a Skew Brownian Motion (SBM). We know
the transition probabilities of the SBM on a uniform grid, and the
average time it spends on each of its cells. A random walk can then
be built, that converges to /X/ in square root of /h/. A second
scheme, that is more general, is proposed still for the dimension
one. A non uniform grid on the real line is given, whose cells have a
size proportional to /h/. Both the transition probabilities of /X/ on
this grid, and the average time it spends on each of its cells, can
be related to the solutions of proper elliptic PDE problems, using
the Feynman-Kac formula. A time-space random walk can then be built,
that converges to /X/ again in square root of /h/. Next some
directions to adapt this approach to the two-dimensional case are
given. Finally numerical exemples illustrate the studied schemes.
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Pierre Etore. APPROXIMATION DE PROCESSUS DE DIFFUSION À COEFFICIENTS DISCONTINUS EN DIMENSION UN
ET APPLICATIONS À LA SIMULATION. Mathématiques [math]. Université Henri Poincaré - Nancy 1, 2006. Français. ⟨NNT : 2006NAN10160⟩. ⟨tel-01746567v2⟩

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