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Summary. We study the approximation problem of 1Ef(XT) by IEf(X~), where (Xt) is the solution of a stochastic differential equation, (Xt n) is defined by the Euler discretization scheme with step T/n, and f is a given function. For smooth f's, Yalay and Yubaro have shown that the error ]Ef(XT) -f(X~) can be expanded in powers of l/n, which permits to constntct Romberg extrapolation procedures to accelerate the convergence rate. Here, we prove that the expansion exists also when f is only supposed measurable and bounded, under an additional nondegeneracy condition of H6rmander type for the infinitesimal generator of (Xt) : to obtain this result, we use the stochastic variations calculus. In the second part of this work, we will consider the density of the law of X~ and compare it to the density of the law of XT.

Abstract. In this paper we introduce and analyze a stochastic particle method for the McKean-Vlasov and the Burgers equation; the construction and error analysis are based upon the theory of the propagation of chaos for interacting particle systems.Our objective is three-fold. First, we consider a McKean-Vlasov equation in [0, T ] × R with sufficiently smooth kernels, and the PDEs giving the distribution function and the density of the measure µt, the solution to the McKean-Vlasov equation. The simulation of the stochastic system with N particles provides a discrete measure which approximates µ k∆t for each time k∆t (where ∆t is a discretization step of the time interval [0, T ]). An integration (resp. smoothing) of this discrete measure provides approximations of the distribution function (resp. density) of µ k∆t . We show that the convergencetive distribution function at time T , and of order O ε 2 + 1 εof the density at time T (Ω is the underlying probability space, ε is a smoothing parameter). Our second objective is to show that our particle method can be modified to solve the Burgers equation with a nonmonotonic initial condition, without modifying the convergence rateThis part extends earlier work of ours, where we have limited ourselves to monotonic initial conditions. Finally, we present numerical experiments which confirm our theoretical estimates and illustrate the numerical efficiency of the method when the viscosity coefficient is very small.

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