Publications

2012

• Singular Forward-Backward Stochastic Differential Equations and Emissions Derivatives
  
  • Carmona René
  • Delarue François
  • Espinosa Gilles-Edouard
  • Touzi Nizar
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2012, 23, pp.1086--1128. We introduce two simple models of forward-backward stochastic differential equations with a singular terminal condition and we explain how and why they appear naturally as models for the valuation of CO2 emission allowances. Single phase cap-and-trade schemes lead readily to terminal conditions given by indicator functions of the forward component, and using fine partial differential equations estimates, we show that the existence theory of these equations, as well as the properties of the candidates for solution, depend strongly upon the characteristics of the forward dynamics. Finally, we give a first order Taylor expansion and show how to numerically calibrate some of these models for the purpose of CO2 option pricing.
• Coupling policy iteration with semi-definite relaxation to compute accurate numerical invariants in static analysis
  
  • Adjé A.
  • Gaubert Stéphane
  • Goubault E.
  Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2012, 8 (1), pp.1:01, 32. We introduce a new domain for finding precise numerical invariants of pro- grams by abstract interpretation. This domain, which consists of sub-level sets of non- linear functions, generalizes the domain of linear templates introduced by Manna, Sankara- narayanan, and Sipma. In the case of quadratic templates, we use Shor's semi-definite relaxation to derive safe and computable abstractions of semantic functionals, and we show that the abstract fixpoint can be over-approximated by coupling policy iteration and semi-definite programming. We demonstrate the interest of our approach on a series of examples (filters, integration schemes) including a degenerate one (symplectic scheme). (10.2168/LMCS-8(1:1)2012)
  DOI : 10.2168/LMCS-8(1:1)2012
• Existence and qualitative properties of isoperimetric sets in periodic media
  
  • Chambolle Antonin
  • Goldman Michael
  • Novaga Matteo
  , 2013. We review and extend here some recent results on the existence of minimal surfaces and isoperimetric sets in non homogeneous and anisotropic periodic media. We also describe the qualitative properties of the homogenized surface tension, also known as stable norm (or minimal action) in Weak KAM theory. In particular we investigate its strict convexity and differentiability properties.
• Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb')
  
  • Colonna Jean-François
  , 2012. Close-up on a pseudo-octonionic Mandelbrot set (a 'Mandelbulb') (Agrandissement d'un ensemble de Mandelbrot dans l'ensemble des pseudo-octonions (un 'Mandelbulb'))
• Faddeev eigenfunctions for point potentials in two dimensions
  
  • Grinevich Piotr
  • Novikov Roman
  Physics Letters A, Elsevier, 2012, 376, pp.1102-1106. We present explicit formulas for the Faddeev eigenfunctions and related generalized scattering data for point (delta-type) potentials in two dimensions. In particular, we obtain the first explicit examples of such eigenfunctions with contour singularity in spectral parameter at a fixed real energy. (10.1016/j.physleta.2012.02.025)
  DOI : 10.1016/j.physleta.2012.02.025
• Instability in the Gel'fand inverse problem at high energies
  
  • Isaev Mikhail
  Applicable Analysis, Taylor & Francis, 2012, pp.DOI:10.1080/00036811.2012.731501. We give an instability estimate for the Gel'fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceeding stability results on inverse problems of such a type. (10.1080/00036811.2012.73150)
  DOI : 10.1080/00036811.2012.73150
• Disentangling dispersal, vicariance and adaptive radiation patterns: A case study using armyworms in the pest genus Spodoptera (Lepidoptera: Noctuidae)
  
  • Kergoat Gael G.
  • Prowell Dorothy P.
  • Le Ru Bruno P.
  • Mitchell Andrew
  • Dumas Pascaline
  • Clamens Anne Laure
  • Condamine Fabien L.
  • Silvain Jean-Francois
  Molecular Phylogenetics and Evolution, Elsevier, 2012, 65 (3), pp.855-870. Thanks to the recent development of integrative approaches that combine dated phylogenies with models of biogeographic evolution, it is becoming more feasible to assess the roles of dispersal and vicariance in creating complex patterns of geographical distribution. However, the historical biogeography of taxa with good dispersal abilities, like birds or flying insects, still remains largely unknown because of the lack of complete phylogenies accompanied by robust estimates of divergence times. In this study, we investigate the evolution and historical biogeography of the globally distributed pest genus Spodoptera (Lepidoptera: Noctuidae) using complete taxon sampling and an extensive set of analyses. Through the analysis of a combined morphological and molecular dataset, we provide the first robust phylogenetic framework for this widespread and economically important group of moths. Historical biogeography approaches indicate that dispersal events have been the driving force in the biogeographic history of the group. One of the most interesting findings of this study is the probable occurrence of two symmetric long-distance dispersal events between the Afrotropical and the Neotropical region, which appear to have occurred in the late Miocene. Even more remarkably, our dated phylogenies reveal that the diversification of the clade that includes specialist grass feeders has followed closely the expansion of grasslands in the Miocene, similar to the adaptive radiation of specialist grazing mammals during the same period. (10.1016/j.ympev.2012.08.006)
  DOI : 10.1016/j.ympev.2012.08.006
• Quasi-stationary distributions and population processes
  
  • Méléard Sylvie
  • Villemonais Denis
  Probability Surveys, Institute of Mathematical Statistics (IMS), 2012, 9, pp.340-410. This survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures. (10.1214/11-PS191)
  DOI : 10.1214/11-PS191
• Tropical linear-fractional programming and parametric mean payoff games
  
  • Gaubert Stéphane
  • Katz R.D.
  • Sergeev S.N.
  Journal of Symbolic Computation, Elsevier, 2012, 47 (12), pp.1447-1478. Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to study the tropical analogue of the classical linear-fractional programming problem. We construct an associated parametric mean payoff game problem, and show that the optimality of a given point, or the unboundedness of the problem, can be certified by exhibiting a strategy for one of the players having certain infinitesimal properties (involving the value of the game and its derivative) that we characterize combinatorially. We use this idea to design a Newton-like algorithm to solve tropical linear-fractional programming problems, by reduction to a sequence of auxiliary mean payoff game problems (10.1016/j.jsc.2011.12.049)
  DOI : 10.1016/j.jsc.2011.12.049
• Credit Risk with asymmetric information on the default threshold
  
  • Hillairet Caroline
  • Jiao Ying
  Stochastics: An International Journal of Probability and Stochastic Processes, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2012, 84 (2-3), pp.183-198. We study the impact of asymmetric information in a general credit model where the default is triggered when a fundamental diff usion process of the firm passes below a random threshold. Inspired by some recent technical default events during the fi nancial crisis, we consider the role of the firm's managers who choose the level of the default threshold and have complete information. However, other investors on the market only have partial observations either on the process or on the threshold. We specify the accessible information for di fferent types of investors. Besides the framework of progressive enlargement of fi ltrations usually adopted in the credit risk modelling, we also combine the results on initial enlargement of filtrations to deal with the uncertainty on the default threshold. We consider several types of investors who have di fferent information levels and we compute the default probabilities in each case. Numerical illustrations show that the insiders who have extra information on the default threshold obtain better estimations of the default probability compared to the standard market investors. (10.1080/17442508.2011.575944)
  DOI : 10.1080/17442508.2011.575944
• Domain theory and mirror properties in inverse semigroups
  
  • Poncet Paul
  Semigroup Forum, Springer Verlag, 2012, 84 (3), pp.434-446. Inverse semigroups are a class of semigroups whose structure induces a compatible partial order. This partial order is examined so as to establish mirror properties between an inverse semigroup and the semilattice of its idempotent elements, such as continuity in the sense of domain theory. (10.1007/s00233-012-9392-4)
  DOI : 10.1007/s00233-012-9392-4
• Large liquidity expansion of super-hedging costs
  
  • Possamaï Dylan
  • Soner Mete H.
  • Touzi Nizar
  Asymptotic Analysis, IOS Press, 2012, 79 (1-2), pp.45-64. We consider a financial market with liquidity cost as in Çetin, Jarrow and Protter [2004], where the supply function S{\epsilon}(s,{\nu}) depends on a parameter {\epsilon}\geq0 with S0(s,{\nu})=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, Soner and Touzi [2010], of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of {\epsilon}. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option. (10.3233/ASY-2011-1089)
  DOI : 10.3233/ASY-2011-1089
• Consistency result for a non monotone scheme for anisotropic mean curvature flow
  
  • Bonnetier Eric
  • Bretin Elie
  • Chambolle Antonin
  Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, European Mathematical Society, 2012, 14 (1), pp.1-35. In this paper, we propose a new scheme for anisotropic motion by mean curvature in $R^d$. The scheme consists of a phase-field approximation of the motion, where the nonlinear diffusive terms in the corresponding anisotropic Allen-Cahn equation are linearized in the Fourier space. In real space, this corresponds to the convolution with a specific kernel of the form $$K_{\phi,t}(x)=F^{−1}[e^{−4\pi^2t\phi^0(ξ)}](x).$$ We analyse the resulting scheme, following the work of Ishii-Pires-Souganidis on the convergence of the Bence-Merriman-Osher algorithm for isotropic motion by mean curvature. The main difficulty here, is that the kernel $K_{\phi,t}$ is not positive and that its moments of order 2 are not in $L^1(R^d)$. Still, we can show that in one sense the scheme is consistent with the anisotropic mean curvature flow. (10.4171/IFB/272)
  DOI : 10.4171/IFB/272
• Tropical polyhedra are equivalent to mean payoff games
  
  • Akian Marianne
  • Gaubert Stéphane
  • Guterman A.
  International Journal of Algebra and Computation, World Scientific Publishing, 2012, 22 (1), pp.1250001, 43. We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory. (10.1142/S0218196711006674)
  DOI : 10.1142/S0218196711006674
• Heat transfer in incompressible magnetic fluid
  
  • Amirat Youcef
  • Hamdache Kamel
  Journal of Mathematical Fluid Mechanics, Springer Verlag, 2012, 14, pp.217--247.
• Understanding, Modeling and Managing Longevity Risk: Key Issues and Main Challenges
  
  • Barrieu Pauline
  • Bensusan Harry
  • El Karoui Nicole
  • Hillairet Caroline
  • Loisel Stéphane
  • Ravanelli Claudia
  • Salhi Yahia
  Scandinavian Actuarial Journal, Taylor & Francis (Routledge), 2012, 2012 (3), pp.203-231. This article investigates the latest developments in longevity risk modelling, and explores the key risk management challenges for both the financial and insurance industries. The article discusses key definitions that are crucial for the enhancement of the way longevity risk is understood; providing a global view of the practical issues for longevity-linked insurance and pension products that have evolved concurrently with the steady increase in life expectancy since 1960s. In addition, the article frames the recent and forthcoming developments that are expected to action industry-wide changes as more effective regulation, designed to better assess and efficiently manage inherited risks, is adopted. Simultaneously, the evolution of longevity is intensifying the need for capital markets to be used to manage and transfer the risk through what are known as Insurance-Linked Securities (ILS). Thus, the article will examine the emerging scenarios, and will finally highlight some important potential developments for longevity risk management from a financial perspective with reference to the most relevant modelling and pricing practices in the banking industry.
• Preliminary report on the darkling beetles (Coleoptera, Tenebrionidae) collected during the SANTO 2006 expedition to Vanuatu and description of a new species from the genus [i]Uloma [/i]Dejean, 1821
  
  • Soldati Laurent
  • Kergoat Gael
  • Condamine Fabien L.
  Zoosystema, Museum Nationale d'Histoire Naturelle Paris, 2012, 34 (2), pp.305-317. An annotated list of Tenebrionidae Latreille, 1802 (Coleoptera Linnaeus, 1758) collected by one of the authors (LS) during the SANTO 2006 expedition is given. It concerns the period covered by the section “Fallow & Aliens” during October 2006. These faunistic data are completed with the description of a new species: [i]Uloma vanuatensis[/i] L. Soldati n. sp. Furthermore, biogeographic hypotheses on the colonisation of the Vanuatu archipelago are presented to explain the origin of the tenebrionid fauna. (10.5252/z2012n2a8)
  DOI : 10.5252/z2012n2a8
• Optimized analytic reconstruction for SPECT
  
  • Guillement Jean-Pol
  • Novikov Roman
  Journal of Inverse and Ill-posed Problems, De Gruyter, 2012, 20 (4), pp.489–500. We develop optimized analytic reconstruction for the single-photon emission computed tomography (SPECT). This reconstruction is based on : (1) Novikov's exact and Chang's approximate inversion formulas for the attenuated ray transform, (2) filtering techniques, and (3) Morozov type discrepancy principle. Our numerical examples include comparisons with the standard least square and expectation maximization iterative SPECT reconstructions.
• Approximation and relaxation of perimeter in the Wiener space
  
  • Goldman Michael
  • Novaga Matteo
  Annales de l'Institut Henri Poincaré, Presses universitaires de France — PUF, 2012. We characterize the relaxation of the perimeter in an infinite dimensional Wiener space, with respect to the weak L^2-topology. We also show that the rescaled Allen-Cahn functionals approximate this relaxed functional in the sense of Gamma-convergence.
• Large graph limit for an SIR process in random network with heterogeneous connectivity
  
  • Decreusefond Laurent
  • Dhersin Jean-Stephane
  • Moyal Pascal
  • Tran Viet Chi
  The Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2012, 22 (2), pp.541-575. We consider an SIR epidemic model propagating on a Configuration Model network, where the degree distribution of the vertices is given and where the edges are randomly matched. The evolution of the epidemic is summed up into three measure-valued equations that describe the degrees of the susceptible individuals and the number of edges from an infectious or removed individual to the set of susceptibles. These three degree distributions are sufficient to describe the course of the disease. The limit in large population is investigated. As a corollary, this provides a rigorous proof of the equations obtained by Volz (2008). (10.1214/11-AAP773)
  DOI : 10.1214/11-AAP773
• Multigrid methods for two-player zero-sum stochastic games
  
  • Akian Marianne
  • Detournay Sylvie
  Numerical Linear Algebra with Applications, Wiley, 2012, 19 (2), pp.313--342. We present a fast numerical algorithm for large scale zero-sum stochastic games with perfect information, which combines policy iteration and algebraic multigrid methods. This algorithm can be applied either to a true finite state space zero-sum two player game or to the discretization of an Isaacs equation. We present numerical tests on discretizations of Isaacs equations or variational inequalities. We also present a full multi-level policy iteration, similar to FMG, which allows to improve substantially the computation time for solving some variational inequalities. (10.1002/nla.1815)
  DOI : 10.1002/nla.1815
• Biogeographic and diversification patterns of Neotropical Troidini butterflies (Papilionidae) support a museum model of diversity dynamics for Amazonia
  
  • Condamine Fabien F.
  • Silva-Brandão Karina L.
  • Kergoat Gael G.
  • Sperling Felix A.H.
  BMC Evolutionary Biology, BioMed Central, 2012, 12, pp.17 p.. BACKGROUND: The temporal and geographical diversification of Neotropical insects remains poorly understood because of the complex changes in geological and climatic conditions that occurred during the Cenozoic. To better understand extant patterns in Neotropical biodiversity, we investigated the evolutionary history of three Neotropical swallowtail Troidini genera (Papilionidae). First, DNA-based species delimitation analyses were conducted to assess species boundaries within Neotropical Troidini using an enlarged fragment of the standard barcode gene. Molecularly delineated species were then used to infer a time-calibrated species-level phylogeny based on a three-gene dataset and Bayesian dating analyses. The corresponding chronogram was used to explore their temporal and geographical diversification through distinct likelihood-based methods. RESULTS: The phylogeny for Neotropical Troidini was well resolved and strongly supported. Molecular dating and biogeographic analyses indicate that the extant lineages of Neotropical Troidini have a late Eocene (33-42 Ma) origin in North America. Two independent lineages (Battus and Euryades + Parides) reached South America via the GAARlandia temporary connection, and later became extinct in North America. They only began substantive diversification during the early Miocene in Amazonia. Macroevolutionary analysis supports the "museum model" of diversification, rather than Pleistocene refugia, as the best explanation for the diversification of these lineages. CONCLUSIONS: This study demonstrates that: (i) current Neotropical biodiversity may have originated ex situ; (ii) the GAARlandia bridge was important in facilitating invasions of South America; (iii) colonization of Amazonia initiated the crown diversification of these swallowtails; and (iv) Amazonia is not only a species-rich region but also acted as a sanctuary for the dynamics of this diversity. In particular, Amazonia probably allowed the persistence of old lineages and contributed to the steady accumulation of diversity over time with constant net diversification rates, a result that contrasts with previous studies on other South American butterflies (10.1186/1471-2148-12-82)
  DOI : 10.1186/1471-2148-12-82
• Homogenization of a One-Dimensional Spectral Problem for a Singularly Perturbed Elliptic Operator with Neumann Boundary Conditions
  
  • Allaire Grégoire
  • Capdeboscq Y.
  • Puel Marjolaine
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2012, 17, pp.1-31. We study the asymptotic behavior of the first eigenvalue and eigenfunctionof a one-dimensional periodic elliptic operator with Neumann boundaryconditions. The second order elliptic equation is not self-adjointand is singularly perturbed since, denoting by $\epsilon$ the period,each derivative is scaled by an $\epsilon$ factor.The main difficulty is that the domain size is not an integer multipleof the period. More precisely, for a domain of size $1$ and a givenfractional part $0\leq\delta<1$, we consider a sequence of periods$\epsilon_n=1/(n+\delta)$ with $n\in \mathbb{N}$. In other words, the domaincontains $n$ entire periodic cells and a fraction $\delta$ of a cellcut by the domain boundary. According to the value of the fractionalpart $\delta$, different asymptotic behaviors are possible: in somecases an homogenized limit is obtained, while in other cases thefirst eigenfunction is exponentially localized at one of theextreme points of the domain. (10.3934/dcdsb.2012.17.1)
  DOI : 10.3934/dcdsb.2012.17.1
• Homogenization and concentration for a diffusion equation with large convection in a bounded domain
  
  • Allaire Grégoire
  • Pankratova I.
  • Piatnitski Andrey
  Journal of Functional Analysis, Elsevier, 2012, 262 (1), pp.300-330. We consider the homogenization of a non-stationary convection–diffusion equation posed in a bounded domain with periodically oscillating coefficients and homogeneous Dirichlet boundary conditions. Assuming that the convection term is large, we give the asymptotic profile of the solution and determine its rate of decay. In particular, it allows us to characterize the “hot spot”, i.e., the precise asymptotic location of the solution maximum which lies close to the domain boundary and is also the point of concentration. Due to the competition between convection and diffusion, the position of the “hot spot” is not always intuitive as exemplified in some numerical tests. (10.1016/j.jfa.2011.09.014)
  DOI : 10.1016/j.jfa.2011.09.014
• Shape Optimization of a Sodium Fast Reactor Core
  
  • Dombre Emmanuel
  • Allaire Grégoire
  • Pantz Olivier
  , 2012.