Initial and Final Backward and Forward Discrete Time Non-homogeneous Semi-Markov Credit Risk Models

In this paper we show how it is possible to construct an efficient Migration models in the study of credit risk problems presented in Jarrow et al. (Rev Financ Stud 10:481–523, 1997) with Markov environment. Recently it was introduced the semi-Markov process in the migration models (D’Amico et al. Decis Econ Finan 28:79–93, 2005a). The introduction of semi-Markov processes permits to overtake some of the Markov constraints given by the dependence of transition probabilities on the duration into a rating category. In this paper, it is shown how it is possible to take into account simultaneously backward and forward processes at beginning and at the end of the time in which the credit risk model is observed. With such a generalization, it is possible to consider what happens inside the time after the first transition and before the last transition where the problem is studied. This paper generalizes other papers presented before. The model is presented in a discrete time environment.     

On the Classification of <Emphasis Type="Italic">N</Emphasis>-Points Concentrating Solutions for Mean Field Equations and the Symmetry Properties of the <Emphasis Type="Italic">N</Emphasis>-Vortex Singular Hamiltonian on the Unit Disk

Motivated by the analysis of the multiple bubbling phenomenon (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7–8):1241–1265, 2004) for a singular mean field equation on the unit disk (Bartolucci and Montefusco in Nonlinearity 19:611–631, 2006), for any N≥3 we characterize a subset of the 2π/N-symmetric part of the critical set of the N-vortex singular Hamiltonian. In particular we prove that this critical subset is of saddle type. As a consequence of our result, and motivated by a recently posed open problem (Bartolucci et al. in Commun. Partial Differ. Equ. 29(7–8):1241–1265, 2004), we can prove the existence of a multiple bubbling sequence of solutions for the singular mean field equation.     

Drug Release Kinetics from Biodegradable Polymers via Partial Differential Equations Models

In order to achieve prescribed drug release kinetics some authors have been investigating bi-phasic and possibly multi-phasic releases from blends of biodegradable polymers. Recently, experimental data for the release of paclitaxel have been published by Lao et al. (Lao and Venkatraman in J. Control. Release 130:9–14, 2008; Lao et al. in Eur. J. Pharm. Biopharm. 70:796–803, 2008). In Blanchet et al. (SIAM J. Appl. Math. 71(6):2269–2286, 2011) we validated a two-parameter quadratic ordinary differential equation (ODE) model against their experimental data from three representative neat polymers. In this paper we provide a gradient flow interpretation of the ODE model. A three-dimensional partial differential equation (PDE) model for the drug release in their experimental set up is introduced and its parameters are related to the ones of the ODE model. The gradient flow interpretation is extended to the study of the asymptotic concentrations that are solutions of the PDE model to determine the range of parameters that are suitable to simulate complete or partial drug release.     

Virtual control regularization of state constrained linear quadratic optimal control problems

A numerical method for linear quadratic optimal control problems with pure state constraints is analyzed. Using the virtual control concept introduced by Cherednichenko et al. (Inverse Probl. 24:1–21, 2008) and Krumbiegel and R?sch (Control Cybern. 37(2):369–392, 2008), the state constrained optimal control problem is embedded into a family of optimal control problems with mixed control-state constraints using a regularization parameter α>0. It is shown that the solutions of the problems with mixed control-state constraints converge to the solution of the state constrained problem in the L ² norm as α tends to zero. The regularized problems can be solved by a semi-smooth Newton method for every α>0 and thus the solution of the original state constrained problem can be approximated arbitrarily close as α approaches zero. Two numerical examples with benchmark problems are provided.     

Orbital stability of spherical galactic models

We consider the three dimensional gravitational Vlasov Poisson system which is a canonical model in astrophysics to describe the dynamics of galactic clusters. A well known conjecture (Binney, Tremaine in Galactic Dynamics, Princeton University Press, Princeton, 1987) is the stability of spherical models which are nonincreasing radially symmetric steady states solutions. This conjecture was proved at the linear level by several authors in the continuation of the breakthrough work by Antonov (Sov. Astron. 4:859–867, 1961). In the previous work (Lemou et al. in A new variational approach to the stability of gravitational systems, submitted, 2011), we derived the stability of anisotropic models under spherically symmetric perturbations using fundamental monotonicity properties of the Hamiltonian under suitable generalized symmetric rearrangements first observed in the physics literature (Lynden-Bell in Mon. Not. R. Astron. Soc. 144:189–217, 1969; Gardner in Phys. Fluids 6:839–840, 1963; Wiechen et al. in Mon. Not. R. Astron. Soc. 223:623–646, 1988; Aly in Mon. Not. R. Astron. Soc. 241:15, 1989). In this work, we show how this approach combined with a new generalized Antonov type coercivity property implies the orbital stability of spherical models under general perturbations.     

Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation

In this article, we continue our study of a system described by a class of initial boundary value problem (IBVP) of the Korteweg-de Vries (KdV) equation and the KdV Burgers (KdVB) equation posed on a finite interval with nonhomogeneous boundary conditions. While the system is known to be locally well-posed (Kramer et al. , [2010]; Rivas et al. in Math. Control Relat. Fields 1:61–81, [2011]) and its small amplitude solutions are known to exist globally, it is not clear whether its large amplitude solutions would blow up in finite time or not. This problem is addressed in this article from control theory point of view: look for some appropriate feedback control laws (with boundary value functions as control inputs) to ensure that the finite time blow-up phenomena would never occur. In this article, a simple, but nonlinear, feedback control law is proposed and the resulting closed-loop system is shown not only to be globally well-posed, but also to be locally exponentially stable for the KdV equation and globally exponentially stable for the KdVB equation.     

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.     

A Note on the Subcritical Two Dimensional Keller-Segel System

The existence of solution for the 2D-Keller-Segel system in the subcritical case, i.e. when the initial mass is less than 8π, is reproved. Instead of using the entropy in the free energy and free energy dissipation, which was used in the proofs (Blanchet et al. in SIAM J. Numer. Anal. 46:691–721, 2008; Electron. J. Differ. Equ. Conf. 44:32, 2006 (electronic)), the potential energy term is fully utilized by adapting Delort’s theory on 2D incompressible Euler equation (Delort in J. Am. Math. Soc. 4:553–386, 1991).     

Spiral Anchoring in Media with Multiple Inhomogeneities: A Dynamical System Approach

The spiral is one of nature’s more ubiquitous shapes: It can be seen in various media, from galactic geometry to cardiac tissue. Mathematically, spiral waves arise as solutions to reaction–diffusion partial differential equations (RDS). In the literature, various experimentally observed dynamical states and bifurcations of spiral waves have been explained using the underlying Euclidean symmetry of the RDS—see for example (Barkley in Phys. Rev. Lett. 68:2090–2093, 1992; Phys. Rev. Lett. 76:164–167, 1994; Sandstede et al. in C. R. Acad. Sci. 324:153–158, 1997; J. Differ. Equ. 141:122–149, 1997; J. Nonlinear Sci. 9:439–478, 1999), or additionally using the concept of forced Euclidean symmetry-breaking for situations where an inhomogeneity or anisotropy is present—see (LeBlanc in Nonlinearity 15:1179–1203, 2002; LeBlanc and Wulff in J. Nonlinear Sci. 10:569–601, 2000). In this paper, we further investigate the role of medium inhomogeneities on spiral wave dynamics by considering the effects of several localized sites of inhomogeneity. Using a model-independent approach based on n>1 simultaneous translational symmetry-breaking perturbations of the dynamics near rotating waves, we fully characterize the local anchoring behavior of the spiral wave in the n-dimensional parameter space of relative “amplitudes” of the individual perturbations. For the case n=2, we supplement the local anchoring results with a classification of the generic one-parameter bifurcation diagrams of anchored states which can be obtained by circling the origin of the two-dimensional amplitude parameter space. Numerical examples are given to illustrate our various results.     

Fractional Intertwinings Between Two Markov Semigroups

We define the notion of α-intertwining between two Markov Feller semigroups on and we give some examples. The 1-intertwining, in particular, is merely the intertwining via the first derivative operator. It can be used in the study of the existence of pseudo-inverses, a notion recently introduced by Madan et al. (2008) and Roynette and Yor (2008).      

Stability of Indices in the KKT Conditions and Metric Regularity in Convex Semi-Infinite Optimization

This paper deals with a parametric family of convex semi-infinite optimization problems for which linear perturbations of the objective function and continuous perturbations of the right-hand side of the constraint system are allowed. In this context, Cánovas et al. (SIAM J. Optim. 18:717–732, [2007]) introduced a sufficient condition (called ENC in the present paper) for the strong Lipschitz stability of the optimal set mapping. Now, we show that ENC also entails high stability for the minimal subsets of indices involved in the KKT conditions, yielding a nice behavior not only for the optimal set mapping, but also for its inverse. Roughly speaking, points near optimal solutions are optimal for proximal parameters. In particular, this fact leads us to a remarkable simplification of a certain expression for the (metric) regularity modulus given in Cánovas et al. (J. Glob. Optim. 41:1–13, [2008]) (and based on Ioffe (Usp. Mat. Nauk 55(3):103–162, [2000]; Control Cybern. 32:543–554, [2003])), which provides a key step in further research oriented to find more computable expressions of this regularity modulus. This research was partially supported by Grants MTM2005-08572-C03 (01-02) and MTM2006-27491-E (MEC, Spain, and FEDER, E.U.), ACOMP06/117-203 and ACOMP/2007/247-292 (Generalitat Valenciana, Spain), and CIO (UMH, Spain).     

A numerical algorithm for singular optimal LQ control systems

A numerical algorithm to obtain the consistent conditions satisfied by singular arcs for singular linear–quadratic optimal control problems is presented. The algorithm is based on the Presymplectic Constraint Algorithm (PCA) by Gotay-Nester (Gotay et al., J Math Phys 19:2388–2399, 1978; Volckaert and Aeyels 1999) that allows to solve presymplectic Hamiltonian systems and that provides a geometrical framework to the Dirac-Bergmann theory of constraints for singular Lagrangian systems (Dirac, Can J Math 2:129–148, 1950). The numerical implementation of the algorithm is based on the singular value decomposition that, on each step, allows to construct a semi-explicit system. Several examples and experiments are discussed, among them a family of arbitrary large singular LQ systems with index 2 and a family of examples of arbitrary large index, all of them exhibiting stable behaviour. Research partially supported by MEC grant MTM2004-07090-C03-03. SIMUMAT-CM, UC3M-MTM-05-028 and CCG06-UC3M/ESP-0850.     

Stability and Instance Optimality for Gaussian Measurements in Compressed Sensing

In compressed sensing, we seek to gain information about a vector x∈ℝ ^N from d ≪ N nonadaptive linear measurements. Candes, Donoho, Tao et al. (see, e.g., Candes, Proc. Intl. Congress Math., Madrid, 2006; Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006; Donoho, IEEE Trans. Inf. Theory 52:1289–1306, 2006) proposed to seek a good approximation to x via ℓ ₁ minimization. In this paper, we show that in the case of Gaussian measurements, ℓ ₁ minimization recovers the signal well from inaccurate measurements, thus improving the result from Candes et al. (Commun. Pure Appl. Math. 59:1207–1223, 2006). We also show that this numerically friendly algorithm (see Candes et al., Commun. Pure Appl. Math. 59:1207–1223, 2006) with overwhelming probability recovers the signal with accuracy, comparable to the accuracy of the best k-term approximation in the Euclidean norm when k∼d/ln N.     

On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects

We consider a class of dynamic advertising problems under uncertainty in the presence of carryover and distributed forgetting effects, generalizing the classical model of Nerlove and Arrow (Economica 29:129–142, 1962). In particular, we allow the dynamics of the product goodwill to depend on its past values, as well as previous advertising levels. Building on previous work (Gozzi and Marinelli in Lect. Notes Pure Appl. Math., vol. 245, pp. 133–148, 2006), the optimal advertising model is formulated as an infinite-dimensional stochastic control problem. We obtain (partial) regularity as well as approximation results for the corresponding value function. Under specific structural assumptions, we study the effects of delays on the value function and optimal strategy. In the absence of carryover effects, since the value function and the optimal advertising policy can be characterized in terms of the solution of the associated HJB equation, we obtain sharper characterizations of the optimal policy.     

Corrigendum to “The distribution of the number of arrivals in a subinterval of a busy period of a single server queue”

The purpose of this corrigendum is two-fold. First, we acknowledge that two results in our paper (Novak et al. in Queueing Syst. 53:105–114, 2006) can be obtained from earlier results of Prabhu and Bhat. Second, we make corrections to Theorem 2.2, Corollary 2.1 and Theorem 4.2 of Novak et al. (Queueing Syst. 53:105–114, 2006).      

On the Surrogate Gradient Algorithm for Lagrangian Relaxation

When applied to large-scale separable optimization problems, the recently developed surrogate subgradient method for Lagrangian relaxation (Zhao et al.: J. Optim. Theory Appl. 100, 699–712, 1999) does not need to solve optimally all the subproblems to update the multipliers, as the traditional subgradient method requires. Based on it, the penalty surrogate subgradient algorithm was further developed to address the homogenous solution issue (Guan et al.: J. Optim. Theory Appl. 113, 65–82, 2002; Zhai et al.: IEEE Trans. Power Syst. 17, 1250–1257, 2002). There were flaws in the proofs of Zhao et al., Guan et al., and Zhai et al.: for problems with inequality constraints, projection is necessary to keep the multipliers nonnegative; however, the effects of projection were not properly considered. This note corrects the flaw, completes the proofs, and asserts the correctness of the methods. This work is supported by the NSFC Grant Nos. 60274011, 60574067, the NCET program (No. NCET-04-0094) of China. The third author was supported in part by US National Science Foundation under Grants ECS-0323685 and DMI-0423607.     

Economic value of weather forecasting: the role of risk aversion

Extreme meteorological events have increased over the last decades and it is widely accepted that it is due to climate change (IPCC, Climate Change 2007, Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, 2007; Beniston et al., Clim. Change 81:71–95, 2007). Some of these extremes, like drought or frost episodes, largely affect agricultural outputs, and risk management becomes crucial. The goal of this paper it is to analyze farmers’ decisions about risk management, taking into account climatological and meteorological information. We consider a situation in which the farmer, as part of crop management, has available technology to protect the harvest from weather effects. This approach has been used by Murphy et al. (Mon. Weather Rev. 113:801–813, 1985), Katz and Murphy (J. Forecast. 9:75–86, 1990 and Economic Value of Weather and Climate Forecasts, pp. 183–217, Cambridge University Press, Cambridge, 1997) and others in the case when the farmer maximizes the expected returns. In our model, we introduce the attitude towards risk. Thus we can evaluate how the optimal decision is affected by the absolute risk aversion coefficient of Arrow and Pratt, and compute the economic value of the information in this context, while proposing a measure to estimate the amount of money that the farmer is willing to pay for this information in terms of the certainty equivalent.     

On Integrability of the Camassa–Holm Equation and Its Invariants

Using geometrical approach exposed in (Kersten et al. in J. Geom. Phys. 50:273–302, [2004] and Acta Appl. Math. 90:143–178, [2005]), we explore the Camassa–Holm equation (both in its initial scalar form, and in the form of 2×2-system). We describe Hamiltonian and symplectic structures, recursion operators and infinite series of symmetries and conservation laws (local and nonlocal). This work was supported in part by the NWO–RFBR grant 047.017.015 and RFBR–Consortium E.I.N.S.T.E.I.N. grant 06-01-92060.     

Decomposition-based Method for Sparse Semidefinite Relaxations of Polynomial Optimization Problems

We consider polynomial optimization problems pervaded by a sparsity pattern. It has been shown in Lasserre (SIAM J. Optim. 17(3):822–843, 2006) and Waki et al. (SIAM J. Optim. 17(1):218–248, 2006) that the optimal solution of a polynomial programming problem with structured sparsity can be computed by solving a series of semidefinite relaxations that possess the same kind of sparsity. We aim at solving the former relaxations with a decomposition-based method, which partitions the relaxations according to their sparsity pattern. The decomposition-based method that we propose is an extension to semidefinite programming of the Benders decomposition for linear programs (Benders, Comput. Manag. Sci. 2(1):3–19, 2005).     

Constant Approximation Algorithms for Embedding Graph Metrics into Trees and Outerplanar Graphs

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of Bǎdoiu et al. (Proceedings of the eighteenth annual ACM–SIAM symposium on discrete algorithms (SODA 2007), pp. 512–521, 2007) and Bǎdoiu et al. (Proceedings of the 11th international workshop on approximation algorithms for combinatorial optimization problems (APPROX 2008), Springer, Berlin, pp. 21–34, 2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of a metric relaxed minor and show that if G contains an α-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is ≥α. Then, for H=K _2,3, we present an algorithm which either finds an α-relaxed minor, or produces an O(α)-embedding into an outerplanar metric.