Explicit characterization of the super-replication strategy in financial markets with partial transaction costs

We consider a continuous time multivariate financial market with proportional transaction costs and study the problem of finding the minimal initial capital needed to hedge, without risk, European-type contingent claims. The model is similar to the one considered in Bouchard and Touzi [B. Bouchard, N. Touzi, Explicit solution of the multivariate super-replication problem under transaction costs, The Annals of Applied Probability 10 (3) (2000) 685–708] except that some of the assets can be exchanged freely, i.e. without paying transaction costs. In this context, we generalize the result of the above paper and prove that the super-replication price is given by the cost of the cheapest hedging strategy in which the number of non-freely exchangeable assets is kept constant over time. Our proof relies on the introduction of a new auxiliary control problem whose value function can be interpreted as the super-hedging price in a model with unbounded stochastic volatility (in the directions where transaction costs are non-zero). In particular, it confirms the usual intuition that transaction costs play a similar role to stochastic volatility.     

Real-variables characterization of generalized Stieltjes functions

We obtain a characterization of generalized Stieltjes functions of any order λ>0$\lambda >0$ in terms of inequalities for their derivatives on (0,∞)$(0,\infty )$. When λ=1$\lambda =1$, this provides a new and simple proof of a characterization of Stieltjes functions first obtained by Widder in 1938.     

Intersection bodies and generalized cosine transforms

The paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis. A new concept of λ-intersection body, that unifies some known classes of geometric objects, is introduced. A parallel between trace theorems in function theory, restriction onto lower-dimensional subspaces of the spherical Radon transforms and the generalized cosine transforms, and sections of λ-intersection bodies is established. New integral formulas for different classes of cosine transforms are obtained and examples of λ-intersection bodies are given. We also revisit some known facts in this area and give them new simple proofs.     

The group inverse of a companion matrix

A complete characterization is given for the group inverse of a companion matrix over an arbitrary ring to exist. Formulae are given for the actual group inverse and some consequences are drawn.     

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.     

Conjugate function method for numerical conformal mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.     

Local asymptotic laws for the Brownian convex hull

Summary We prove a general theorem for the precise rate at which the convex hull of Brownian motion gets created. The latter result relates large deviation theory to P. Lévy's geometric proof of Strassen's law of the iterated logarithm. This also answers a question of S. Evans. Moreover, we give a partial solution to a question of J. Hammersley and P. Lévy regarding the slowness of the growth of the hull process. Several examples, some classical and some new, are given to illustrate the theorems. Finally, we present applications to the convex hull of random walks ind dimensions.     

Existence and uniqueness of pseudo-almost periodic solutions to some classes of partial evolution equations

Some sufficient conditions for the existence and uniqueness of pseudo-almost periodic (mild) solutions to some classes of partial evolution equations are given. We then make use of our abstract results to discuss the existence of pseudo-almost periodic solutions to some partial differential equations.     

Finding exact solutions of nonlinear PDEs using the natural decomposition method

In this work, we implement the natural decomposition method (NDM) to solve nonlinear partial differential equations. We apply the NDM to obtain exact solutions for three applications of nonlinear partial differential equations. The new method is a combination of the natural transform method and the Adomian decomposition method. We prove some of the properties that are related to the natural transform method. The results are compared with existing solutions obtained by other methods, and one can conclude that the NDM is easy to use and efficient. Copyright © 2016 John Wiley & Sons, Ltd.     

Spectral factorization in the disk algebra

Every strictly positive function f, given on the unit circle of the complex plane, defines an outer function. This article investigates the behavior of these outer functions on the boundary of the unit disk. It is shown that even if the given function f on the boundary is continuous, the corresponding outer function is generally not continuous on the closure of the unit disk. Moreover, any subset E∈ [-π ,π) of Lebesgue measure zero is a valid divergence set for outer functions of some continuous functions f. These results are applied to study the solutions of non-linear boundary-value problems and the factorization of spectral density functions.     

Some inequalities for the Poincaré metric of plane domains

In this paper, the Poincaré (or hyperbolic) metric and the associated distance are investigated for a plane domain based on the detailed properties of those for the particular domain In particular, another proof of a recent result of Gardiner and Lakic [7] is given with explicit constant. This and some other constants in this paper involve particular values of complete elliptic integrals and related special functions. A concrete estimate for the hyperbolic distance near a boundary point is also given, from which refinements of Littlewood’s theorem are derived.This research was carried out during the first-named author’s visit to the University of Helsinki under the exchange programme of scientists between the Academy of Finland and the JSPS.     

Mappings of higher order and nonlinear equations in some spaces of almost periodic functions

In the first part of the paper we examine mappings of higher order from a general point of view, that is, in normed spaces of bounded real-valued functions defined on R$\mathbb{R}$. Particular attention is paid to the relation of such mappings with the so-called autonomous superposition operators. Next we investigate mappings of higher order in Banach spaces of almost periodic functions and their perturbations. We also give necessary and sufficient conditions guaranteeing that a nonautonomous superposition operator acts in the space of almost periodic functions in the sense of Levitan and is uniformly continuous. In the Banach space of bounded almost periodic functions in the sense of Levitan we discuss mappings of higher order and a convolution operator. Some applications to nonlinear differential and integral equations are given.     

On factorization structures and epidense hulls

Characterizations of epidense subcategories of topological categories and of existence of epidense hulls have been described in [2, 3, 4]. In this paper a similar characterization is given in a much more general setting; for example the category need not have products. The relationship between finite factorization structures and existence of epidense hulls is investigated. It is found to be analogous to the relationship between general factorization structures and epireflective hulls.     

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz-Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved.