On the polynomial τ-functions for the KP hierarchy and the bispectral property

We show that the -functions obtained from Schur polynomials lead to wave functions w(x ₁, x ₂, ... ; k) that possess the following bispectral property: There exists a differential operator B{k,_k}, independent of x ₁ , such that B{k,_k}w = {x ₁}w, where {x ₁} is independent of k. This extends for the KP hierarchy some earlier results of J. J. Duistermaat and F. A. Grünbaum for the rational solutions of KdV and of P. Wright for certain rational solutions of the generalized KdV equations.     

Convex regularization of local volatility models from option prices: Convergence analysis and rates

We study a convex regularization of the local volatility surface identification problem for the Black-Scholes partial differential equation from prices of European call options. This is a highly nonlinear ill-posed problem which in practice is subject to different noise levels associated to bid-ask spreads and sampling errors. We analyze, in appropriate function spaces, different properties of the parameter-to-solution map that assigns to a given volatility surface the corresponding option prices. Using such properties, we show stability and convergence of the regularized solutions in terms of the Bregman distance with respect to a class of convex regularization functionals when the noise level goes to zero.We improve convergence rates available in the literature for the volatility identification problem. Furthermore, in the present context, we relate convex regularization with the notion of exponential families in Statistics. Finally, we connect convex regularization functionals with convex risk measures through Fenchel conjugation. We do this by showing that if the source condition for the regularization functional is satisfied, then convex risk measures can be constructed.     

On the Inverse Problem for Scattering of Electromagnetic Radiation by a Periodic Structure

We consider a smooth perturbation  δε( x , y , z )  of a constant background permittivity  ε=ε₀  that varies periodically with x , does not depend on y , and is supported on a finite-length interval in z . We investigate the theoretical and numerical determination of such perturbation from (several) fixed frequency y -invariant electromagnetic waves.
By varying the direction and frequency of the probing radiation a scattering matrix is defined. By using an invariant-imbedding technique we derive an operator Riccati equation for such scattering matrix. We obtain a theoretical uniqueness result for the problem of determining the perturbation from the scattering matrix.
We also investigate a numerical method for performing such reconstruction using multi-frequency information of the truncated scattering matrix. This relies on ideas of regularization and recursive linearization. Numerical experiments are presented validating such approach.     

On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy

We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones. Received: 23 July 1996 / Accepted: 6 January 1997     

Towards a generalization of Dupire's equation for several assets

We pose the problem of generalizing Dupire's equation for the price of call options on a basket of underlying assets. We present an analogue of Dupire's equation that holds in the case of several underlying assets provided the volatility is time dependent but not asset-price dependent. We deduce it from a relation that seems to be of interest on its own.     

Tangent Graeffe iteration

Summary.   Graeffe iteration was the choice algorithm for solving univariate polynomials in the XIX-th and early XX-th century. In this paper, a new variation of Graeffe iteration is given, suitable to IEEE floating-point arithmetics of modern digital computers. We prove that under a certain generic assumption the proposed algorithm converges. We also estimate the error after N iterations and the running cost. The main ideas from which this algorithm is built are: classical Graeffe iteration and Newton Diagrams, changes of scale (renormalization), and replacement of a difference technique by a differentiation one. The algorithm was implemented successfully and a number of numerical experiments are displayed. Received May 29, 1998 / Revised version received September 13, 1999 / Published online April 5, 2001     

Bundle bispectrality for matrix differential equations

We consider the fundamental solutions of a wide class of first order systems with polynomial dependence on the spectral parameter and rational matrix potentials. Such matrix potentials are rational solutions of a large class of integrable nonlinear equations, which play an important role in different mathematical physics problems. The concept of bispectrality, which was originally introduced by Grünbaum, is extended in a natural way for the systems under consideration and their bispectrality is derived via the representation of the fundamental solutions. This bispectrality is preserved under the flows of the corresponding integrable nonlinear equations. For the case of Dirac type (canonical) systems the complete characterization of the bispectral potentials under consideration is obtained in terms of the system's spectral function.     

The Bi-Hamiltonian Theory of the Harry Dym Equation

We describe how the Harry Dym equation fits into the the bi-Hamiltonian formalism for the Korteweg–de Vries equation and other soliton equations. This is achieved using a certain Poisson pencil constructed from two compatible Poisson structures. We obtain an analogue of the Kadomtsev–Petviashivili hierarchy whose reduction leads to the Harry Dym hierarchy. We call such a system the HD–KP hierarchy. We then construct an infinite system of ordinary differential equations (in infinitely many variables) that is equivalent to the HD–KP hierarchy. Its role is analogous to the role of the Central System in the Kadomtsev–Petviashivili hierarchy.