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Register a new userThe Malliavin-Stein method for Hawkes functionals
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In this paper, following Nourdin-Peccatis methodology, we combine the Malliavin calculus and Steins method to provide general bounds on the Wasserstein distance between functionals of a compound Hawkes process and a given Gaussian density. To achieve this, we rely on the Poisson embedding representation of an Hawkes process to provide a Malliavin calculus for the Hawkes processes, and more generally for compound Hawkes processes. As an application, we close a gap in the literature by providing the first Berry-Esseen bounds associated to Central Limit Theorems for the compound Hawkes process.
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In this paper, we provide upper bounds on the d2 distance between a large class of functionals of a multivariate compound Hawkes process and a given Gaussian vector. This is proven using Malliavins calculus defined on an underlying Poisson embedding. The upper bound is then used to infer the speed of convergence of Central Limit Theorems for the multivariate compound Hawkes process with exponential kernels as the observation time T goes to infinity.
In this paper, we construct a Malliavin derivative for functionals of square-integrable Levy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representation for the maximum of a Levy process.
We combine Steins method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion.
We consider the nonparametric functional estimation of the drift of a Gaussian process via minimax and Bayes estimators. In this context, we construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and superharmonic functionals on Gaussian space. Our results are illustrated by numerical simulations and extend the construction of James--Stein type estimators for Gaussian processes by Berger and Wolpert [J. Multivariate Anal. 13 (1983) 401--424].
Suppose that ${u(t,, x)}_{t >0, x inmathbb{R}^d}$ is the solution to a $d$-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalangs condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} int_{[0,N]^d} g(u(t,,x)), mathrm{d} x$, as $Nrightarrowinfty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Steins method for normal approximations. Our results include a central limit theorem for the {it Hopf-Cole} solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.
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