Incomplete financial markets model with nominal assets: Variational approach

We deal with the analysis of the general equilibrium model with incomplete financial markets and nominal assets. We assume that there are 2 periods of time, say today and tomorrow. We define a consumption, portfolio holding, commodity and asset price vector as an equilibrium vector associated with a given economy if at those prices and economies households maximize utility under a budget constraints and markets clear. While the path breaking proofs of existence by Cass [6] and Werner [25] use a fixed point argument, we provide an independent existence proof in terms of variational inequalities (about the variational approach for the analysis of general equilibrium models see for example [9] and [10]). The analysis presented in this paper indicates that the variational inequality approach promises to be applicable in many specifications of the incomplete market model.     

Monotone Methods for Markovian Equilibrium in Dynamic Economies

In this paper, we provide an overview of an emerging class of monotone map methods in analyzing distorted equilibrium in dynamic economies. In particular, we focus on proving the existence and characterization of competitive equilibrium in non-optimal versions of the optimal growth models. We suggest two alternative methods: an Euler equation method for a smooth, strongly concave environment, and a value function method for a non-smooth supermodular environment. We are able to extend this analysis to study models that allow for unbounded growth or a labor–leisure choice.     

Fixed point theorems in <Emphasis Type="Italic">b</Emphasis>-metric spaces with applications to differential equations

In this paper, we present some fixed point theorems for a class of contractive mappings in b-metric spaces. We verify the T-stability of Picard’s iteration and the P property for such mappings. We also give an example to support our assertions. In addition, by using our results, we obtain the existence and uniqueness of solution to some ordinary differential equations with initial value conditions. Further, we provide the precise mathematical expressions of solutions to such equations.     

Vectorial Ekeland variational principle for systems of equilibrium problems and its applications

For a family of vector-valued bifunctions,we introduce the notion of sequentially lower monotonity,which is strictly weaker than the lower semi-continuity of the second variables of the bifunctions.Then,we give a general version of vectorial Ekeland variational principle(briefly,denoted by EVP) for a system of equilibrium problems,where the sequentially lower monotone objective bifunction family is defined on products of sequentially lower complete spaces(concerning sequentially lower complete spaces,see Zhu et al(2013)),and taking values in a quasi-ordered locally convex space.Besides,the perturbation consists of a subset of the ordering cone and a family {p_i}_(i∈I) of negative functions satisfying for each i∈I,p_i(x_i,y_i) = 0 if and only if x_i=y_i.From the general version,we can deduce several particular equilibrium versions of EVP,which can be applied to show the existence of solutions for countable systems of equilibrium problems.In particular,we obtain a general existence result of solutions for countable systems of equilibrium problems in the setting of sequentially lower complete spaces.By weakening the compactness of domains and the lower semi-continuity of objective bifunctions,we extend and improve some known existence results of solutions for countable system of equilibrium problems in the setting of complete metric spaces(or Fréchet spaces).When the domains are non-compact,by using the theory of angelic spaces(see Floret(1980)),we generalize some existence results on solutions for countable systems of equilibrium problems by extending the framework from reflexive Banach spaces to the strong duals of weakly compactly generated spaces.     

Variational Formulation of a General Equilibrium Model with Incomplete Financial Markets and Numeraire Assets: Existence

We present a general equilibrium model with incomplete financial markets and numeraire assets. We assume that there are 2 periods of time, say today and tomorrow. In period 0, households exchange goods and assets and then consumption takes place; in period 1, one of S possible states of nature occurs. In each of them, assets pay their returns, which are measured in units of a given physical good, i.e., the numeraire commodity; households exchange goods; finally, consumption takes place. We define a consumption, portfolio holding, commodity and asset price vector as an equilibrium vector associated with a given economy, if at those prices and economies households maximize, and market clears. While the existence proof by Geneakoplos and Polemarchakis (Essays in honor of K.J. Arrow, vol 3, Cambridge University Press, Cambridge, pp 65–95, 1986) uses a fixed point argument, we provide an independent existence result in terms of variational inequalities. That approach allows us to get the desired existence result under some different and more general or realistic assumptions than those usually made in the literature.     

Nonhomogeneous Markov processes

We present the foundations of the theory of nonhomogeneous Markov processes in general state spaces and we give a survey of the fundamental papers in this topic. We consider the following questions:

1. The existence of transition functions for a Markov process.
2. The construction of regularization of processes.
3. The properties of right and left processes: the strict Markov property, the behavior of excessive functions, etc.
4. The relation of right and left processes with dual homogeneous processes and the application of the results of the nonhomogeneous theory to dual homogeneous processes, etc.

     

Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces

We use methods of harmonic analysis and group representation theory to study the spectral properties of the abstract parabolic operator \({\mathcal{L} = -{\rm d}/{\rm d}t + A}\) in homogeneous function spaces. We provide sufficient conditions for invertibility of such operators in terms of the spectral properties of the operator A and the semigroup generated by A. We introduce a homogeneous space of functions with absolutely summable spectrum and prove a generalization of the Gearhart–Prüss Theorem for such spaces. We use the results to prove existence and uniqueness of solutions of a certain class of non-linear equations.     

Existence of competitive equilibrium in coalition production economies with a continuum of agents

Aumann (Econometrica 34: 1–17, 1966) established the existence of competitive equilibria in exchange economies with a continuum of traders. This result has been extended to special production economies under the condition that the production sets are countably additive or superadditive in the literature. In this paper, we extend Aumann’s result to coalition production economies with a continuum of traders in which each coalition can have a different production set and the production sets need not to be additive or superadditive. Our new existence theorem for competitive equilibrium implies the classical result by Hildenbrand (Econometrica 38: 608–623, 1970), and implies those by Sondermann (J Econ Theory 8: 259–291, 1974) and Greenberg et al. (J Math Econ 6: 31–41, 1979) under the assumption that the production sets are convex.     

Fixed point theorems in new generalized metric spaces

The aim of our paper is to present new fixed point theorems under very general contractive conditions in generalized metric spaces which were recently introduced by Jleli and Samet in [Fixed Point Theory Appl. 2015 (2015), doi: 10.1186/s13663-015-0312-7]. Although these spaces are not endowed with a triangle inequality, these spaces extend some well known abstract metric spaces (for example, b-metric spaces, Hitzler–Seda metric spaces, modular spaces with the Fatou property, etc.). We handle several types of contractive conditions. The main theorems we present involve a reflexive and transitive binary relation that is not necessarily a partial order. We give a counterexample to a recent fixed point result of Jleli and Samet. Our results extend and unify recent results in the context of partially ordered abstract metric spaces.     

Maximal elements in noncompact spaces with application to equilibria

A new maximal theorem for -majorized correspondences in noncompact spaces is presented and applied to obtain an equilibrium existence theorem for noncompact abstract economies. The corresponding results of Borglin and Keiding (1976), Yannelis and Prabhakar (1983), Ding and Tan (1993), Yuan and Tarafdar (1996), and Ding and Yuan (1998) are generalized by our results.

     

Gradient flow for the Boltzmann entropy and Cheeger’s energy on time-dependent metric measure spaces

We introduce notions of dynamic gradient flows on time-dependent metric spaces as well as on time-dependent Hilbert spaces. We prove existence of solutions for a class of time-dependent energy functionals in both settings. In particular, in the case when each underlying space satisfies a lower Ricci curvature bound in the sense of Lott, Sturm and Villani, we provide time-discrete approximations of the time-dependent heat flows introduced in Kopfer and Sturm (Heat flows on time-dependent metric measure spaces and super-Ricci flows, 2017. arXiv:1611.02570).     

Nonlinear degenerate elliptic problems with {{W^{1,1}_{0}(\Omega)}} solutions

We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique ${{W^{1,1}_{0}(\Omega)}}$ distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.     

On a Regularized Family of Models for Homogeneous Incompressible Two-Phase Flows

We consider a general family of regularized models for incompressible two-phase flows based on the Allen–Cahn formulation in \(n\) -dimensional compact Riemannian manifolds for \(n=2,3\) . The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes- \(\alpha \) -like model, the Leray- \(\alpha \) model, the modified Leray- \(\alpha \) model, the simplified Bardina model, the Navier–Stokes–Voight model, and the Navier–Stokes model) for the fluid velocity \(u\) suitably coupled with a convective Allen–Cahn equation for the order (phase) parameter \(\phi \) . We give a unified analysis of the entire three-parameter family of two-phase models using only abstract mapping properties of the principal dissipation and smoothing operators and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We establish existence, stability, and regularity results and some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the \(\alpha \rightarrow 0\) limit in \(\alpha \) models. Then we show the existence of a global attractor and exponential attractor for our general model and establish precise conditions under which each trajectory \(\left( u,\phi \right) \) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality. We also derive new results on the existence of global and exponential attractors for the regularized family of Navier–Stokes equations and magnetohydrodynamics models that improve and complement the results of Holst et al. (J Nonlinear Sci 20(5):523–567, 2010). Finally, our analysis is applied to certain regularized Ericksen–Leslie models for the hydrodynamics of liquid crystals in \(n\) -dimensional compact Riemannian manifolds.     

Double solid twistor spaces: the case of arbitrary signature

In a recent paper ([9]) we constructed a series of new Moishezon twistor spaces which are a kind of variant of the famous LeBrun twistor spaces. In this paper we explicitly give projective models of another series of Moishezon twistor spaces on n CP ² for arbitrary n≥3, which can be regarded as a generalization of the twistor spaces of ‘double solid type’ on 3CP ² studied by Kreußler, Kurke, Poon and the author. Similarly to the twistor spaces of ‘double solid type’ on 3CP ², projective models of the present twistor spaces have a natural structure of double covering of a CP ²-bundle over CP ¹. We explicitly give a defining polynomial of the branch divisor of the double covering, whose restriction to fibers is degree four. If n≥4 these are new twistor spaces, to the best of the author’s knowledge. We also compute the dimension of the moduli space of these twistor spaces. Differently from [9], the present investigation is based on analysis of pluri-(half-)anticanonical systems of the twistor spaces.     

New Average Optimality Conditions for Semi-Markov Decision Processes in Borel Spaces

This paper deals with semi-Markov decision processes under the average expected criterion. The state and action spaces are Borel spaces, and the cost/reward function is allowed to be unbounded from above and from below. We give another set of conditions, under which the existence of an optimal (deterministic) stationary policy is proven by a new technique of two average optimality inequalities. Our conditions are slightly weaker than those in the existing literature, and some new sufficient conditions for the verifications of our assumptions are imposed on the primitive data of the model. Finally, we illustrate our results with three examples.     

Full characterizations of minimax inequality,fixed point theorem,saddle point theorem,and KKM principle in arbitrary topological spaces

This paper provides necessary and sufficient conditions for the existence of solutions for some important problems from optimization and non-linear analysis by replacing two typical conditions—continuity and quasiconcavity with a unique condition, weakening topological vector spaces to arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, \(\gamma \)-recursive transfer lower semicontinuity, which fully characterizes the existence of \(\gamma \)-equilibrium of minimax inequality without imposing any restrictions on topological space. The result is then used to provide full characterizations of fixed point theorem, saddle point theorem, and KKM principle.     

Generic elements in isometry groups of Polish ultrametric spaces

This paper presents a study of generic elements in full isometry groups of Polish ultrametric spaces. We obtain a complete characterization of Polish ultrametric spaces X whose isometry group Iso(X) has a neighborhood basis at the identity consisting of open subgroups with ample generics. It also gives a characterization of the existence of an open subgroup in Iso(X) with a comeager conjugacy class.We also study the transfinite sequence defined by the projection of a Polish ultrametric space X on the ultrametric space of orbits of X under the action of Iso(X).     

Retraction algorithms for solving variational inequalities,pseudomonotone equilibrium problems,and fixed-point problems in Banach spaces

In this paper, using sunny generalized nonexpansive retractions which are different from the metric projection and generalized metric projection in Banach spaces, we present new extragradient and line search algorithms for finding the solution of a J-variational inequality whose constraint set is the common elements of the set of fixed points of a family of generalized nonexpansive mappings and the set of solutions of a pseudomonotone J-equilibrium problem for a J -α-inverse-strongly monotone operator in a Banach space. To prove strong convergence of generated iterates in the extragradient method, we introduce a ? _?-Lipschitz-type condition and assume that the equilibrium bifunction satisfies this condition. This condition is unnecessary when the line search method is used instead of the extragradient method. Using FMINCON optimization toolbox in MATLAB, we give some numerical examples and compare them with several existence results in literature to illustrate the usability of our results.     

The computational content of Walras’ existence theorem

In [Y. Tanaka, Undecidability of the Uzawa equivalence theorem and LLPO, Appl. Math. Comput. 201 (2008) 378-383] Yasuhito Tanaka showed that Walras’ existence theorem implies the nonconstructive lesser limited principle of omniscience (LLPO); it follows that Walras’ existence theorem does not admit a constructive proof. We give a constructive proof of an approximate version of Walras’ existence theorem from which the full theorem can be recovered with an application of LLPO. We then push Uzawa’s equivalence theorem to the level of approximate solutions, before considering economies with at most one equilibrium.     

On Vector Quasi-Equilibrium Problems with Set-Valued Maps

In this paper, we introduce a new class of vector quasi-equilibrium problems with set-valued maps. Almost all the vector equilibrium models of the Blum-Oettli type in the literature are special cases of our new class of equilibrium problems under consideration. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the -diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for such vector equilibrium problems.