Contributions to multilevel splitting for rare events, and applications to air traffic

Damien Jacquemart 1, 2
1 ASPI - Applications of interacting particle systems to statistics
IRMAR - Institut de Recherche Mathématique de Rennes, Inria Rennes – Bretagne Atlantique
Abstract : The thesis deals with the design and mathematical analysis of reliable and accurate Monte Carlo methods in order to estimate the (very small) probability that a Markov process reaches a critical region of the state space before a deterministic final time. The underlying idea behind the multilevel splitting methods studied here is to design an embedded sequence of intermediate more and more critical regions, in such a way that reaching an intermediate region, given that the previous intermediate region has already been reached, is not so rare. In practice, trajectories are propagated, selected and replicated as soon as the next intermediate region is reached, and it is easy to accurately estimate the transition probability between two successive intermediate regions. The bias due to time discretization of the Markov process trajectories is corrected using perturbed intermediate regions as proposed by Gobet and Menozzi. An adaptive version would consist in the automatic design of the intermediate regions, using empirical quantiles. However, it is often difficult if not impossible to remember where (in which state) and when (at which time instant) did each successful trajectory reach the empirically defined intermediate region. The contribution of the thesis consists in using a first population of pilot trajectories to define the next threshold, in using a second population of trajectories to estimate the probability of exceeding this empirically defined threshold, and in iterating these two steps (definition of the next threshold, and evaluation of the transition probability) until the critical region is reached. The convergence of this adaptive two-step algorithm is studied in the asymptotic framework of a large number of trajectories. Ideally, the intermediate regions should be defined in terms of the spatial and temporal variables jointly (for example, as the set of states and times for which a scalar function of the state exceeds a time-dependent threshold). The alternate point of view proposed in the thesis is to keep intermediate regions as simple as possible, defined in terms of the spatial variable only, and to make sure that trajectories that manage to exceed a threshold at an early time instant are more replicated than trajectories that exceed the same threshold at a later time instant. The resulting algorithm combines importance sampling and multilevel splitting. Its preformance is evaluated in the asymptotic framework of a large number of trajectories, and in particular a central limit theorem is obtained for the relative approximation error.
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Damien Jacquemart. Contributions to multilevel splitting for rare events, and applications to air traffic. Probability [math.PR]. Université Rennes 1, 2014. English. ⟨NNT : 2014REN1S186⟩. ⟨tel-01111540v2⟩

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