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Nicolas Marie

Publications and source records attributed to Nicolas Marie.

At least 37 records · Page 2Linked to original sources
arXiv2021

Nonparametric Estimation for I.I.D. Paths of Fractional SDE

This paper deals with nonparametric estimators of the drift function $b$ computed from independent continuous observations, on a compact time interval, of the solution of a stochastic differential equation driven by the fractional Brownian motion (fSDE). First, a risk bound is established on a Skorokhod's integral based least squares oracle $\widehat b$ of $b$. Thanks to the relationship between the solution of the fSDE and its derivative with respect to the initial condition, a risk bound is deduced on a calculable approximation of $\widehat b$. Another bound is directly established on an estimator of $b'$ for comparison. The consistency and rates of convergence are established for these estimators in the case of the compactly supported trigonometric basis or the $\mathbb R$-supported Hermite basis.

math.ST↗
arXiv2021

On a Nadaraya-Watson Estimator with Two Bandwidths

In a regression model, we write the Nadaraya-Watson estimator of the regression function as the quotient of two kernel estimators, and propose a bandwidth selection method for both the numerator and the denominator. We prove risk bounds for both data driven estimators and for the resulting ratio. The simulation study confirms that both estimators have good performances, compared to the ones obtained by cross-validation selection of the bandwidth. However, unexpectedly, the single-bandwidth cross-validation estimator is found to be much better than the ratio of the previous two good estimators, in the small noise context. However, the two methods have similar performances in models with large noise.

math.ST↗
arXiv2021

Kernel Selection in Nonparametric Regression

In the regression model $Y = b(X) +σ(X)\varepsilon$, where $X$ has a density $f$, this paper deals with an oracle inequality for an estimator of $bf$, involving a kernel in the sense of Lerasle et al. (2016), selected via the PCO method. In addition to the bandwidth selection for kernel-based estimators already studied in Lacour, Massart and Rivoirard (2017) and Comte and Marie (2020), the dimension selection for anisotropic projection estimators of $f$ and $bf$ is covered.

math.ST↗
arXiv2020

Almost Periodic and Periodic Solutions of Differential Equations Driven by the Fractional Brownian Motion with Statistical Application

We show that the unique solution to a semilinear stochastic differential equation with almost periodic coefficients driven by a fractional Brownian motion is almost periodic in a sense related to random dynamical systems. This type of almost periodicity allows for the construction of a consistent estimator of the drift parameter in the almost periodic and periodic cases.

math.PR↗
arXiv2019

Bandwidth Selection for the Wolverton-Wagner Estimator

For $n$ independent random variables having the same Hölder continuous density, this paper deals with controls of the Wolverton-Wagner's estimator MSE and MISE. Then, for a bandwidth $h_n(β)$, estimators of $β$ are obtained by a Goldenshluger-Lepski type method and a Lacour-Massart-Rivoirard type method. Some numerical experiments are provided for this last method.

math.ST↗
arXiv2019

Matrix factorization for multivariate time series analysis

Matrix factorization is a powerful data analysis tool. It has been used in multivariate time series analysis, leading to the decomposition of the series in a small set of latent factors. However, little is known on the statistical performances of matrix factorization for time series. In this paper, we extend the results known for matrix estimation in the i.i.d setting to time series. Moreover, we prove that when the series exhibit some additional structure like periodicity or smoothness, it is possible to improve on the classical rates of convergence.

math.ST↗
arXiv2019

Nonparametric Estimation in Fractional SDE

This paper deals with the consistency and a rate of convergence for a Nadaraya-Watson estimator of the drift function of a stochastic differential equation driven by an additive fractional noise. The results of this paper are obtained via both some long-time behavior properties of Hairer and some properties of the Skorokhod integral with respect to the fractional Brownian motion. These results are illustrated on the fractional Ornstein-Uhlenbeck process.

math.PR↗
arXiv2018

Sweeping Processes Perturbed by Rough Signals

This paper deals with the existence, the uniqueness and an approximation scheme of the solution to sweeping processes perturbed by a continuous signal of finite $p$-variation with $p\in [1,3[$. It covers pathwise stochastic noises directed by a fractional Brownian motion of Hurst parameter greater than $1/3$.

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↗
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

On a Constrained Fractional Stochastic Volatility Model

This paper deals with an extension of the so-called Black-Scholes model in which the volatility is modeled by a linear combination of the components of the solution of a differential equation driven by a fractional Brownian motion of Hurst parameter greater than $1/4$. In order to ensure the positiveness of the volatility, the coefficients of that equation satisfy a viability condition. The absence of arbitrages, the completeness of the market and a pricing formula are established.

math.PR↗
arXiv2015

Ergodicity of a Generalized Jacobi's Equation and Applications

Consider a $1$-dimensional centered Gaussian process $W$ with $α$-Hölder continuous paths on the compact intervals of $\mathbb R_+$ ($α\in ]0,1[$) and $W_0 = 0$, and $X$ the local solution in rough paths sense of Jacobi's equation driven by the signal $W$. The global existence and the uniqueness of the solution are proved via a change of variable taking into account the singularities of the vector field, because it doesn't satisfy the non-explosion condition. The regularity of the associated Itô map is studied. By using these deterministic results, Jacobi's equation is studied on probabilistic side : an ergodic theorem in L. Arnold's random dynamical systems framework, and the existence of an explicit density with respect to Lebesgue's measure for each $X_t$, $t > 0$ are proved. The paper concludes on a generalization of Morris-Lecar's neuron model, where the normalized conductance of the $\textrm{K}^+$ current is the solution of a generalized Jacobi's equation.

math.PR↗
arXiv2015

Singular Equations Driven by an Additive Noise and Applications

In the pathwise stochastic calculus framework, the paper deals with the general study of equations driven by an additive Gaussian noise, with a drift function having an infinite limit at point zero. An ergodic theorem and the convergence of the implicit Euler scheme are proved. The Malliavin calculus is used to study the absolute continuity of the distribution of the solution. An estimation procedure of the parameters of the random component of the model is provided. The properties are transferred on a class of singular stochastic differential equations driven by a multiplicative noise. A fractional Heston model is introduced.

math.PR↗
arXiv2015

A Skew Normal Model of the Dose-Effect Relation in Pharmacology

This paper deals with a skew-normal model of the relation between a dose $d > 0$ and a quantitative measure $E(d)$ of an effect of the administered drug. Precisely, $E(d)$ is a measure of the therapeutic response or a measure of a side-effect. Some existing and additional properties of the logistic functions are proved, and a skew-normal model of the escape time of rats under an experimental antidepressant medication is provided.

stat.ME↗
arXiv2014

A Pathwise Fractional one Compartment Intra-Veinous Bolus Model

Extending deterministic compartments pharmacokinetic models as diffusions seems not realistic on biological side because paths of these stochastic processes are not smooth enough. In order to extend one compartment intra-veinous bolus models, this paper suggests to modelize the concentration process $C$ by a class of stochastic differential equations driven by a fractional Brownian motion of Hurst parameter belonging to $]1/2,1[$. The first part of the paper provides probabilistic and statistical results on the concentration process $C$ : the distribution of $C$, a control of the uniform distance between $C$ and the solution of the associated ordinary differential equation, an ergodic theorem for the concentration process and its application to the estimation of the elimination constant, and consistent estimators of the driving signal's Hurst parameter and of the volatility constant. The second part of the paper provides applications of these theoretical results on simulated concentration datas : a qualitative procedure for choosing parameters on small sets of observations, and simulations of the estimators of the elimination constant and of the driving signal's Hurst parameter. The relationship between the estimations quality and the size/length of the sample is discussed.

math.PR↗
arXiv2014

A Generalized Mean-Reverting Equation and Applications

Consider a mean-reverting equation, generalized in the sense it is driven by a 1-dimensional centered Gaussian process with Hölder continuous paths on $[0,T]$ ($T > 0$). Taking that equation in rough paths sense only gives local existence of the solution because the non-explosion condition is not satisfied in general. Under natural assumptions, by using specific methods, we show the global existence and uniqueness of the solution, its integrability, the continuity and differentiability of the associated Itô map, and we provide an $L^p$-converging approximation with a rate of convergence ($p\geq 1$). The regularity of the Itô map ensures a large deviation principle, and the existence of a density with respect to the Lebesgue measure, for the solution of that generalized mean-reverting equation. Finally, we study a generalized mean-reverting pharmacokinetic model.

math.PR↗
arXiv2014

Sensitivities via Rough Paths

Motivated by a problematic coming from mathematical finance, this paper is devoted to existing and additional results of continuity and differentiability of the Itô map associated to rough differential equations. These regularity results together with Malliavin calculus are applied to sensitivities analysis for stochastic differential equations driven by multidimensional Gaussian processes with continuous paths, especially fractional Brownian motions. Precisely, in that framework, results on computation of greeks for Itô's stochastic differential equations are extended. An application in mathematical finance, and simulations, are provided.

math.PR↗