Publications

2010

• Merton Problem with Taxes: Characterization, computation and Approximation
  
  • Ben Tahar Imen
  • Touzi Nizar
  • Soner Mete H.
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2010, 1, pp.366-395. We formulate a computationally tractable extension of the classical Merton optimal consumptioninvestment problem to include the capital gains taxes. This is the continuous-time version of the model introduced by Dammon, Spatt, and Zhang [Rev. Financ. Stud., 14 (2001), pp. 583-616]. In this model the tax basis is computed as the average cost of the stocks in the investor's portfolio. This average rule introduces only one additional state variable, namely the tax basis. Since the other tax rules such as the first in first out rule require the knowledge of all past transactions, the average model is computationally much easier. We emphasize the linear taxation rule, which allows for tax credits when capital gains losses are experienced. In this context wash sales are optimal, and we prove it rigorously. Our main contributions are a first order explicit approximation of the value function of the problem and a unique characterization by means of the corresponding dynamic programming equation. The latter characterization builds on technical results isolated in the accompanying paper [I. Ben Tahar, H. M. Soner, and N. Touzi, SIAM J. Control Optim., 46 (2007), pp. 1779-1801]. We also suggest a numerical computation technique based on a combination of finite differences and the Howard iteration algorithm. Finally, we provide some numerical results on the welfare consequences of taxes and the quality of the first order approximation. (10.1137/080742178)
  DOI : 10.1137/080742178
• A global stability estimate for the Gel'fand-Calderon inverse problem in two dimensions
  
  • Novikov Roman
  • Santacesaria Matteo
  J. Inv. Ill-Posed Problems, 2010, 18, pp.765-785. We prove a global logarithmic stability estimate for the Gel'fand-Calderon inverse problem on a two-dimensional domain. (10.1515/JIIP.2011.003)
  DOI : 10.1515/JIIP.2011.003
• What happens after a default : The conditional density approach
  
  • El Karoui Nicole
  • Jeanblanc M.
  • Jiao Y.
  Stochastic Processes and their Applications, Elsevier, 2010, 120 (7), pp.1011-1032. We present a general model for default times, making precise the role of the intensity process, and showing that this process allows for a knowledge of the conditional distribution of the default only "before the default". This lack of information is crucial while working in a multi-default setting. In a single default case, the knowledge of the intensity process does not allow us to compute the price of defaultable claims, except in the case where the immersion property is satisfied. We propose in this paper a density approach for default times. The density process will give a full characterization of the links between the default time and the reference filtration, in particular "after the default time". We also investigate the description of martingales in the full filtration in terms of martingales in the reference filtration, and the impact of Girsanov transformation on the density and intensity processes, and on the immersion property. (10.1016/j.spa.2010.02.003)
  DOI : 10.1016/j.spa.2010.02.003
• Inverse impedance boundary problem via the conformal mapping method: the case of small impedances
  
  • Ben Hassen Fehmi
  • Boukari Yosra
  • Haddar Houssem
  Revue Africaine de Recherche en Informatique et Mathématiques Appliquées, African Society in Digital Science, 2010, 13, pp.47-62.
• A matrix interpolation between classical and free max operations: I. The univariate case
  
  • Benaych-Georges Florent
  • Cabanal-Duvillard Thierry
  Journal of Theoretical Probability, Springer, 2010, 23 (2), pp.447-465. Recently, Ben Arous and Voiculescu considered taking the maximum of two free random variables and brought to light a deep analogy with the operation of taking the maximum of two independent random variables. We present here a new insight on this analogy: its concrete realization based on random matrices giving an interpolation between classical and free settings. (10.1007/s10959-009-0210-1)
  DOI : 10.1007/s10959-009-0210-1
• Conformal mapping and impedance tomography
  
  • Haddar Houssem
  • Kress Rainer
  Inverse Problems, IOP Publishing, 2010, 26 (7), pp.074002, 18. (10.1088/0266-5611/26/7/074002)
  DOI : 10.1088/0266-5611/26/7/074002
• Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices
  
  • Benaych-Georges Florent
  • Guionnet Alice
  • Maïda Mylène
  , 2010. Consider a deterministic self-adjoint matrix X_n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X_n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-perturbed model and fluctuate in the same scale. We generalize these results to the case when X_n is random and get similar behavior when we deform some classical models such as Wigner or Wishart matrices with rather general entries or the so-called matrix models.
• Quadratic growth conditions in optimal control problems
  
  • Bonnans Joseph Frederic
  • Osmolovskii Nikolai P.
  Contemporary mathematics, American Mathematical Society, 2010, 514, pp.85--98.
• Localization of high frequency waves propagating in a locally periodic medium
  
  • Allaire Grégoire
  • Friz Luis
  Proceedings of the Royal Society of Edinburgh: Section A, Mathematics, Royal Society of Edinburgh, 2010, 140A, pp.897-926. We study the homogenization and localization of high frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a point x with length scale √ ε. We assume that (ξ, x) is a stationary point in the phase space of the Hamiltonian λ(ξ, x), i.e., of the corresponding Bloch eigenvalue. Upon rescaling at size √ ε we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ, x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions.
• Second-order analysis of optimal control problems with control and initial-final state constraints
  
  • Bonnans J. Frederic
  • Osmolovskii Nikolai P.
  Journal of Convex Analysis, Heldermann, 2010, 17 (3), pp.885-913. This paper provides an analysis of Pontryagine minima satisfying a quadratic growth condition, for optimal control problems of ordinary differential equations with constraints on initial-final state, as well as control constraints satisfying the uniform positive linear independence condition.