The challenge in managing new financial risks: adopting an heuristic or theoretical approach

The financial crisis began with the collapse of Lehman Brothers and the subprime asset backed securities debacle. Credit risk was turned into liquidity risk, resulting in a lack of confidence among financial institutions. In this article, we will propose a way to model liquidity risk and the credit risk in best practices. We will show that liquidity risk is a new type of risk and the current way to deal with it is based solely on observed variables without any theoretical link. We propose an heuristic approach to combine the numerous liquidity risk indicators with a logistic regression for the first time. In regards to credit risk, several articles prove that the best practice is to use an option model to appreciate this risk. We will present our methodology using stochastic diffusion for the interest rate because currently the yield curves aren’t liquid. This approach is more relevant because the basis model in prior publications has a constant interest rate or a forward rate. Both models allow a better understanding of liquidity and credit risks and the further development of research deals with the link between these two financial risks.     

Nevanlinna counting function and Carleson function of analytic maps

We show that the maximal Nevanlinna counting function and the Carleson function of analytic self-maps of the unit disk are equivalent, up to constants.     

Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L ₂ boundary data. The coefficients A may depend on all variables, but are assumed to be close to coefficients A ₀ that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖A−A ₀‖ _C defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour results under finiteness of ‖A−A ₀‖ _C . Our methods yield full characterization of weak solutions, whose gradients have L ₂ estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled in L ₂ by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension 3 or higher. The existence of a proof a priori to well-posedness, is also a new fact.     

Spin and Majorana polarization in topological superconducting wires

We study a one-dimensional wire with strong Rashba and Dresselhaus spin-orbit coupling (SOC), which supports Majorana fermions when subject to a Zeeman magnetic field and in the proximity of a superconductor. Using both analytical and numerical techniques we calculate the electronic spin texture of the Majorana end states. We find that the spin polarization of these states depends on the relative magnitude of the Rashba and Dresselhaus SOC components. Moreover, we define and calculate a local "Majorana polarization" and "Majorana density" and argue that they can be used as order parameters to characterize the topological transition between the trivial system and the system exhibiting Majorana bound modes. We find that the local Majorana polarization is correlated to the transverse spin polarization, and we propose to test the presence of Majorana fermions in a 1D system by a spin-polarized density of states measurement.     

Z-scan measurement of the nonlinear refractive index of graphene

Under strong laser illumination, few-layer graphene exhibits both a transmittance increase due to saturable absorption and a nonlinear phase shift. Here, we unambiguously distinguish these two nonlinear optical effects and identify both real and imaginary parts of the complex nonlinear refractive index of graphene. We show that graphene possesses a giant nonlinear refractive index n(2)?10(-7) cm(2) W(-1), almost 9 orders of magnitude larger than bulk dielectrics. We find that the nonlinear refractive index decreases with increasing excitation flux but slower than the absorption. This suggests that graphene may be a very promising nonlinear medium, paving the way for graphene-based nonlinear photonics.     

Automatic determination of the number of targets present when using the time reversal operator

Acoustical time reversal mirrors have been shown to provide a highly accurate means of studying and focusing on acoustical sources. The DORT method is a derivation of the time reversal process, which allows for focusing on multiple targets. An important step in this process is the determination of the number of targets or sources present. This is achieved by examining the eigenvalues of the time reversal operator (TRO). The number of significant eigenvalues is then chosen as the number of sources present. However, as mentioned in [N. Mordant, C. Prada, and M. Fink, J. Acoust. Soc. Am. 105, 2634-2642 (1999) and C. Prada, M. Tanter, and M. Fink, in Proceedings of the IEEE Symposium, 1997, pp. 679-683], factors such as low signal to noise ratio (SNR), small data sample, array configuration and the target location may result in the eigenvalues corresponding to the targets no longer being distinguishable from the background noise eigenvalues. This paper proposes a robust method of automatically determining the number of targets even in the presence of a small number of snapshots. For white Gaussian noise, the profile of the ordered eigenvalues is seen to fit an exponential law. The observed eigenvalues are then compared to this model and a mismatch is detected between the observed profile and the noise-only model. The index of the mismatch gives the number of scatterers present.     

Definability of Geometric Properties in Algebraically Closed Fields

We prove that there exists no sentence F of the language of rings with an extra binary predicat I₂ satisfying the following property: for every definable set X ? ?², X is connected if and only if (?, X) ? F, where I₂ is interpreted by X. We conjecture that the same result holds for closed subset of ?². We prove some results motivated by this conjecture.