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We provide a probabilistic representations of the solution of some semilinear hyperbolic and high-order PDEs based on branching diffusions. These representations pave the way for a Monte-Carlo approximation of the solution, thus bypassing the curse of dimensionality. We illustrate the numerical implications in the context of some popular PDEs in physics such as nonlinear Klein-Gordon equation, a simplified scalar version of the Yang-Mills equation, a fourth-order nonlinear beam equation and the Gross-Pitaevskii PDE as an example of nonlinear Schrödinger equations.

International audienceIn this paper, we propose a new approach to study the stability of the optimal filter w.r.t. its initial condition in introducing a "robust filter," which approximates the optimal filter uniformly in time. This approach allows us to prove, in some cases, when the signal is nonergodic, the stability of the optimal filter in mean over the observations and the uniform convergence in mean over the observations of a special interacting particle filter to the optimal filter

We study the stability of the optimal filter w.r.t. its initial condition and w.r.t. the model for the hidden state and the observations in a general hidden Markov model, using the Hilbert projective metric. These stability results are then used to prove, under some mixing assumption, the uniform convergence to the optimal filter of several particle filters, such as the interacting particle filter and some other original particle filters. IntroductionThe stability of the optimal filter has become recently an active research area. Ocone and Pardoux have proved in [26] that the filter forgets its initial condition in the L p sense, without stating any rate of convergence. Recently, a new approach has been proposed using the Hilbert projective metric. This metric allows to get rid of the normalization constant in the Bayes formula, and reduces the problem to studying the linear equation satisfied by the unnormalized optimal filter. Using the Hilbert metric, stability results w.r.t. the initial condition have been proved by Atar and Zeitouni in [4], and some stability result w.r.t. the model have been proved by Le Gland and Mevel in [19,20] Independently, Del Moral and Guionnet have adopted in [9], for the same class of HMM, another approach based on semi-group techniques and on the Dobrushin ergodic coefficient, to derive stability results w.r.t. the initial condition, which are used to prove uniform convergence of the interacting particle system (IPS) approximation to the optimal predictor. New approaches have been proposed recently, to prove the stability of the optimal filter w.r.t. its initial condition, in the case of a noncompact state space, see e.g. Atar [1], Atar, Viens and Zeitouni [2], Budhiraja and Ocone [6,7].In this article, we use the approach based on the Hilbert metric to study the asymptotic behavior of the optimal filter, and to prove as in [9] the uniform convergence of several particle filters, such as the interacting particle filter (IPF) and some other original particle filters.A common assumption to prove stability results, see e.g. in [9, Theorem 2.4], is that the Markov transition kernels are mixing, which implies that the hidden state sequence is ergodic. Our results are obtained under the assumption that the nonnegative kernels describing the evolution of the unnormalized optimal filter, and incorporating simultaneously the Markov transition kernels and the likelihood functions, are mixing. This is a weaker assumption, see Proposition 3.9, which allows to consider some cases, similar to the case studied in [6], where the hidden state sequence is not ergodic, see Example 3.10. This point of view is further developped by Le Gland and Oudjane in [22] and by Oudjane and Rubenthaler in [28]. Our main contribution is to study also the stability of the optimal filter w.r.t. the model, when the local error is propagated by mixing kernels, and can be estimated in the Hilbert metric, in the total variation norm, or in a weaker distance suitable for random probability distributions.AMS 1991 subject...

An important part of the Smart Grid literature on residential Demand Response deals with game-theoretic consumption models. Among those papers, the hourly billing model is of special interest as an intuitive and fair mechanism. We focus on this model and answer to several theoretical and practical questions. First, we prove the uniqueness of the consumption profile corresponding to the Nash equilibrium, and we analyze its efficiency by providing a bound on the Price of Anarchy. Next, we address the computational issue of the equilibrium profile by providing two algorithms: the cycling best response dynamics and a projected gradient descent method, and by giving an upper bound on their convergence rate to the equilibrium. Last, we simulate this demand response framework in a stochastic environment where the parameters depend on forecasts. We show numerically the relevance of an online demand response procedure, which reduces the impact of inaccurate forecasts.