Convexity Hypothesis Research Articles

An Aggregation Theorem for the Valuation of Equity Under Linear Information Dynamics

We state an Aggregation Theorem which shows that the recursion value of equity is functionally proportional to its adaptation value. Since the recursion value of equity is equal to its book value plus the expected present value of its abnormal earnings, it follows that the adaptation value of equity can normally be determined by a process of simple quadrature. We demonstrate the application of the Aggregation Theorem using two stochastic processes. The first uses the linear information dynamics of the Ohlson (1995) model. The second uses linear information dynamics based on the Cox, Ingersoll and Ross (1985)‘square root’ process. Both these processes lead to closed form expressions for the adaptation and overall market value of equity. There are, however, many other processes which are compatible with the Aggregation Theorem. These all show that the market value of equity will be a highly convex function of its recursion value. The empirical evidence we report for UK companies largely supports the convexity hypothesis.

An extension to a theorem of Jörgens, Calabi, and Pogorelov

Communications on Pure and Applied MathematicsVolume 56, Issue 5 p. 549-583 An extension to a theorem of Jörgens, Calabi, and Pogorelov L. Caffarelli, L. Caffarelli caffarel@math.utexas.edu University of Texas, Department of Mathematics, Austin, TX 78712Search for more papers by this authorYanYan Li, YanYan Li yyli@math.rutgers.edu Rutgers–The State University, Department of Mathematics, New Brunswick, NJ 08903Search for more papers by this author L. Caffarelli, L. Caffarelli caffarel@math.utexas.edu University of Texas, Department of Mathematics, Austin, TX 78712Search for more papers by this authorYanYan Li, YanYan Li yyli@math.rutgers.edu Rutgers–The State University, Department of Mathematics, New Brunswick, NJ 08903Search for more papers by this author First published: 21 February 2003 https://doi.org/10.1002/cpa.10067Citations: 68AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume56, Issue5May 2003Pages 549-583 RelatedInformation

On Brown's and Newton's methods with convexity hypotheses

In the context of the monotone Newton theorem (MNT) it has been conjectured that discretised Brown iterations converge at least as fast as discretised Newton iterations, because such is the case for analytic iterations. With easily verified hypotheses, it is proved here that Brown analytic iterations converge strictly faster than Newton ones. As a consequence, the same result holds for discretised iterations with conveniently small incremental steps. However, in the general context of the MNT, it may happen that Newton's discretised method converges faster than Brown's, but this situation can be remedied in many cases by conveniently shifting the initial value, so that those hypotheses ensuring the reverse are satisfied. Thus, a fairly effective solution is given to the problem stated initially.

Continuum Limits of Discrete Systems without Convexity Hypotheses

We describe the variational limit of one-dimensional nearest-neighbour systems of interactions, under no structure hypotheses on the discrete energy densities. We show that the continuum limit is characterized by a bulk and a interfacial energy density, which can be explicitly computed from the discrete energies through operations of limit, scaling and regularization that highlight possible bulk oscillations and multiple cracking.

A nonparametric test of the non-convexity of regression

This paper proposes a nonparametric test of the non-convexity of a smooth regression function based on least squares or hybrid splines. By a simple formulation of the convexity hypothesis in the lcass of all polynomial cubic splines, we build a test which has an asymptotic size equal to the nominal level. It is shown that the test is consistent and is robust to nonnormality. The behavior of the test under the local alternatives is studied.

Asymptotic Completeness for Multiparticle Dispersive Charge Transfer Models

We prove the asymptotic completeness of wave operators associated with the scattering of a quantum particle in a field of classical particles in the dispersive case when the free hamiltonian has the formH0=p(D) withpelliptic of degreem⩾1 and satisfying some convexity hypotheses.

Necessary Conditions for Optimal Control Problems Involving Nonlinear Differential Algebraic Equations

Dynamic models which take the form of a coupled set of differential and algebraic equations (DAEs) are widely used in process systems engineering. Necessary conditions of optimality for optimal control problems involving such models are derived. A strong Maximum Principle is obtained under a convexity hypothesis on the velocity set. An example illustrates that the strong Maximal Principle may be violated when this hypothesis is dropped. For problems involving nonconvex velocity sets, however, a weak Maximum Principle is valid.

A statistical approach to the evaluation of characters to estimate evolutionary relationships among the species of the aquatic subterranean genus, Iberobathynella (Crustacea, Syncarida)

The 13 known species of the genus Iberobathynella were studied. Twenty-six characters with usually two or three states were revealed to have low within-species variability but show clear differences among some species. These characters were hypothesized to have states convex on the branching pattern of the phylogenetic lines that gave rise to these 13 species (i.e. be uniquely derived). Each pair of these hypotheses was tested for logical compatibility; then, for each character, a new character was created by choosing equiprobably one of the possible permutations of the 13 species to rename the species in each state. Characters created in this random way would have convex states only by chance, not by evolution. This random character was tested with each of the remaining 25 for logical compatibility as hypotheses of convexity. For each character, one thousand such random characters were created and tested. Sixteen observed characters were compatible with more other observed characters than 90% of their randomly generated counterparts, and so were considered plausibly non-random. They were used to speculate on branching patterns of the phylogenetic lines among the 13 species.

A statistical approach to the evaluation of characters to estimate evolutionary relationships among the species of the aquatic subterranean genus,Iberobathynella(Crustacea, Syncarida)

The 13 known species of the genusIberobathynellawere studied. Twenty-six characters with usually two or three states were revealed to have low within-species variability but show clear differences among some species. These characters were hypothesized to have states convex on the branching pattern of the phylogenetic lines that gave rise to these 13 species (i.e. be uniquely derived). Each pair of these hypotheses was tested for logical compatibility; then, for each character, a new character was created by choosing equiprobably one of the possible permutations of the 13 species to rename the species in each state. Characters created in this random way would have convex states only by chance, not by evolution. This random character was tested with each of the remaining 25 for logical compatibility as hypotheses of convexity. For each character, one thousand such random characters were created and tested. Sixteen observed characters were compatible with more other observed characters than 90% of their randomly generated counterparts, and so were considered plausibly non-random. They were used to speculate on branching patterns of the phylogenetic lines among the 13 species.

Comparison theorems for a third order method

It is proved that monotone convergence of a third order bracketing method for nonlinear systems with convexity hypotheses is improved whenever the starting points are chosen componentwise as close to the root as possible.

Optimal Control and Admissible Relaxation of Uncertain Nonlinear Elliptic Systems

In this article we study the optimal control of uncertain systems monitored by nonlinear elliptic equations. First under convexity hypotheses, we show that the min–max problem has a solution. Then we drop the convexity assumptions and we pass to the larger relaxed system. We show that this always has a solution under very general hypotheses on the data. We also produce conditions that guarantee that the relaxation is admissible; i.e., the relaxed problem is the “closure” of the original one.

Optimal control problems involving optimal economic growth with infinite horizon

We examine the existence of optimal solutions for a very general optimal economic growth problem with unbounded planning time horizon. A convexity hypothesis about a certain orientor field plays a crucial role in the proof. Then we conduct sensitivity analysis and determine the relation between optimal pairs and the initial capital accumulation. Finally we see what happens if the convexity hypothesis we have made no longer exists. This leads us to the introduction of the relaxed system and its comparison with the original one.

On the baroclinic dynamics, hamiltonian formulation and general stability characteristics of density-driven surface currents and fronts over a sloping continental shelf

A new theory is presented to describe the baroclinic dynamics of density-driven currents and fronts over a sloping continental shelf. The frontal dynamics is geostrophic to leading-order but not quasi-geostrophic since the dynamic frontal height is not small in comparison with the scale frontal thickness. The evolution of the underlying slope water is modelled quasigeostrophically and includes the influence of a background vorticity gradient due to the sloping bottom . The two layers are coupled together via baroclinic vortex-tube stretching associated with the perturbeddensity-driven current. The current dynamics includes the advection of mean flow vorticity . The model equations are obtained in a formal asymptotic expansion of there levant two-layer shallow-water equation sand boundary conditions. It is shown that the governing equations for the model can be put in to non-canonical hamiltonian form. A comprehensive analysis of the general linear and nonlinear stability characteristics of the governing equations is given. The normal mode problem associated with steady along-shore currents is studied and sufficient stability and necessary instability conditions are presented. It is shown that a zero in the frontal vorticity gradient is not needed for instability. Jump conditions for the perturbation frontal thickness are systematically derived associated with the continuity of pressure and normal mass flux for steady frontal configurations that possess discontinuities in the velocity or vorticity, and rigorous regularity conditions are obtained for the perturbation thickness on outcroppings. The formal stability of arbitrary steady currents is studied. It is shown how to obtain general steady current solutions as a variational solution to a suitably constrained hamiltonian. General criteria are obtained for establishing the linear stability of these steady density-driven currents in the sense of Liapunov. In the limit of steady parallel along-shore flow, the formal stability results reduce to the sufficient conditions found for the normal modes. Finally, the nonlinear stability of steady density-driven currents and fronts is studied. Based on the formal stability analysis, appropriate convexity hypothesis are found that rigorously establish nonlinear stability of steady currents in the sense of Liapunov, and establish nonlinear saturation bounds on the perturbation flow with respect to a potential enstrophy/energy norm .

Weak Sharp Minima in Mathematical Programming

The notion of a sharp, or strongly unique, minimum is extended to include the possibility of a nonunique solution set. These minima will be called weak sharp minima. Conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases. These conditions are also shown to be sufficient under the appropriate convexity hypotheses. The existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem. In particular, a result of Mangasarian and Meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty. Consequences for the convergence theory of algorithms are also examined, especially conditions yielding finite termination.

Nonconvex and relaxed infinite-horizon optimal control problems

In this paper, we consider the Lagrange problem of optimal control defined on an unbounded time interval in which the traditional convexity hypotheses are not met. Models of this form have been introduced into the economics literature to investigate the exploitation of a renewable resource and to treat various aspects of continuous-time investment. An additional distinguishing feature in the models considered is that we do not assume a priori that the objective functional (described by an improper integral) is finite, and so we are led to consider the weaker notions of overtaking and weakly overtaking optimality. To treat these models, we introduce a relaxed optimal control problem through the introduction of chattering controls. This leads us naturally to consider the relationship between the original problem and the convexified relaxed problem. In particular, we show that the relaxed problem may be viewed as a limiting case for the original problem. We also present several examples demonstrating the applicability of our results.

Optimal control and relaxation of nonlinear elliptic systems

In this paper we study the optimal control of systems driven by nonlinear elliptic partial differential equations. First, with the aid of an appropriate convexity hypothesis we establish the existence of optimal admissible pairs. Then we drop the convexity hypothesis and we pass to the larger relaxed system. First we consider a relaxed system based on the Gamkrelidze-Warga approach, in which the controls are transition probabilities. We show that this relaxed problem has always had a solution and the value of the problem is that of the original one. We also introduce two alternative formulations of the relaxed problem (one of them control free), which we show that they are both equivalent to the first one. Then we compare those relaxed problems, with that of Buttazzo which is based on the Λ-regularization of the “extended” cost functional. Finally, using a powerful multiplier rule of Ioffe-Tichomirov, we derive necessary conditions for optimality in systems with inequality state constraints.

Optimal control and relaxation of nonlinear elliptic systems

In this paper we study the optimal control of systems driven by nonlinear elliptic partial differential equations. First with the aid of an appropriate convexity hypothesis we establish the existence of optimal admissible pairs. Then we drop the convexity hypothesis and we pass to the larger relaxed system. First we consider a relaxed system based on the Gamkrelidze-Warga approach, in which the controls are transition probabilities. We show that this relaxed problem has always had a solution and the value of the problem is that of the original one. We also introduce two alternative formulations of the relaxed problem (one of them control free), which we show that they are both equivalent to the first one. Then we compare those relaxed problems, with that of Buttazzo which is based on the Γ-regularization of the “extended” cost functional. Finally using a powerful multiplier rule of Ioffe-Tichomirov, we derive necessary conditions for optimality in systems with inequality state constraints.

Separation and von Neumann intersection theorems

We give some separation theorems to extend the intersection theorem of von Neumann, Fan, and others, omitting hypotheses of convexity and local convexity for one of the coordinate spaces.

Optimality conditions and duality for a class of continuous-time programming problems with nonlinear operator equality and inequality constraints

Using a minmax approach, we establish saddle point optimality conditions and Lagrangian duality relations under generalized convexity hypotheses for a class of continuous-time programming problems with nonlinear operator equality and inequality constraints. We also discuss the implications of these results for differentiable problems.

Optimal control of nonlinear evolution inclusions

In this paper, we study the optimal control of nonlinear evolution inclusions. First, we prove the existence of admissible trajectories and then we show that the set that they form is relatively sequentially compact and in certain cases sequentially compact in an appropriate function space. Then, with the help of a convexity hypothesis and using Cesari's approach, we solve a general Lagrange optimal control problem. After that, we drop the convexity hypothesis and pass to the relaxed system, for which we prove the existence of optimal controls, we show that it has a value equal to that of the original one, and also we prove that the original trajectories are dense in an appropriate topology to the relaxed ones. Finally, we present an example of a nonlinear parabolic optimal control that illustrates the applicability of our results.