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## Über dieses Buch

The series is designed to bring together those mathematicians who are seriously interested in getting new challenging stimuli from economic theories with those economists who are seeking effective mathematical tools for their research. A lot of economic problems can be formulated as constrained optimizations and equilibration of their solutions. Various mathematical theories have been supplying economists with indispensable machineries for these problems arising in economic theory. Conversely, mathematicians have been stimulated by various mathematical difficulties raised by economic theories.

## Inhaltsverzeichnis

### Local Risk-Minimization for Barndorff-Nielsen and Shephard Models with Volatility Risk Premium

We derive representations of locally risk-minimizing strategies of call and put options for Barndorff-Nielsen and Shephard models: jump type stochastic volatility models whose squared volatility process is given by a non-Gaussian Ornstein-Uhlenbeck process. The general form of Barndorff-Nielsen and Shephard models includes two parameters: volatility risk premium β and leverage effect ρ. Arai and Suzuki (Local risk minimization for Barndorff-Nielsen and Shephard models. submitted. Available at http://​arxiv.​org/​pdf/​1503.​08589v1) dealt with the same problem under constraint $$\beta = -\frac{1} {2}$$. In this paper, we relax the restriction on β; and restrict ρ to 0 instead. We introduce a Malliavin calculus under the minimal martingale measure to solve the problem.
Takuji Arai

### On a Fractional Differential Inclusion in Banach Space Under Weak Compactness Condition

We consider a class of boundary value problem in a separable Banach space governed by a fractional differential inclusion with integral boundary conditions
$$\displaystyle{\left \{\begin{array}{lll} w\text{-}D^{\alpha }u(t) \in F(t,u(t),w\text{-}D^{\alpha -1}u(t)), t \in [0,1] \\ I^{\beta }u(t)\vert _{t=0} = 0, u(1) =\int _{ 0}^{1}u(t)dt\end{array} \right.}$$
where α ∈ ]1, 2], $$\beta \in ]0,\infty [$$ are given constant and w-D γ is the fractional w-R.L derivative of order γ ∈ {α − 1, α}, F is a convex weakly compact valued mapping. Topological properties of the solutions set are presented. Applications to control problems and further variants are provided.
C. Castaing, C. Godet-Thobie, L. X. Truong, F. Z. Mostefai

### On First-Order Partial Differential Equations: An Existence Theorem and Its Applications

In this paper, we present an equivalence theorem between the existence of a global solution of a standard first-order partial differential equation and the extendability of the solution of corresponding ordinary differential equation. Moreover, we use this result to produce existence theorems on partial differential equation, and apply this theorem to the integrability problem in consumer theory.
Yuhki Hosoya

### Real Radicals and Finite Convergence of Polynomial Optimization Problems

Polynomial optimization appears various areas of mathematics. Although it is a fully nonlinear nonconvex optimization problems, there are numerical algorithms to approximate the global optimal value by generating sequences of semidefinite programming relaxations. In this paper, we study how real radicals of ideals have roles in duality theory and finite convergence property. Especially, duality theory is considered in the case that the truncated quadratic module is not necessarily closed. We will also try to explain the results by giving concrete examples.
Yoshiyuki Sekiguchi

### On Differentiated and Indivisible Commodities: An Expository Re-framing of Mas-Colell’s 1975 Model

With a pure exchange economy and its Walrasian equilibrium formalized as a distribution on the space of consumer characteristics, Mas-Colell (J Math Econ 2:263–296, 1975) showed the existence of equilibrium in a pure exchange economy with differentiated and indivisible commodities. We present a variant of Mas-Colell’s theorem; but more than for its own sake, we use it to expose and illustrate recent techniques due to Keisler-Sun (Adv Math 221:1584–1607, 2009), as developed in Khan-Rath-Yu-Zhang (On the equivalence of large individualized and distributionalized games. Johns Hopkins University, mimeo, 2015), to translate a result on a large distributionalized economy (LDE) to a large individualized economy (LIE), when the former can be represented by a saturated or super-atomless measure space of consumers, as formalized in Keisler-Sun (Adv Math 221:1584–1607, 2009) and Podczeck (J Math Econ 44:836–852, 2008) respectively. This also leads us to identify, hitherto unnoticed, open problems concerning symmetrization of distributionalized equilibria of economies in their distributionalized formulations. In relating our result to the antecedent literature, we bring into salience the notions of (i)“overriding desirability of the indivisible commodity,” as in Hicks (A revision of demand theory. Clarendon Press, Oxford, 1956), Mas-Colell (J Econ Theory 16:443–456, 1977) and Yamazaki (Econometrica 46:541–555, 1978; Econometrica 49:639–654, 1981), and of (ii) “bounded marginal rates of substitution,” as in Jones (J Math Econ 12:119–138, 1983; Econometrica 52:507–530, 1984) and Ostroy-Zame (Econometrica 62:593–633, 1994). Our work also relies heavily on the technical notion of Gelfand integration.
M. Ali Khan, Takashi Suzuki

### Survey of the Theory of Extremal Problems

In the paper some general principles of the theory of extremum are considered, and basing of these principles we give a survey of fundamental results on the foundation of the theory, conditions of extrema and existence of solutions.
V. Tikhomirov

### Fourier Analysis of Periodic Weakly Stationary Processes: A Note on Slutsky’s Observation

The periodic behavior of a specific weakly stationary stochastic process (w.s.p.) is examined from a viewpoint of classical Fourier analysis.(1) A w.s.p. has a spectral measure which is absolutely continuous with respect to the Lebesgue measure if and only if it is a moving average of a white noise. (2) A periodic or almost periodic w.s.p. must have a “discrete” spectral measure. Combining these two, we can conclude that any moving average of a white noise can neither be periodic nor almost periodic.However any w.s.p. can be approximated by a sequence of almost periodic w.s.p.’s in some specific sense.
Toru Maruyama

### Backmatter

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