Estimation of the Brownian dimension of a continuous It\^{o} process

In this paper, we consider a $d$-dimensional continuous It\^{o} process which is observed at $n$ regularly spaced times on a given time interval $[0,T]$. This process is driven by a multidimensional Wiener process and our aim is to provide asymptotic statistical procedures which give the minimal dimension of the driving Wiener process, which is between 0 (a pure drift) and $d$. We exhibit several different procedures, all similar to asymptotic testing hypotheses.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6190 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons"

In this note, we clarify the well-posedness of the limit equations to the mean-field $N$-neuron models proposed in Baladron et al. and we prove the associated propagation of chaos property. We also complete the modeling issue in Baladron et al. by discussing the well-posedness of the stochastic differential equations which govern the behavior of the ion channels and the amount of available neurotransmitters

Mean--field limit of a particle approximation of the one-dimensional parabolic--parabolic Keller-Segel model without smoothing

In this work, we prove the well--posedness of a singularly interacting stochastic particle system and we establish propagation of chaos result towards the one-dimensional parabolic-parabolic Keller-Segel model

Liquidity costs: a new numerical methodology and an empirical study

We consider rate swaps which pay a fixed rate against a floating rate in presence of bid-ask spread costs. Even for simple models of bid-ask spread costs, there is no explicit strategy optimizing an expected function of the hedging error. We here propose an efficient algorithm based on the stochastic gradient method to compute an approximate optimal strategy without solving a stochastic control problem. We validate our algorithm by numerical experiments. We also develop several variants of the algorithm and discuss their performances in terms of the numerical parameters and the liquidity cost

Mean-field limit of a stochastic particle system smoothly interacting through threshold hitting-times and applications to neural networks with dendritic component

In this article we study the convergence of a stochastic particle system that interacts through threshold hitting times towards a novel equation of McKean-Vlasov type. The particle system is motivated by an original model for the behavior of a network of neurons, in which a classical noisy integrate-and-fire model is coupled with a cable equation to describe the dendritic structure of each neuron

Approximation et simulation de modèles stochastiques en Mécanique

Nous présenterons quelques idées générales sur les problèmes d'approximation numérique de modèles stochastiques en mécanique, ainsi que quelques problèmes encore ouverts que posent spécifiquement certains de ces modèles. En particulier seront abordées des questions telles que les différentes méthodes de discrétisation et leurs vitesses de convergence, les approximations de moments et de régimes stationnaires

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Simulation and numerical analysis of stochastic differential systems : a review

Projet MEFISTOWe present methods of approximating quantities related to the solutions of stochastic differential systems based on the simulation of time-discrete Markov chains. The motivations come from random mechanics and the numerical integration of certains deterministic P.D.E.'s by probabilistic algorithms. We state theoretical results concerning the rates of convergence of these methods. We give results of numerical tests, and we describe an application of this approach to an engineering problem (the study of stability of the motion of a helicopter blade)