arXiv subjects

Céline Labart

Publications and source records attributed to Céline Labart.

At least 19 recordsLinked to original sources

arXiv2025

Locally Lipschitz Path Dependent FBSDEs with Unbounded Terminal Conditions in Brownian and L{\'e}vy Settings

This paper is dedicated to the analysis of forward backward stochastic differential equations driven by a L{\'e}vy process. We assume that the generator and the terminal condition are path-dependent and satisfy a local Lipschitz condition. We study solvability and Malliavin differentiability of such BSDEs. The proof of the existence and uniqueness is done in three steps. First of all, we truncate and localize the terminal condition and the generator. Then we use an iteration argument to get bounds for the solutions of the truncated BSDE (independent from the level of truncation). Finally, we let the level of truncation tend to infinity. A stability result ends the proof. The Malliavin differentiability result is based on a recent characterisation for the Malliavin Sobolev space D 1,2 by S. Geiss and Zhou.

math.PR↗

arXiv2024

Convergence rate for random walk approximations of mean field BSDEs

We study the rate of convergence w.r.t.~a Wasserstein type distance for random walk approximations of mean field BSDEs. Our method does not use the particle method but instead a freezing technique. We extend results by Briand, Ch. Geiss, S. Geiss, and Labart [Bernoulli, 27(2) 2021] about the rate of convergence of a Donsker-type theorem for BSDEs from the classical setting to the mean field setting. In this connection the mean field setting leads to new phenomena and requires new techniques that should be of independent interest: The H\"older continuous terminal condition causes a singularity in time of the generator when seen as a generator in the non-mean field setting. To handle this singularity we introduce a concept of modified H\"older continuity by which we are able to achieve, up to a logarithmic term, the same polynomial approximation rates as in the classical non-mean field setting (in fact, already when approximating the Brownian motion itself a logarithmic term is necessary). Moreover, the exploited freezing technique of the mean field terms yields to the problem to handle the quantitative behavior of several different generators. Using BMO-techniques we obtain the rate of convergence for the integrated gradient process in the scale of Lorentz (type) spaces of exponential type.

math.PR↗

arXiv2023

Distance Rank Score: Unsupervised filter method for feature selection on imbalanced dataset

This paper presents a new filter method for unsupervised feature selection. This method is particularly effective on imbalanced multi-class dataset, as in case of clusters of different anomaly types. Existing methods usually involve the variance of the features, which is not suitable when the different types of observations are not represented equally. Our method, based on Spearman's Rank Correlation between distances on the observations and on feature values, avoids this drawback. The performance of the method is measured on several clustering problems and is compared with existing filter methods suitable for unsupervised data.

stat.ML↗

arXiv2022

Explainable multi-class anomaly detection on functional data

In this paper we describe an approach for anomaly detection and its explainability in multivariate functional data. The anomaly detection procedure consists of transforming the series into a vector of features and using an Isolation forest algorithm. The explainable procedure is based on the computation of the SHAP coefficients and on the use of a supervised decision tree. We apply it on simulated data to measure the performance of our method and on real data coming from industry.

stat.ML↗

arXiv2020

Extreme Inundation Statistics on a Composite Beach

The runup of initial Gaussian narrow-banded and wide-banded wave fields and its statistical characteristics are investigated using direct numerical simulations, based on the nonlinear shallow water equations. The bathymetry consists of the section of a constant depth, which is matched with the beach of constant slope. To address different levels of nonlinearity, the time series with five different significant wave heights are considered. The selected wave parameters allow also seeing the effects of wave breaking on wave statistics. The total physical time of each simulated time-series is 1000 hours (~360000 wave periods). The statistics of calculated wave runup heights are discussed with respect to the wave nonlinearity, wave breaking and the bandwidth of the incoming wave field. The conditional Weibull distribution is suggested as a model for the description of extreme runup heights and assessment of extreme inundations.

physics.ao-ph↗

arXiv2020

Mean square rate of convergence for random walk approximation of forward-backward SDEs

Let (Y, Z) denote the solution to a forward-backward SDE. If one constructs a random walk B n from the underlying Brownian motion B by Skorohod embedding, one can show L 2 convergence of the corresponding solutions (Y n , Z n) to (Y, Z). We estimate the rate of convergence in dependence of smoothness properties, especially for a terminal condition function in C 2,$α$. The proof relies on an approximative representation of Z n and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the PDE associated to the FBSDE as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by stochastic methods.

math.PR↗

arXiv2019

Donsker-Type Theorem for BSDEs: Rate of Convergence

In this paper, we study in the Markovian case the rate of convergence in the Wasserstein distance of an approximation of the solution to a BSDE given by a BSDE which is driven by a scaled random walk as introduced in Briand, Delyon and M{é}min (Electron. Comm. Probab. 6(2001),1-14).

math.PR↗

arXiv2018

Random walk approximation of BSDEs with H{ö}lder continuous terminal condition

In this paper we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally H{ö}lder continuous function of the Brownian motion. We state the rate of the L 2-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space.

math.PR↗

arXiv2018

Mean Reflected Stochastic Differential Equations with Jumps

This paper is devoted to the study of reflected Stochastic Differential Equations with jumps when the constraint is not on the paths of the solution but acts on the law of the solution. This type of reflected equations have been introduced recently by Briand, Elie and Hu [BEH18] in the context of BSDEs, when no jumps occur. In [BCdRGL16], the authors study a numerical scheme based on particle systems to approximate these reflected SDEs. In this paper, we prove existence and uniqueness of solutions to this kind of reflected SDEs with jumps and we generalize the results obtained in [BCdRGL16] to this context.

math.PR↗

arXiv2017

Simulation of BSDEs with jumps by Wiener Chaos Expansion

We present an algorithm to solve BSDEs with jumps based on Wiener Chaos Expansion and Picard's iterations. This paper extends the results given in Briand-Labart (2014) to the case of BSDEs with jumps. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. Concerning the error, we derive explicit bounds with respect to the number of chaos, the discretization time step and the number of Monte Carlo simulations. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.

math.PR↗

arXiv2017

Particles Systems and Numerical Schemes for Mean Reflected Stochastic Differential Equations

This paper is devoted to the study of reflected Stochastic Differential Equations when the constraint is not on the paths of the solution but acts on the law of the solution. These reflected equations have been introduced recently by Briand, Elie and Hu (arXiv:1605.06301) in the context of risk measures. Our main objective is to provide an approximation of solutions to these reflected SDEs with the help of interacting particles systems. This approximation allows to design a numerical scheme for this kind of equations.

math.PR↗

arXiv2016

Numerical approximation of doubly reflected BSDEs with jumps and RCLL obstacles

We study a discrete time approximation scheme for the solution of a doubly reflected Backward Stochastic Differential Equation (DBBSDE in short) with jumps, driven by a Brownian motion and an independent compensated Poisson process. Moreover, we suppose that the obstacles are right continuous and left limited (RCLL) processes with predictable and totally inaccessible jumps and satisfy Mokobodski's condition. Our main contribution consists in the construction of an implementable numerical sheme, based on two random binomial trees and the penalization method, which is shown to converge to the solution of the DBBSDE. Finally, we illustrate the theoretical results with some numerical examples in the case of general jumps.

math.PR↗

arXiv2015

Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles

We introduce a discrete time reflected scheme to solve doubly reflected Backward Stochastic Differential Equations with jumps (in short DRBSDEs), driven by a Brownian motion and an independent compensated Poisson process. As in Dumitrescu-Labart (2014), we approximate the Brownian motion and the Poisson process by two random walks, but contrary to this paper, we discretize directly the DRBSDE, without using a penalization step. This gives us a fully implementable scheme, which only depends on one parameter of approximation: the number of time steps $n$ (contrary to the scheme proposed in Dumitrescu-Labart (2014), which also depends on the penalization parameter). We prove the convergence of the scheme, and give some numerical examples.

math.PR↗

arXiv2014

Simulation of BSDEs by Wiener chaos expansion

We present an algorithm to solve BSDEs based on Wiener chaos expansion and Picard's iterations. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. We use the Malliavin derivative to compute $Z$. Concerning the error, we derive explicit bounds with respect to the number of chaos and the discretization time step. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.

math.PR↗

arXiv2013

An asymptotical method to estimate the parameters of a deteriorating system under condition-based maintenance

In this paper, we develop a new method to estimate the parameters of a deteriorating system under perfect condition-based maintenance. This method is based on the asymptotical behavior of the system, which is studied by using the renewal process theory. We obtain a Central Limit Theorem (CLT in the following) for the parameters. We compare the accuracy and the speed of the method with the maximum likelihood one (ML method in the following) on different examples.

math.ST↗

arXiv2013

Stochastic Local Intensity Loss Models with Interacting Particle Systems

It is well-known from the work of Schönbucher (2005) that the marginal laws of a loss process can be matched by a unit increasing time inhomogeneous Markov process, whose deterministic jump intensity is called local intensity. The Stochastic Local Intensity (SLI) models such as the one proposed by Arnsdorf and Halperin (2008) allow to get a stochastic jump intensity while keeping the same marginal laws. These models involve a non-linear SDE with jumps. The first contribution of this paper is to prove the existence and uniqueness of such processes. This is made by means of an interacting particle system, whose convergence rate towards the non-linear SDE is analyzed. Second, this approach provides a powerful way to compute pathwise expectations with the SLI model: we show that the computational cost is roughly the same as a crude Monte-Carlo algorithm for standard SDEs.

math.PR↗

arXiv2011

Long wave runup on random beaches

The estimation of the maximum wave sunup height is a problem of practical importance. Most of the analytical and numerical studies are limited to a constant slope plain shore and to the classical Nonlinear Shallow Water (NSW) equations. However, in nature the shore is characterized by some roughness. In order to take into account the effects of the bottom rugosity various ad-hoc friction terms are usually used. In this paper we study the effect of the roughness of the bottom on the maximum runup height. A stochastic model is proposed to describe the bottom irregularity and its effect is quantified using Monte-Carlo simulations. For the discretization of the NSW equations we employ modern finite volume schemes. Moreover, the results of the random bottom model are compared with the more conventional approaches.

physics.class-ph↗

arXiv2011

A Parallel Algorithm for solving BSDEs - Application to the pricing and hedging of American options

We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our algorithm based on Gobet and Labart (2010) exploits the link between BSDEs and non linear partial differential equations (PDEs in short) and hence enables to solve high dimensional non linear PDEs. In this work, we apply it to the pricing and hedging of American options in high dimensional local volatility models, which remains very computationally demanding. We have tested our algorithm up to dimension 10 on a cluster of 512 CPUs and we obtained linear speedups which proves the scalability of our implementation

math.PR↗