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Laure Coutin

Publications and source records attributed to Laure Coutin.

At least 19 recordsLinked to original sources
arXiv2026

Fixed-Point Estimation of the Drift Parameter in Stochastic Differential Equations Driven by Rough Multiplicative Fractional Noise

We investigate the problem of estimating the drift parameter from $N$ independent copies of the solution of a stochastic differential equation driven by a multiplicative fractional Brownian noise with Hurst parameter $H\in (1/3,1)$. Building on a least-squares-type object involving the Skorokhod integral, a key challenge consists in approximating this unobservable quantity with a computable fixed-point estimator, which requires addressing the correction induced by replacing the Skorokhod integral with its pathwise counterpart. To this end, a crucial technical contribution of this work is the reformulation of the Malliavin derivative of the process in a way that does not depend explicitly on the driving noise, enabling control of the approximation error in the multiplicative setting. For the case $H\in (1/3,1/2]$, we further exploit results on two-dimensional Young integrals to manage the more intricate correction term that appears. As a result, we establish the well-posedness of a fixed-point estimator for any $H\in (1/3,1)$, together with both an asymptotic confidence interval and a non-asymptotic risk bound. Finally, a numerical study illustrates the good practical performance of the proposed estimator.

math.ST↗
arXiv2025

Quantification of limit theorems for Hawkes processes

In this article, we fill a gap in the literature regarding quantitative functional central limit theorems (qfCLT) for Hawkes processes by providing an upper bound for the convergence of a nearly unstable Hawkes process toward a Cox-Ingersoll-Ross (CIR) process. Note that in this case no speed of convergence has been established even for one-dimensional marginals; we provide in this paper a control in terms of a supremum norm in $2$-Wasserstein distance. To do so, we make use of the so-called Poisson imbedding representation and provide a qfCLT formulation in terms of a Brownian sheet. Incidentally, we construct an optimal coupling between a rescaled bi-dimensional Poisson random measure and a Brownian sheet with respect to the $2$-Wasserstein distance and analyze the asymptotic quality of this coupling in detail.

math.PR↗
arXiv2025

Normal approximation of Functionals of Point Processes: Application to Hawkes Processes

In this paper, we derive an explicit upper bound for the Wasserstein distance between a functional of point processes and a Gaussian distribution. Using Stein's method in conjunction with Malliavin's calculus and the Poisson embedding representation, our result applies to a variety of point processes including discrete and continuous Hawkes processes. In particular, we establish an explicit convergence rate for stable continuous non-linear Hawkes processes and for discrete Hawkes processes. Finally, we obtain an upper bound in the context of nearly unstable Hawkes processes.

math.PR↗
arXiv2025

Weak uniqueness for the PDE governing the joint law of a diffusion and its running supremum

In a previous work [8], it was shown that the joint law of a diffusion process and the running supremum of its first component is absolutely continuous, and that its density satisfies a non standard weak partial differential equation (PDE). In this paper, we establish the uniqueness of the solution to this PDE, providing a more complete understanding of the system's behavior and further validating the approach introduced in [8].

math.AP↗
arXiv2024

Invertibility of functionals of the Poisson process and applications

Following previous investigations by {Ü}st{ü}nel [22] about the invertibility of some transformations on the Wiener space, we find some entropic conditions under which a random change of time is invertible on the Poisson space. As a consequence, we provide a new construction of Hawkes processes. We also establish a new variational representation of the entropy.

math.PR↗
arXiv2024

Functional approximation of the marked Hawkes risk process

The marked Hawkes risk process is a compound point process for which the occurrence and amplitude of past events impact the future. Thanks to its autoregressive properties, it found applications in various fields such as neuosciences, social networks and insurance.Since data in real life is acquired over a discrete time grid, we propose a strong discrete-time approximation of the continuous-time Hawkes risk process obtained be embedding from the same Poisson measure. We then prove trajectorial convergence results both in some fractional Sobolev spaces and in the Skorokhod space, hence extending the theorems proven in the literature. We also provide upper bounds on the convergence speed with explicit dependence on the size of the discretisation step, the time horizon and the regularity of the kernel.

math.PR↗
arXiv2024

Rate of Convergence in the Functional Central Limit Theorem for Stable Processes

In this article, we quantify the functional convergence of the rescaled random walk with heavy tails to a stable process.This generalizes the Generalized Central Limit Theorem for stable random variables infinite dimension. We show that provided we have a control between the randomwalk or the limiting stable process and their respective affine interpolation, we canlift the rate of convergence obtained for multivariate distributions to a rateof convergence in some functional spaces.

math.PR↗
arXiv2023

PDE for the joint law of the pair of a continuous diffusion and its running maximum

Let X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any t > 0, the density (with respect to the d + 1-dimensional Lebesgue measure) of the pair (Mt, Xt) is a weak solution of a Fokker-Planck partial differential equation on the closed set {(m, x) $\in$ R d+1, m $\ge$ x 1}, using an integral expansion of this density.

math.PR↗
arXiv2022

On a Set-Valued Young Integral with Applications to Differential Inclusions

We present a new Aumann-like integral for a Hölder multifunction with respect to a Hölder signal, based on the Young integral of a particular set of Hölder selections. This restricted Aumann integral has continuity properties that allow for numerical approximation as well as an existence theorem for an abstract stochastic differential inclusion. This is applied to concrete examples of first order and second order stochastic differential inclusions directed by fractional Brownian motion.

math.PR↗
arXiv2021

Diffusive limits of Lipschitz functionals of Poisson measures

Continuous Time Markov Chains, Hawkes processes and many other interesting processes can be described as solution of stochastic differential equations driven by Poisson measures. Previous works, using the Stein's method, give the convergence rate of a sequence of renormalized Poisson measures towards the Brownian motion in several distances, constructed on the model of the Kantorovitch-Rubinstein (or Wasserstein-1) distance. We show that many operations (like time change, convolution) on continuous functions are Lipschitz continuous to extend these quantified convergences to diffuse limits of Markov processes and long-time behavior of Hawkes processes.

math.PR↗
arXiv2021

Rates of convergence for the number of zeros of random trigonometric polynomials

In this paper, we quantify the rate of convergence between the distribution of number of zeros of random trigonometric polynomials (RTP) with i.i.d. centered random coefficients and the number of zeros of a stationary centered Gaussian process G, whose covariance function is given by the sinc function. First, we find the convergence of the RTP towards G in the Wasserstein-1 distance, which in turn is a consequence of Donsker Theorem. Then, we use this result to derive the rate of convergence between their respective number of zeros. Since the number of real zeros of the RTP is not a continuous function, we use the Kac-Rice formula to express it as the limit of an integral and, in this way, we approximate it by locally Lipschitz continuous functions.

math.PR↗
arXiv2018

Stein's method for rough paths

The original Donsker theorem says that a standard random walk converges in distribution to a Brownian motion in the space of continuous functions. It has recently been extended to enriched random walks and enriched Brownian motion. We use the Stein-Dirichlet method to precise the rate of this convergence in the topology of fractional Sobolev spaces.

math.PR↗
arXiv2018

On a Fractional Stochastic Hodgkin-Huxley Model

The model studied in this paper is a stochastic extension of the so-called neuron model introduced by Hodgkin and Huxley. In the sense of rough paths, the model is perturbed by a multiplicative noise driven by a fractional Brownian motion, with a vector field satisfying the viability condition of Coutin and Marie for $\mathbb R\times [0,1]^3$. An application to the modeling of the membrane potential of nerve fibers damaged by a neuropathy is provided.

math.PR↗
arXiv2017

Sensitivity of rough differential equations: an approach through the Omega lemma

The It{ô} map assigns the solution of a Rough Differential Equation, a generalization of an Ordinary Differential Equation driven by an irregular path, when existence and uniqueness hold. By studying how a path is transformed through the vector field which is integrated, we prove that the It{ô} map is H{ö}lder or Lipschitz continuous with respect to all its parameters. This result unifies and weaken the hypotheses of the regularity results already established in the literature.

math.PR↗
arXiv2017

Volterra differential equations with singular kernels

Motivated by the potential applications to the fractional Brownianmotion, we study Volterra stochasticdifferential of the form~:\begin{equation}X\_t = x+ \int\_0^tK(t,s)b(s,X\_s)ds + \int\_0^tK(t,s) σ(s,X\_s)\,dB\_s ,\tag{E} \label{eq:sdefbm}\end{equation}where $(B\_s, \, s\in [0,1])$ is a one-dimensional standard Brownianmotion and $(K(t,s), \, t,s \in [0,1])$ is a deterministic kernelwhose properties will be precised below but for which we don't assumeany boundedness property.

math.PR↗
arXiv2016

Invariance for Rough Differential Equations

In 1990, in Itô's stochastic calculus framework, Aubin and Da Prato established a necessary and sufficient condition of invariance of a nonempty compact or convex subset $C$ of $\mathbb R^d$ ($d\in\mathbb N^*$) for stochastic differential equations (SDE) driven by a Brownian motion. In Lyons rough paths framework, this paper deals with an extension of Aubin and Da Prato's results to rough differential equations. A comparison theorem is provided, and the special case of differential equations driven by a fractional Brownian motion is detailed.

math.PR↗
arXiv2016

Joint law of the hitting time, overshoot and undershoot for a Lévy process

Let be $(X_t, t\geq 0)$ be a Lévy process which is the sum of a Brownian motion with drift and a compound Poisson process. We consider the first passage time $τ_x$ at a fixed level $x>0$ by $(X_t, t\geq 0)$ and $K_x:= X_{τ_x}-x$ the overshoot and $L_x:= x-X_{τ_x^-}$ the undershoot. We first study the regularity of the density of the first passage time. Secondly, we calculate the joint law of $(τ_x, K_x, L_x).$

math.PR↗