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Consider a problem of Markovian trajectories of particles for which you are trying to estimate the probability of a event.
Under the assumption that you can represent this event as the last event of a nested sequence of events, it is possible to design a splitting algorithm to estimate the probability of the last event in an efficient way. Moreover you can obtain a sequence of trajectories which realize this particular event, giving access to statistical representation of quantities conditionally to realize the event.
In this talk I will present the "Adaptive Multilevel Splitting" algorithm and its application to various toy models. I will explain why it creates an unbiased estimator of a probability, and I will give results obtained from numerical simulations.
Consider a problem of Markovian trajectories of particles for which you are trying to estimate the probability of a event.
Under the assumption that you can represent this event as the last event of a nested sequence of events, it is possible to design a splitting algorithm to estimate the probability of the last event in an efficient way. Moreover you can obtain a sequence of trajectories which realize this particular event, giving access to ...

60J22 ; 65C35 ; 65C05 ; 65C40

We propose a novel projection-based particle method for solving the McKean-Vlasov stochastic differential equations. Our approach is based on a projection-type estimation of the marginal density of the solution in each time step.
The projection-based particle method leads in many situation to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms.
We derive strong convergence rates and rates of density estimation. The convergence analysis in the case of linearly growing coefficients turns out to be rather challenging and requires some new type of averaging technique.
This case is exemplified by explicit solutions to a class of McKean-Vlasov equations with affine drift.
The performance of the proposed algorithm is illustrated by several numerical examples.
We propose a novel projection-based particle method for solving the McKean-Vlasov stochastic differential equations. Our approach is based on a projection-type estimation of the marginal density of the solution in each time step.
The projection-based particle method leads in many situation to a significant reduction of numerical complexity compared to the widely used kernel density estimation algorithms.
We derive strong convergence rates and ...

65C30 ; 65C35

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