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Nobusumi Sagara

Publications and source records attributed to Nobusumi Sagara.

12 recordsLinked to original sources
arXiv2021

Fuzzy Core Equivalence in Large Economies: A Role for the Infinite-Dimensional Lyapunov Theorem

We present the equivalence between the fuzzy core and the core under minimal assumptions. Due to the exact version of the Lyapunov convexity theorem in Banach spaces, we clarify that the additional structure of commodity spaces and preferences is unnecessary whenever the measure space of agents is "saturated". As a spin-off of the above equivalence, we obtain the coincidence of the core, the fuzzy core, and the Schmeidler's restricted core under minimal assumptions. The coincidence of the fuzzy core and the restricted core has not been articulated anywhere.

econ.TH↗
arXiv2020

Value Functions and Optimality Conditions for Nonconvex Variational Problems with an Infinite Horizon in Banach Spaces

We investigate the value function of an infinite horizon variational problem in the infinite-dimensional setting. Firstly, we provide an upper estimate of its Dini--Hadamard subdifferential in terms of the Clarke subdifferential of the Lipschitz continuous integrand and the Clarke normal cone to the graph of the set-valued mapping describing dynamics. Secondly, we derive a necessary condition for optimality in the form of an adjoint inclusion that grasps a connection between the Euler--Lagrange condition and the maximum principle.

math.OC↗
arXiv2019

Optimality Conditions for Nonconvex Variational Problems with Integral Constraints in Banach Spaces

This paper exemplifies that saturation is an indispensable structure on measure spaces to obtain the existence and characterization of solutions to nonconvex variational problems with integral constraints in Banach spaces and their dual spaces. We provide a characterization of optimality via the maximum principle for the Hamiltonian and an existence result without the purification of relaxed controls, in which the Lyapunov convexity theorem in infinite dimensions under the saturation hypothesis on the underlying measure space plays a crucial role. We also demonstrate that the existence of solutions for certain class of primitives is necessary and sufficient for the measure space to be saturated.

math.OC↗
arXiv2019

Subdifferentials of Value Functions in Nonconvex Dynamic Programming for Nonstationary Stochastic Processes

The main goal of this paper is to apply the machinery of variational analysis and generalized differentiation to study infinite horizon stochastic dynamic programming (DP) with discrete time in the Banach space setting without convexity assumptions. Unlike to standard stochastic DP with stationary Markov processes, we investigate here stochastic DP in $L^p$ spaces to deal with nonstationary stochastic processes, which describe a more flexible learning procedure for the decision-maker. Our main concern is to calculate generalized subgradients of the corresponding value function and to derive necessary conditions for optimality in terms of the stochastic Euler inclusion under appropriate Lipschitzian assumptions. The usage of the subdifferential formula for integral functionals on $L^p$ spaces allows us, in particular, to find verifiable conditions to ensure smoothness of the value function without any convexity and/or interiority assumptions.

math.OC↗
arXiv2018

Vector Representation of Preferences on $σ$-Algebras and Fair Division in Saturated Measure Spaces

The purpose of this paper is twofold. First, we axiomatize preference relations on a $σ$-algebra of a saturated measure space represented by a vector measure and furnish a utility representation in terms of a nonadditive measure satisfying the appropriate requirement of continuity and convexity. Second, we investigate the fair division problems in which each individual has nonadditive preferences on a $σ$-algebra invoking our utility representation result. We show the existence of individually rational Pareto optimal partitions, Walrasian equilibria, core partitions, and Pareto optimal envy-free partitions.

math.FA↗
arXiv2017

Recursive Variational Problems in Nonreflexive Banach Spaces with an Infinite Horizon: An Existence Result

We investigate variational problems with recursive integral functionals governed by infinite-dimensional differential inclusions with an infinite horizon and present an existence result in the setting of nonreflexive Banach spaces. We find an optimal solution in a Sobolev space taking values in a Banach space under the Cesari type condition. We also investigate sufficient conditions for the existence of solutions to the initial value problem for the differential inclusion.

math.OC↗
arXiv2017

Relaxation and Purification for Nonconvex Variational Problems in Dual Banach Spaces: The Minimization Principle in Saturated Measure Spaces

We formulate bang-bang, purification, and minimization principles in dual Banach spaces with Gelfand integrals and provide a complete characterization of the saturation property of finite measure spaces. We also present a new application of the relaxation technique to large economies with infinite-dimensional commodity spaces, where the space of agents is modeled as a finite measure space. We propose a "relaxation" of large economies, which is regarded as a reasonable convexification of original economies. Under the saturation hypothesis, the relaxation and purification techniques enable us to prove the existence of Pareto optimal allocations without convexity assumptions.

math.FA↗
arXiv2017

Fatou's Lemma, Galerkin Approximations and the Existence of Walrasian Equilibria in Infinite Dimensions

This essay has three objectives: (i) to report recent generalizations of Fatou's lemma to multi-functions taking values in a Banach space, and framed in terms of both Bochner and Gelfand integration; (ii) to delineate the importance of Galerkin approximations in Walrasian general equilibrium theory with a continuum of agents and commodities; and thereby (iii) to present two new results on the existence of a Walrasian equilibrium in economies where the continuum of agents is formalized as a saturated measure space.

math.FA↗
arXiv2016

Relaxed Large Economies with Infinite-Dimensional Commodity Spaces: The Existence of Walrasian Equilibria

Whereas "convexification by aggregation" is a well-understood procedure in mathematical economics, "convexification by randomization" has largely been limited to theories of statistical decision-making, optimal control and non-cooperative games. In this paper, in the context of classical Walrasian general equilibrium theory, we offer a comprehensive treatment of {\it relaxed economies} and their {\it relaxed Walrasian equilibria}: our results pertain to a setting with a finite or a continuum of agents, and a continuum of commodities modeled either as an ordered separable Banach space or as an $L^\infty$-space. As a substantive consequence, we demonstrate that the convexity hypothesis can be removed from the original large economy under the saturation hypothesis, and that existing results in the antecedent literature can be effortlessly recovered.

math.OC↗
arXiv2016

Subdifferentials of Nonconvex Integral Functionals in Banach Spaces with Applications to Stochastic Dynamic Programming

The paper concerns the investigation of nonconvex and nondifferentiable integral functionals on general Banach spaces, which may not be reflexive and/or separable. Considering two major subdifferentials of variational analysis, we derive nonsmooth versions of the Leibniz rule on subdifferentiation under the integral sign, where the integral of the subdifferential set-valued mappings generated by Lipschitzian integrands is understood in the Gelfand sense. Besides examining integration over complete measure spaces and also over those with nonatomic measures, our special attention is drawn to a stronger version of measure nonatomicity, known as saturation, to invoke the recent results of the Lyapunov convexity theorem type for the Gelfand integral of the subdifferential mappings. The main results are applied to the subdifferential study of the optimal value functions and deriving the corresponding necessary optimality conditions in nonconvex problems of stochastic dynamic programming with discrete time on the infinite horizon.

math.OC↗
arXiv2015

Maharam-Types and Lyapunov's Theorem for Vector Measures on Locally Convex Spaces without Control Measures

We formulate the saturation property for vector measures in lcHs as a nonseparability condition on the derived Boolean $σ$-algebras by drawing on the topological structure of vector measure algebras. We exploit a Pettis-like notion of vector integration in lcHs, the Bourbaki--Kluvánek--Lewis integral, to derive an exact version of the Lyapunov convexity theorem in lcHs without the BDS property. We apply our Lyapunov convexity theorem to the bang-bang principle in Lyapunov control systems in lcHs to provide a further characterization of the saturation property.

math.FA↗