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In this paper, we study the distribution of firm size by using a model based on Sato's paper in 1970, and proved the static distribution of firm size satisfies Pareto distribution in its upper tail. [ABSTRACT FROM AUTHOR]

We study the minimal hedging risk for a bounded European contingent claim when we use a convex risk measure. We find the infimum of hedging risk by using a kind of min-max theorem. Also we show that this infimum is again regarded as a convex risk measure. [ABSTRACT FROM AUTHOR]

When preferences may not be homothetic but satisfy other regularity conditions such as monotonicity, the market excess demand function is characterized by continuity and Walras' law on almost entire region of the price simplex. In particular, Mas-Colell (1977) shows that for a continuous function f defined on the interior of the price simplex satisfying Walras' law and the boundary condition, there exists an exchange economy ℰ whose excess demand function is approximately equal to f and the equilibrium price set of ℰ is exactly equal to the one of f. This paper shows that if f may be finite on the boundary of the price simplex, ℰ can be chosen so that the equilibrium price set of ℰ is approximately equal to the one of f. Theorem 3 in Wong (1997), showing the equivalence between Brouwer's fixed-point theorem and Arrow-Debreu's equilibrium existence theorem, follows from this result. [ABSTRACT FROM AUTHOR]

We study the filtering problem in which a system process Xt(ε) and a observing process Yt(ε) depend on the parameter ε, and Xt(ε) converges to a deterministic function Xt(0) as ε ↓ 0. We give an asymptotic expansion formula in Lp for the conditional expectation of a function of Xt(ε) under the σ-field generated by the process Ys(ε), 0 ≤ s ≤ t. [ABSTRACT FROM AUTHOR]

The modern theory of games initiated by John von Neumann with the minimax theorem in 1928 has now grown to be an indispensable analytical framework for social sciences, and economics in particular. In this paper, we shall review the early history of game theory from von Neumann to John F. Nash, the founder of the non-cooperative game theory, including Émile Borel, Hugo Steinhaus and Oskar Morgenstern, thereby pointing out a hint of why game theory has come to be widely applied in economics. [ABSTRACT FROM AUTHOR]

S. Kusuoka [K01, Theorem 4] gave an interesting dual characterization of law invariant coherent risk measures, satisfying the Fatou property. The latter property was introduced by F. Delbaen [D 02]. In the present note we extend Kusuoka's characterization in two directions, the first one being rather standard, while the second one is somewhat surprising. Firstly we generalize — similarly as M. Fritelli and E. Rossaza Gianin [FG 05] — from the notion of coherent risk measures to the more general notion of convex risk measures as introduced by H. Föllmer and A. Schied [FS 04]. Secondly — and more importantly — we show that the hypothesis of Fatou property may actually be dropped as it is automatically implied by the hypothesis of law invariance. We also introduce the notion of the Lebesgue property of a convex risk measure, where the inequality in the definition of the Fatou property is replaced by an equality, and give some dual characterizations of this property. [ABSTRACT FROM AUTHOR]

In this paper, we will give exact calculations of probability disrtributions of continuous rank processes in Brownian case, Brownian motion with drift case, Pinned Brownian motion case. These calculations are based on the results of the joint distributions of Brownian motion and its soujourn time. Also we give new exotic options using rank statistics and calculate the price of rank options. [ABSTRACT FROM AUTHOR]

In this paper we study options on a unit-type closed-end investment fund. These options are included among the exotic options, because the underlying asset of the options is the value process of the investment fund and therefore depends on a fund manager (= an option writer)'s action. We prove that a fair price of such option is represented as the value function of the associated stochastic exit time control problem. Using Hajek's mean comparison theorem, we find an explicit form of the fair option premium in the case of a constant volatility. We also characterize the fair option premium as a limit of a sequence of classical solutions to the associated Hamilton-Jacobi-Bellman equations with a classical Dirichlet boundary condition in the case of a diffusion market model. [ABSTRACT FROM AUTHOR]

In my recent study, I have developed a Bertrand-Edgeworth game in which the price and the allocation identical to those in a long-run competitive equilibrium can be supported as a Nash equilibrium even if the number of sellers is small. Because that study focuses on markets with free entry, it has not directly examined the Bertrand-Edgeworth debate, concerning a market without free entry. Focusing on such a market, the present study demonstrates that the standard Bertrand equilibrium can be supported as a Nash equilibrium in Yano’s game. [ABSTRACT FROM AUTHOR]

We investigate coalition structures formed by an external licensor of a patented innovation and firms operating in oligopolistic markets, and study licensing agreements reached as the bargaining outcomes under those coalition structures. The following are shown. (I) If the goods are not so substitutive in Bertrand competition, the grand coalition forms and the core is not empty, whereas in Cournot competition the core is empty under any coalition structure. (II) When the optimal number of licensees is one in the Bertrand duopoly with differentiated goods, the bargaining set gives more revenue to the licensor than licensing by means of upfront fee expects to give. [ABSTRACT FROM AUTHOR]

In the present paper, we propose a new computational technique with the Asymptotic Expansion (AE) approach to achieve variance reduction of the Monte-Carlo integration appearing especially in finance. We extend the algorithm developed by Takahashi and Yoshida (2003) to the second order asymptotics. Moreover, we apply the AE to approximate time dependent differentials of the target value in Newton (1994)’s scheme. Our numerical examples include pricing of average and basket options when the underlying state variables follow Constant Elasticity of Variance (CEV) processes. [ABSTRACT FROM AUTHOR]

In this paper we develop the theory of potential of cooperative games for semivalues, characterize congestion models that are defined by semivalues, and suggest an application of these results to combinatorial auctions, which may explain the success of the Iowa electronic market. [ABSTRACT FROM AUTHOR]

We propose a prepayment model of mortgage based on a structural approach in order to analyze prepayment risk of mortgage-backed securities (MBS). We introduce a continuous process named prepayment cost process. Specifically, each mortgager’s prepayment time is defined by the first time when her or his prepayment cost process falls below zero, but prepayment cost processes are supposed to be unobservable in the market. We also introduce a risk unique to each loan pool of mortgages, called a loan pool risk (LPR), and we regard LPR as a systematic risk other than interest rate. Using the model, we discuss the conditional distribution of prepayment times and a risk-neutral valuation of pass-through MBS. It is shown that each mortgager’s conditional non-prepayment probability and the posterior distribution of LPR play quite important roles in our study. [ABSTRACT FROM AUTHOR]

This paper initiates a comprehensive investigation of discounted optimal growth in the two-sector RSS model as a specific instance of the general theory of resource allocation associated with Brock, Gale and McKenzie. For an interval of values of a parameter ξ formalizing the marginal technical rate of transformation, under zero consumption, of machines from one period to the next, we show that the optimal policy in the discounted case remains identical to that in the undiscounted case irrespective of the discount factor. For two particular cases of ξ outside the said interval, we give a complete characterization of the optimal policy function, and of a variety of subsets that extend the facet notions formulated by McKenzie. Methodologically, this essay is a further rehabilitation of the geometric apparatus introduced by the authors for the undiscounted case. [ABSTRACT FROM AUTHOR]

This paper discusses the nature of optimal and stable networks in the link formation game. Players are directly or indirectly connected in each network, and players’ incentives to form new links depend upon the relative importance of these links. In this paper, instead of introducing a production function for each direct or indirect link to determine a player’s payoffs and the network value, we define the value of each network (and its sub-networks) directly, and determine players’ payoffs according to the Shapley value. To make the analysis tractable, we pay a special attention to the convexity or concavity of the underling networks, and in another model, study how the optimal and the pairwise stable networks change as costs of link formation change. We will show that special networks such as the star or circle form networks, as well as the complete network, can be both optimal and stable. [ABSTRACT FROM AUTHOR]

The first part of this paper provides conceptual discussions of key ingredients. The basic one-shot model of Bayesian society is presented first; it synthesizes Harsanyi’s (1967/68) Bayesian game and Aumann and Peleg’s (1960) non-side-payment game (NTU game). Jackson’s (1991) observation of the equivalence of the two widely adopted formulations of incomplete information is briefly reviewed. Two required meaningful conditions on an endogenously determined strategy are discussed: (1) Radner’s (1968) measurability condition with respect to an information structure, and (2) d’Aspremont and Gérard-Varet’s (1979) Bayesian incentive compatibility. Two descriptive interim solution concepts, the Bayesian incentive-compatible coarse core and the Bayesian incentive-compatible interim core, are discussed. The second part presents some results on the two interim solutions for the Bayesian pure exchange economy (a specific instance of the Bayesian society) in the private information case. The Bayesian incentive-compatible coarse core is nonemepty. The Bayesian incentive-compatible interim core may be empty, as pointed out by Hahn and Yannelis (1997). Sufficient conditions for its nonemptiness are explicitly formulated. It is unlikely that these two positive results are extended to the general framework of Bayesian society. The materials here are taken from Ichiishi and Yamazaki (forthcoming). [ABSTRACT FROM AUTHOR]

A large deviations control problem is treated for a long term optimal investment on a financial market with a bank account and a risky stock, both of which are affected by a stochastic factor described as Cox-Ingersoll-Ross’s interest rates. The solution is presented in explicit form by investigating the effective domain of the associated risk-sensitive control problem in risk-seeking case. [ABSTRACT FROM AUTHOR]

We explore two variants of the Bargaining Set in a simple majority game on income distributions in order to understand the apparent stability of tax schedules in democratic societies, despite the fact that the core of such games is empty (no majority Condorcet winner). Those variants are sharper than in the literature (Mas-Colell (1989), Shitovitz (1989), Zhou (1994)), by requiring that counterobjections try to guarantee their initial income levels to all members of the minority who stand to lose in an objection. A first variant defines as usual an income disbribution to be stable if there is no objection against it that is “justified”, i.e. for which there is no counterobjection satisfying the above requirement. A second variant allows objecting majorities to look one more step ahead. An objection is “weakly justified” if, whenever there is a counterobjection, the objecting majority can beat it while guaranteeing their income levels to all of its members. An income distribution is strongly stable if there is no weakly justified objection against it. These two variants generate sharper solution sets than when applied to large market games as in Mas-Colell (1989), Shitovitz (1989). Stable income distributions can indeed be characterized by their degrees of inequality. An income distribution is stable if and only if its Lorenz curve has no point in common with the graph C of f: [1/2,1] → [0,1], with f(b) = 1 − 1/(2b), for b > 1/2. It is strongly stable if and only if it is the egalitarian one. [ABSTRACT FROM AUTHOR]

In the absence of a world government, stopping the advance of global warming requires implementation of self-enforcing treaties among the countries of the world. In the language of game theory, such treaties are Nash equilibria of an underlying dynamic “climate change game.” In this paper, we report on the progress of a project to formulate and analyze models of such a game. The players are the sovereign countries of the world (say the roughly 200 members of the United Nations). The rules of this game are determined by the laws of physics and chemistry, and by the economic resources of the various countries. An important property of our models is the large multiplicity of equilibria. Indeed, this property enables us to find “Pareto-improving” equilibria, i.e., that improve the outcome for every country relative to the “business-as-usual equilibrium” we seem to be in at the present time. In each model we describe the set of equilibria, the business-as-usual equilibrium, and equilibria that are Pareto-improving relative to business-as-usual. Since much of the global wanning is caused by the accumulation of greenhouse gases (GHGs) in the earth’s atmosphere, and the GHGs dissipate very slowly, an appropriate model must be in the form of a dynamic game, with state variables that change over time as a consequence of the actions of the individual countries. Thus, the state variables include the global stock of GHG and the state of the relevant technology in each country. [ABSTRACT FROM AUTHOR]

We investigate the possibility of a dynamic explanation of the equilibrium ideas in terms of the perfect Bayesian equilibrium (PBE) (or sequential equilibrium). In particular, we take an equilibrium outcome which has been found by means of a static optimizing behavior and ask the following question. Can this outcome be supported, (or implemented), as a PBE of an extensive form game of a reasonable form? We provide a positive answer for solution concepts which are incentive compatible and a negative one for those which are not. [ABSTRACT FROM AUTHOR]

For a function acting between Banach spaces, we recall the notions of Hadamard and w-Hadamard differentiability and their relation to the common notions of Gâteaux and Fréchet differentiability. We observe that even for a function F: H → H that is both Hadamard and w-Hadamard differentiable but not Fréchet differentiable at 0 on a real Hilbert space H, there may be bifurcation for the equation F(u) = λu at points λ which do not belong to the spectrum of F′(0). We establish some necessary conditions for λ to be a bifurcation point in such cases and we show how this result can be used in the context of partial differential equations such as $$ - \Delta u\left( x \right) + q\left( x \right)u\left( x \right) = \lambda \left( {e^{\left x \right} u\left( x \right)} \right) for u \in H^2 \left( {\mathbb{R}^N } \right) $$ where this situation occurs. [ABSTRACT FROM AUTHOR]

Kahneman and Tversky asserted a fundamental asymmetry between gains and losses, namely a “reflection effect” which occurs when an individual prefers a sure gain of $pz to an uncertain gain of $z with probability p, while preferring an uncertain loss of $z with probability p to a certain loss of $pz. We focus on this class of choices (actuarially fair), and explore the extent to which the reflection effect, understood as occurring at a range of wealth levels, is compatible with single-self preferences. We decompose the reflection effect into two components, a “probability switch” effect, which is compatible with single-self preferences, and a “translation effect,” which is not. To argue the first point, we analyze two classes of single-self, nonexpected utility preferences, which we label “homothetic” and “weakly homothetic.” In both cases, we characterize the switch effect as well as the dependence of risk attitudes on wealth. We also discuss two types of utility functions of a form reminiscent of expected utility but with distorted probabilities. Type I always distorts the probability of the worst outcome downwards, yielding attraction to small risks for all probabilities. Type II distorts low probabilities upwards, and high probabilities downwards, implying risk aversion when the probability of the worst outcome is low. By combining homothetic or weak homothetic preferences with Type I or Type II distortion functions, we present four explicit examples: All four display a switch effect and, hence, a form of reflection effect consistent single self-preferences. [ABSTRACT FROM AUTHOR]

We consider a financial market with two or more separate components each driven by a Brownian Motion. We look at the problem to hedge a bounded contingent claim in such a way that all the components remain bounded. The problem can also be rephrased as a problem in risk measures. [ABSTRACT FROM AUTHOR]

We study variational convergence for integral functionals defined on LH∞ ([0, 1];dt) × y([0,1]; $$ \mathbb{Y} $$ ) where ℍ is a separable Hilbert space, $$ \mathbb{D} $$ is a Polish space and y[0,1]; $$ \mathbb{D} $$ ) is the space of Young measures on [0,1] × $$ \mathbb{D} $$ , and we investigate its applications to evolution inclusions. We prove the dependence of solutions with respect to the control Young measures and apply it to the study of the value function associated with these control problems. In this framework we then prove that the value function is a viscosity subsolution of the associated HJB equation. Some limiting properties for nonconvex integral functionals in proximal analysis are also investigated. [ABSTRACT FROM AUTHOR]

We consider the model of a stochastic financial exchange economy with finitely many periods. Time and uncertainty are represented by a finite event-tree $$ \mathbb{D} $$ and consumers may have constraints on their portfolios. We provide a general existence result of financial equilibria, which allows to cover several important cases of financial structures in the literature with or without constraints on portfolios. [ABSTRACT FROM AUTHOR]

This paper has two main parts. In the first part, we discuss the existence and uniqueness of the WE2,1-solution uμ,ν of a second order differential equation with two boundary points conditions in a finite dimensional space, governed by controls μ, ν which are measures on a compact metric space. We also discuss the dependence on the controls and the variational properties of the value function Vh(t, μ) := supν∈ℜh(uμ, ν(t)), associated with a bounded lower semicontinuous function h. In the second main part, we discuss the limiting behaviour of a sequence of dynamics governed by second order evolution inclusions with two boundary points conditions. We prove that (up to extracted sequences) the solutions stably converge to a Young measure ν and we show that the limit measure ν satisfies a Fatou-type lemma in Mathematical Economics with variational-type inclusion property. [ABSTRACT FROM AUTHOR]

As a generalization of a result by Kusuoka (2001), we provide the representation of law invariant convex risk measures. Very particular cases of law invariant coherent and convex risk measures are also studied. [ABSTRACT FROM AUTHOR]

We present some observations about links between some classical theories of microeconomics and dualities which have been used in optimization theory and in the study of first-order Hamilton-Jacobi equations. We introduce a variant of the classical indirect utility function called the wary indirect utility function and a variant of the expenditure function. We focus the attention on the links between these functions, observing that they have better relationships with the direct functions than their classical forms and we give economic interpretations of them. [ABSTRACT FROM AUTHOR]

This paper investigates the real indeterminacy of equilibria in an incomplete market model in which there are two periods, with uncertainty in the second, and both real and nominal assets exist. As is well known, the equilibria of a model with real assets behave very differently from the equilibria of a model with nominal assets. Then, what happens if real and nominal assets coexist? In this paper it is demonstrated within a general framework that regardless of the presence of real assets there is generically still real indeterminacy of equilibria whose degree is the same as without the real assets. [ABSTRACT FROM AUTHOR]

A method in demand analysis based on the Monge—Kantorovich duality is developed. We characterize (insatiate) demand functions that are rationalized, in different meanings, by concave utility functions with some additional properties such as upper semi-continuity, continuity, non-decrease, strict concavity, positive homogeneity and so on. The characterizations are some kinds of abstract cyclic monotonicity strengthening revealed preference axioms, and also they may be considered as an extension of the Afriat—Varian theory to an arbitrary (infinite) set of ‘observed data’. Particular attention is paid to the case of smooth functions. [ABSTRACT FROM AUTHOR]