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Mireille Bossy

Publications and source records attributed to Mireille Bossy.

At least 19 recordsLinked to original sources
arXiv2025

Weak rough kernel comparison via PPDEs for integrated Volterra processes

Motivated by applications in physics (e.g., turbulence intermittency) and financial mathematics (e.g., rough volatility), this paper examines a family of integrated stochastic Volterra processes characterized by a small Hurst parameter $H<\tfrac{1}{2}$. We investigate the impact of kernel approximation on the integrated process by examining the resulting weak error. Our findings quantify this error in terms of the $L^1$ norm of the difference between the two kernels, as well as the $L^1$ norm of the difference of the squares of these kernels. Our analysis is based on a path-dependent Feynman-Kac formula and the associated partial differential equation (PPDE), providing a robust and extendible framework for our analysis.

math.PR↗
arXiv2024

Particle resuspension from complex multilayer deposits by laminar flows: statistical analysis and modeling

Particle resuspension refers to the physical process by which solid particles deposited on a surface are, first, detached and, then, entrained away by the action of a fluid flow. In this study, we explore the dynamics of large and heavy spherical particles forming a complex sediment bed which is exposed to a laminar shear flow. For that purpose, we rely on fine-scale simulations based on a fully-resolved flow field around individual particles whose motion is explicitly tracked. Using statistical tools, we characterize several features: (a) the overall bed dynamics (e.g. the average particle velocity as a function of the elevation), (b) the evolution of the top surface of the sediment bed (e.g. distribution of the surface elevation or of the surface slope) and (c) the dynamics of individual particles as they detach from or re-attach to the sediment bed (including the frequency of these events, and the velocity difference / surface angle for each event). These results show that particles detach more frequently around the peaks in the top surface of the sediment bed and that, once detached, they undergo short hops as particles quickly sediment towards the sediment bed. A simple model based on the surface characteristics (including its slope and elevation) is proposed to reproduce the detachment ratio.

physics.flu-dyn↗
arXiv2024

Strong convergence of the exponential Euler scheme for SDEs with superlinear growth coefficients and one-sided Lipschitz drift

We consider the problem of the discrete-time approximation of the solution of a one-dimensional SDE with piecewise locally Lipschitz drift and continuous diffusion coefficients with polynomial growth. In this paper, we study the strong convergence of a (semi-explicit) exponential-Euler scheme previously introduced in Bossy et al. (2021). We show the usual 1/2 rate of convergence for the exponential-Euler scheme when the drift is continuous. When the drift is discontinuous, the convergence rate is penalised by a factor {$ε$} decreasing with the time-step. We examine the case of the diffusion coefficient vanishing at zero, which adds a positivity preservation condition and a convergence analysis that exploits the negative moments and exponential moments of the scheme with the help of change of time technique introduced in Berkaoui et al. (2008). Asymptotic behaviour and theoretical stability of the exponential scheme, as well as numerical experiments, are also presented.

math.NA↗
arXiv2024

On the $\varepsilon$-Euler-Maruyama scheme for time-inhomogeneous jump-driven SDEs

We consider a class of general SDEs with a jump integral term driven by a time-inhomogeneous Poisson random measure. We propose a two-parameters Euler-type scheme for this SDE class and prove an optimal rate for the strong convergence with respect to the $L^p(\Omega)$-norm and for the weak convergence, considering integration over $n$ uniform time-steps. One of the primary issues to address in this context is the approximation of the noise structure when it can no longer be expressed as the increment of random variables. We extend the Asmussen-Rosi\'nski approach to the case of a fully dependent jump coefficient and time-dependent Poisson compensation, handling contribution of jumps smaller than $\varepsilon$ with an appropriate Gaussian substitute and exact simulation for the large jumps contribution. For any $p \geq 2$, under hypotheses required to control the $L^p$-moments of the process, we obtain a strong convergence rate of order $1/p$. Under standard regularity hypotheses on the coefficients, we obtain a weak convergence rate of $1/n+\varepsilon^{3-\beta}$, where $\beta$ is the Blumenthal-Getoor index of the underlying L\'evy measure. We compare this scheme with the Rubenthaler's approach where the jumps smaller than $\varepsilon$ are neglected, providing strong and weak rates of convergence in that case too. The theoretical rates are confirmed by numerical experiments afterwards. We apply this model class for some anomalous diffusion model related to the dynamics of rigid fibres in turbulence.

math.PR↗
arXiv2023

Lagrangian stochastic model for the orientation of inertialess spheroidal particles in turbulent flows: an efficient numerical method for CFD approach

In this work, we propose a model for the orientation of inertialess spheroidal particles suspended in turbulent flows. This model consists in a stochastic version of the Jeffery equation that can be included in a statistical Lagrangian description of particles suspended in a flow. It is compatible and coherent with turbulence models that are widely used in CFD codes for the simulation of the flow field in practical large-scale applications. In this context, we propose and analyze a numerical scheme based on a splitting scheme algorithm that decouples the orientation dynamics into its main contributions: stretching and rotation. We detail its implementation in an open-source CFD software. We analyze the weak and strong convergence of both the global scheme and of each sub-part. Subsequently, the splitting technique yields to a highly efficient hybrid algorithm coupling pure probabilistic and deterministic numerical schemes. Various numerical experiments were implemented and the results were compared with analytical predictions of the model to assess the algorithm efficiency and accuracy.

physics.flu-dyn↗
arXiv2022

Stochastic model for the alignment and tumbling of rigid fibres in two-dimensional turbulent shear flow

Non-spherical particles transported by an anisotropic turbulent flow preferentially align with the mean shear and intermittently tumble when the local strain fluctuates. Such an intricate behaviour is here studied for inertialess, rod-shaped particles embedded in a two-dimensional turbulent flow with homogeneous shear. A Lagrangian stochastic model for the rods angular dynamics is introduced and compared to the results of direct numerical simulations. The model consists in superposing a short-correlated random component to the steady large-scale mean shear, and can thereby be integrated analytically. To reproduce the single-time orientation statistics obtained numerically, it is found that one has to properly account for the combined effect of the mean shear, for anisotropic velocity gradient fluctuations, and for the presence of persistent rotating structures in the flow that bias Lagrangian statistics. The model is then used to address two-time statistics. The notion of tumbling rate is extended to diffusive dynamics by introducing the stationary probability flux of the rods unfolded angle. The model is found to reproduce the long-term effects of an average shear on the mean and the variance of the fibres angular increment. Still, it does not reproduce an intricate behaviour observed in numerics for intermediate times: the unfolded angle is there very similar to a Lévy walk with distributions of increments displaying intermediate power-law tails.

cond-mat.soft↗
arXiv2022

Instantaneous turbulent kinetic energy modelling based on Lagrangian stochastic approach in CFD and application to wind energy

We present the construction of an original stochastic model for the instantaneous turbulent kinetic energy at a given point of a flow, and we validate estimator methods on this model with observational data examples. Motivated by the need for wind energy industry of acquiring relevant statistical information of air motion at a local place, we adopt the Lagrangian description of fluid flows to derive, from the $3$D+time equations of the physics, a $0$D+time-stochastic model for the time series of the instantaneous turbulent kinetic energy at a given position. Specifically, based on the Lagrangian stochastic description of a generic fluid-particles, we derive a family of mean-field dynamics featuring the square norm of the turbulent velocity. By approximating at equilibrium the characteristic nonlinear terms of the dynamics, we recover the so called Cox-Ingersoll-Ross process, which was previously suggested in the literature for modelling wind speed. We then propose a calibration procedure for the parameters employing both direct methods and Bayesian inference. In particular, we show the consistency of the estimators and validate the model through the quantification of uncertainty, with respect to the range of values given in the literature for some physical constants of turbulence modelling.

physics.flu-dyn↗
arXiv2020

On the weak convergence rate of an exponential Euler scheme for SDEs governed by coefficients with superlinear growth

We consider the problem of the approximation of the solution of a one-dimensional SDE with non-globally Lipschitz drift and diffusion coefficients behaving as $x^α$, with $α>1$. We propose an (semi-explicit) exponential-Euler scheme and study its convergence through its weak approximation error. To this aim, we analyze the $C^{1,4}$ regularity of the solution of the associated backward Kolmogorov PDE using its Feynman-Kac representation and the flow derivative of the involved processes. From this, under some suitable hypotheses on the parameters of the model ensuring the control of its positive moments, we recover a rate of weak convergence of order one for the proposed exponential Euler scheme. Finally, numerical experiments are shown in order to support and complement our theoretical result.

math.PR↗
arXiv2019

New particle representations for ergodic McKean-Vlasov SDEs

The aim of this paper is to introduce several new particle representations for \textit{ergodic} McKean-Vlasov SDEs. We construct new algorithms by leveraging recent progress in weak convergence analysis of interacting particle system. We present detailed analysis of errors and associated costs of various estimators, highlighting key differences between long-time simulations of linear (classical SDEs) versus non-linear (Mckean-Vlasov SDEs) process.

math.PR↗
arXiv2018

Synchronization of stochastic mean field networks of Hodgkin-Huxley neurons with noisy channels

In this work we are interested in a mathematical model of the collective behavior of a fully connected network of finitely many neurons, when their number and when time go to infinity. We assume that every neuron follows a stochastic version of the Hodgkin-Huxley model, and that pairs of neurons interact through both electrical and chemical synapses, the global connectivity being of mean field type. When the leak conductance is strictly positive, we prove that if the initial voltages are uniformly bounded and the electrical interaction between neurons is strong enough, then, uniformly in the number of neurons, the whole system synchronizes exponentially fast as time goes to infinity, up to some error controlled by (and vanishing with) the channels noise level. Moreover, we prove that if the random initial condition is exchangeable, on every bounded time interval the propagation of chaos property for this system holds (regardless of the interaction intensities). Combining these results, we deduce that the nonlinear McKean-Vlasov equation describing an infinite network of such neurons concentrates, as time goes to infinity, around the dynamics of a single Hodgkin-Huxley neuron with chemical neurotransmitter channels. Our results are illustrated and complemented with numerical simulations.

math.PR↗
arXiv2018

On the wellposedness of some McKean models with moderated or singular diffusion coefficient

We investigate the well-posedness problem related to two models of nonlinear McKean Stochastic Differential Equations with some local interaction in the diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with moderate interaction, previously studied by Meleard and Jourdain in [16], under slightly weaker assumptions, by showing the existence and uniqueness of a weak solution using a Sobolev regularity framework instead of a Holder one. Second, we study the construction of a Lagrangian Stochastic model endowed with a conditional McKean diffusion term in the velocity dynamics and a nondegenerate diffusion term in the position dynamics.

math.PR↗
arXiv2016

A stochastic approach to colloidal particle collision/agglomeration

Colloidal particles that experience perfectly elastic collisions can be modelled using Langevin processes with specular reflection conditions. The article presents a discretisation scheme and offers a conjecture for the rate of convergence of the bias produced. Numerically, these conjectures are confirmed for the specular reflection scheme but also for the absorption scheme, which models perfect agglomeration.

cond-mat.stat-mech↗
arXiv2016

Modeling the wind circulation around mills with a Lagrangian stochastic approach

This work aims at introducing model methodology and numerical studies related to a Lagrangian stochastic approach applied to the computation of the wind circulation around mills. We adapt the Lagrangian stochastic downscaling method that we have introduced in [3] and [4] to the atmospheric boundary layer and we introduce here a Lagrangian version of the actuator disc methods to take account of the mills. We present our numerical method and numerical experiments in the case of non rotating and rotating actuator disc models. We also present some features of our numerical method, in particular the computation of the probability distribution of the wind in the wake zone, as a byproduct of the fluid particle model and the associated PDF method.

cs.CE↗
arXiv2016

Strong convergence of the symmetrized Milstein scheme for some CEV-like SDEs

In this paper we study the rate of convergence of a symmetrized version of the Milstein scheme applied to the solution of the one dimensional SDE $$X_t = x_0 + \int_{0}^t{b(X_s)ds}+\int_{0}^t{σ|X_s|^αdW_s}, \;x_0>0,\;σ>0,\; α\in[\tfrac{1}{2},1).$$ Assuming $b(0)/σ^2$ big enough, and $b$ smooth, we prove a strong rate of convergence of order one, recovering the classical result of Milstein for SDEs with smooth diffusion coefficient. In contrast with other recent results, our proof does not relies on Lamperti transformation, and it can be applied to a wide class of drift functions. On the downside, our hypothesis on the critical parameter value $b(0)/σ^2$ is more restrictive than others available in the literature. Some numerical experiments and comparison with various other schemes complement our theoretical analysis that also applies for the simple projected Milstein scheme with same convergence rate.

math.PR↗
arXiv2015

Weak convergence analysis of the symmetrized Euler scheme for one dimensional SDEs with diffusion coefficient |x|^a, a in [1/2,1)

In this paper, we are interested in the time discrete approximation of Ef(X(T)) when X is the solution of a stochastic differential equation with a diffusion coefficient function of the form |x|^a. We propose a symmetrized version of the Euler scheme, applied to X. The symmetrized version is very easy to simulate on a computer. For smooth functions f, we prove the Feynman Kac representation u(t,x) = E_{t,x} f(X(T)), for u(t,x) solving the associated Kolmogorov PDE and we obtain the upper-bounds on the spatial derivatives of u up to the order four. Then we show that the weak error of our symmetrized scheme is of order one, as for the classical Euler scheme.

math.PR↗
arXiv2015

Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons"

In this note, we clarify the well-posedness of the limit equations to the mean-field $N$-neuron models proposed in Baladron et al. and we prove the associated propagation of chaos property. We also complete the modeling issue in Baladron et al. by discussing the well-posedness of the stochastic differential equations which govern the behavior of the ion channels and the amount of available neurotransmitters.

math.PR↗
arXiv2015

Game theory analysis for carbon auction market through electricity market coupling

In this paper, we analyze Nash equilibria between electricity producers selling their production on an electricity market and buying CO2 emission allowances on an auction carbon market. The producers' strategies integrate the coupling of the two markets via the cost functions of the electricity production. We set out a clear Nash equilibrium on the power market that can be used to compute equilibrium prices on both markets as well as the related electricity produced and CO2 emissions released.

q-fin.MF↗