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We prove a law of large numbers and a functional central limit theorem for multivariate Hawkes processes observed over a time interval [0, T ] in the limit T → ∞. We further exhibit the asymptotic behaviour of the covariation of the increments of the components of a multivariate Hawkes process, when the observations are imposed by a discrete scheme with(More)
We introduce and analyze numerical methods for the treatment of inverse problems , based on an adaptive wavelet Galerkin discretization. These methods combine the theoretical advantages of the wavelet-vaguelette decomposition (WVD) in terms of optimally adapting to the unknown smoothness of the solution, together with the numerical simplicity of Galerkin(More)
We study two nonlinear methods for statistical linear inverse problems when the operator is not known. The two constructions combine Galerkin regularization and wavelet thresholding. Their performances depend on the underlying structure of the operator, quantified by an index of sparsity. We prove their rate-optimality and adaptivity properties over Besov(More)
We model the growth of a cell population by a piecewise deter-ministic Markov branching tree. Each cell splits into two offsprings at a division rate B(x) that depends on its size x. The size of each cell grows exponentially in time, at a rate that varies for each individual. We show that the mean empirical measure of the model satisfies a(More)
Multifractal analysis of multiplicative random cascades is revis-ited within the framework of mixed asymptotics. In this new framework , statistics are estimated over a sample which size increases as the resolution scale (or the sampling period) becomes finer. This allows one to continuously interpolate between the situation where one studies a single(More)
We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative(More)
We study the problem of estimating the coefficients of a diffu-−1 is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is ill-posed: the minimax rates of convergence for Sobolev constraints and(More)
We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on linear self and mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the(More)
PURPOSE The goal of this work was to assess the effect of comorbidities on medical care use and costs among patients with partial seizure disorder who are also refractory to initial antiepileptic drug (AED) monotherapy. METHODS Retrospective data from the PharMetrics managed care claims database were collected for adult patients treated with AED(More)
Many organisms coordinate cell growth and division through size control mechanisms: cells must reach a critical size to trigger a cell cycle event. Bacterial division is often assumed to be controlled in this way, but experimental evidence to support this assumption is still lacking. Theoretical arguments show that size control is required to maintain size(More)