Publications

2019

• Stochastic Approximation Schemes for Economic Capital and Risk Margin Computations
  
  • Barrera David
  • Crépey Stéphane
  • Diallo Babacar
  • Fort Gersende
  • Gobet Emmanuel
  • Stazhynski Uladzislau
  ESAIM: Proceedings and Surveys, EDP Sciences, 2019, 65, pp.182-218. We consider the problem of the numerical computation of its economic capital by an insurance or a bank, in the form of a value-at-risk or expected shortfall of its loss over a given time horizon. This loss includes the appreciation of the mark-to-model of the liabilities of the firm, which we account for by nested Monte Carlo à la Gordy and Juneja (2010) or by regression à la Broadie, Du, and Moallemi (2015). Using a stochastic approximation point of view on value-at-risk and expected shortfall, we establish the convergence of the resulting economic capital simulation schemes, under mild assumptions that only bear on the theoretical limiting problem at hand, as opposed to assumptions on the approximating problems in Gordy-Juneja (2010) and Broadie-Du-Moallemi (2015). Our economic capital estimates can then be made conditional in a Markov framework and integrated in an outer Monte Carlo simulation to yield the risk margin of the firm, corresponding to a market value margin (MVM) in insurance or to a capital valuation adjustment (KVA) in banking par- lance. This is illustrated numerically by a KVA case study implemented on GPUs. (10.1051/proc/201965182)
  DOI : 10.1051/proc/201965182
• Reflected stochastic differential equations driven by $G$-Brownian motion in non-convex domains
  
  • Lin Yiqing
  • Soumana-Hima Abdoulaye
  Stochastics and Dynamics, World Scientific Publishing, 2019, 19 (3), pp.n°1950025. In this paper, we first review the penalization method for solving deterministic Skorokhod problems in non-convex domains and establish estimates for problems with $\alpha$-H\"older continuous functions. With the help of these results obtained previously for deterministic problems, we pathwisely define the reflected $G$-Brownian motion and prove its existence and uniqueness in a Banach space. Finally, multi-dimensional reflected stochastic differential equations driven by $G$-Brownian motion are investigated via a fixed-point argument. (10.1142/S0219493719500254)
  DOI : 10.1142/S0219493719500254
• High Pressure Flames with Multicomponent Transport
  
  • Giovangigli Vincent
  • Matuszewski Lionel
  • Gaillard Pierre
  , 2019. The thermodynamic formulation and the traditional formulation of multicomponent transport fluxes in high pressure fluids are discussed. The impact of high pressure transport models on mixing layers, premixed plane flames and strained diffusion flames is then investigated. Multicomponent fluxes in diffuse-interface transcritical diffusion flames are further addressed.
• A variational approach to nonlinear and interacting diffusions
  
  • Arnaudon Marc
  • del Moral Pierre
  Stochastic Analysis and Applications, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2019, 37 (5), pp.717-748. The article presents a novel variational calculus to analyze the stability and the propagation of chaos properties of nonlinear and interacting diffusions. This differential methodology combines gradient flow estimates with backward stochastic interpolations, Lyapunov linearization techniques as well as spectral theory. This framework applies to a large class of stochastic models including nonhomogeneous diffusions, as well as stochastic processes evolving on differentiable manifolds, such as constraint-type embedded manifolds on Euclidian spaces and manifolds equipped with some Riemannian metric. We derive uniform as well as almost sure exponential contraction inequalities at the level of the nonlinear diffusion flow, yielding what seems to be the first result of this type for this class of models. Uniform propagation of chaos properties w.r.t. the time parameter is also provided. Illustrations are provided in the context of a class of gradient flow diffusions arising in fluid mechanics and granular media literature. The extended versions of these nonlinear Langevin-type diffusions on Riemannian manifolds are also discussed. (10.1080/07362994.2019.1609985)
  DOI : 10.1080/07362994.2019.1609985
• An explicit Floquet-type representation of Riccati aperiodic exponential semigroups
  
  • Bishop Adrian N
  • del Moral Pierre
  International Journal of Control, Taylor & Francis, 2019, pp.1-9. The article presents a rather surprising Floquet-type representation of time-varying transition matri-ces associated with a class of nonlinear matrix differentialRiccati equations. The main difference withconventional Floquet theory comes from the fact that the underlying flow of the solution matrix is aperi-odic. The monodromy matrix associated with this Floquet representation coincides with the exponential(fundamental) matrix associated with the stabilizing fixedpoint of the Riccati equation. The second partof this article is dedicated to the application of this representation to the stability of matrix differentialRiccati equations. We provide refined global and local contraction inequalities for the Riccati exponentialsemigroup that depend linearly on the spectral norm of the initial condition. These refinements improveupon existing results and are a direct consequence of the Floquet-type representation, yielding whatseems to be the first results of this type for this class of models. (10.1080/00207179.2019.1590647)
  DOI : 10.1080/00207179.2019.1590647
• Hamiltonian models of interacting fermion fields in Quantum Field Theory
  
  • Alvarez Benjamin
  • Faupin Jérémy
  • Guillot Jean-Claude
  Letters in Mathematical Physics, Springer Verlag, 2019, 109 (11), pp.2403-2437. We consider Hamiltonian models representing an arbitrary number of spin 1 / 2 fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a self-adjoint operator on a tensor product of antisymmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated $N_\tau $ estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cutoffs. (10.1007/s11005-019-01193-9)
  DOI : 10.1007/s11005-019-01193-9
• Moutard transforms for the conductivity equation
  
  • Grinevich Piotr G
  • Novikov Roman G
  Letters in Mathematical Physics, Springer Verlag, 2019, 109 (10), pp.2209-2222. We construct Darboux-Moutard type transforms for the two-dimensional conductivity equation. This result continues our recent studies of Darboux-Moutard type transforms for generalized analytic functions. In addition, at least, some of the Darboux-Moutard type transforms of the present work admit direct extension to the conductivity equation in multidimensions. Relations to the Schrödinger equation at zero energy are also shown. (10.1007/s11005-019-01183-x)
  DOI : 10.1007/s11005-019-01183-x
• Relationship between biodiversity and agricultural production
  
  • Brunetti Ilaria
  • Tidball Mabel
  • Couvet Denis
  Natural Resource Modeling, Rocky Mountain Mathematics Consortium, 2019, 32 (2), pp.e12204. The intensification of agriculture is one of the main causes of biodiversity loss. We model the interdependent relationship between agriculture and wild biodiversity providing regulating services to agriculture on farmed land. We suppose that while agriculture has a negative impact on wild biodiversity, the latter can increase agricultural production. Farmers act as myopic agents, who maximize their instantaneous profit without considering the negative effects of their practice on the evolution of biodiversity. Two unexpected results arise (a) a tax on inputs can have a positive effect on yield since it can be considered as a social signal helping farmers to avoid myopic behavior concerning the positive effect of biodiversity on yield; (b) increasing biodiversity productivity, a proposal of ecological intensification, affects negatively the level of biodiversity, a counter‐intuitive result; due to the fact that when biodiversity is more productive, farmers can maintain lower biodiversity to get the same yield. (10.1111/nrm.12204)
  DOI : 10.1111/nrm.12204