Showing papers by "Nicolas Fournier published in 2011"

Simulation and approximation of Lévy-driven stochastic differential equations

Abstract: We consider the approximate Euler scheme for Levy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Levy measure of the driving process behaves like |z |−1−α dz near 0 , for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα . For a similar error when neglecting the small jumps, see [S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Levy process. Stochastic Process. Appl. 103 (2003) 311–349], the computational cost is of order n α /(2−α ) , which is huge when α is close to 2. In the same spirit, we study the problem of the approximation of a Levy-driven S.D.E. by a Brownian S.D.E. when the Levy process has no large jumps. Our results rely on some results of [E. Rio, Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincare Probab. Stat. 45 (2009) 802–817] about the central limit theorem, in the spirit of the famous paper by Komlos-Major-Tsunady [J. Komlos, P. Major and G. Tusnady, An approximation of partial sums of independent rvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111–131].

Regularization properties of the 2D homogeneous Boltzmann equation without cutoff

Abstract: We consider the 2-dimensional spatially homogeneous Boltzmann equation for hard potentials. We assume that the initial condition is a probability measure that has some exponential moments and is not a Dirac mass. We prove some regularization properties: for a class of very hard potentials, the solution instantaneously belongs to H r , for some $${r\in (-1,2)}$$ depending on the parameters of the equation. Our proof relies on the use of a well-suited Malliavin calculus for jump processes.

One-dimensional general forest fire processes

Abstract: We consider the one-dimensional generalized forest fire process: at each site of $\zz$, seeds and matches fall according some i.i.d. stationary renewal processes. When a seed falls on an empty site, a tree grows immediately. When a match falls on an occupied site, a fire starts and destroys immediately the corresponding connected component of occupied sites. Under some quite reasonable assumptions on the renewal processes, we show that when matches become less and less frequent, the process converges, with a correct normalization, to a limit forest fire model. According to the nature of the renewal processes governing seeds, there are four possible limit forest fire models. The four limit processes can be perfectly simulated. This study generalizes consequently a previous result of the authors where seeds and matches were assumed to fall according to Poisson processes.

Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff

Abstract: The subject of this article is the Kac equation without cutoff. We first show that in the asymptotic of grazing collisions, the Kac equation can be approximated by a Fokker-Planck equation. The convergence is uniform in time and we give an explicit rate of convergence. Next, we replace the small collisions by a small diffusion term in order to approximate the solution of the Kac equation and study the resulting error. We finally build a system of stochastic particles undergoing collisions and diffusion, that we can easily simulate, which approximates the solution of the Kac equation without cutoff. We give some estimates on the rate of convergence.

Stability of the stochastic heat equation in $L^1([0,1])$

Abstract: We consider the white-noise driven stochastic heat equation on $[0,1]$ with Lipschitz-continuous drift and diffusion coefficients. We derive an inequality for the $L^1([0,1])$-norm of the difference between two solutions. Using some martingale arguments, we show that this inequality provides some estimates which allow us to study the stability of the solution with respect the initial condition, the uniqueness of the possible invariant distribution and the asymptotic confluence of solutions.