A comparative assessment of different fuzzy regression methods for volatility forecasting

The aim of this paper is to compare different fuzzy regression methods in the assessment of the information content on future realised volatility of option-based volatility forecasts. These methods offer a suitable tool to handle both imprecision of measurements and fuzziness of the relationship among variables. Therefore, they are particularly useful for volatility forecasting, since the variable of interest (realised volatility) is unobservable and a proxy for it is used. Moreover, measurement errors in both realised volatility and volatility forecasts may affect the regression results. We compare both the possibilistic regression method of Tanaka et al. (IEEE Trans Syst Man Cybern 12:903–907, 1982) and the least squares fuzzy regression method of Savic and Pedrycz (Fuzzy Sets Syst 39:51–63, 1991). In our case study, based on intra-daily data of the DAX-index options market, both methods have proved to have advantages and disadvantages. Overall, among the two methods, we prefer the Savic and Pedrycz (Fuzzy Sets Syst 39:51–63, 1991) method, since it contains as special case (the central line) the ordinary least squares regression, is robust to the analysis of the variables in logarithmic terms or in levels, and provides sharper results than the Tanaka et al. (IEEE Trans Syst Man Cybern 12:903–907, 1982) method.     

Modularity and monotonicity of games

The purpose of this paper is twofold. First, we generalize Kajii et al. (J Math Econ 43:218–230, 2007) and provide a condition under which for a game \(v\) , its Möbius inverse is equal to zero within the framework of the \(k\) -modularity of \(v\) for \(k \ge 2\) . This condition is more general than that in Kajii et al. (J Math Econ 43:218–230, 2007). Second, we provide a condition under which for a game \(v\) , its Möbius inverse takes non-negative values, and not just zero. This paper relates the study of totally monotone games to that of \(k\) -monotone games. Furthermore, this paper shows that the modularity of a game is related to \(k\) -additive capacities proposed by Grabisch (Fuzzy Sets Syst 92:167–189, 1997). To illustrate its application in the field of economics, we use these results to characterize a Gini index representation of Ben-Porath and Gilboa (J Econ Theory 64:443–467, 1994). Our results can also be applied to potential functions proposed by Hart and Mas-Colell (Econometrica 57:589–614, 1989) and further analyzed by Ui et al. (Math Methods Oper Res 74:427–443, 2011).     

Remarks on possibilistic variances of fuzzy numbers

Carlsson and Fullér (Fuzzy Sets Syst. 122:315–326, 2001) introduced the definitions of two crisp possibilistic variances of a fuzzy number A, Var(A) and Var′(A). They showed that the subsethood does entail smaller variance in the sense of Var(?). Thus it is natural to ask whether it holds in the sense of Var′(?). Yet we are able to prove that it actually does not hold. Zhang and Wang (Appl. Math. Lett. 20:1167–1173, 2007) had introduced some conditions for which the subsethood does entail smaller variance in the sense of Var′(?). In this paper we give more generalized conditions for which it holds.     

Universal integrals based on copulas

A hierarchical family of integrals based on a fixed copula is introduced and discussed. The extremal members of this family correspond to the inner and outer extension of integrals of basic functions, the copula under consideration being the corresponding multiplication. The limits of the members of the family are just copula-based universal integrals as recently introduced in Klement et al. (IEEE Trans Fuzzy Syst 18:178–187, 2010). For the product copula, the family of integrals considered here contains the Choquet and the Shilkret integral, and it belongs to the class of decomposition integrals proposed in Even and Lehrer (Econ Theory, 2013) as well as to the class of superdecomposition integrals introduced in Mesiar et al. (Superdecomposition integral, 2013). For the upper Fréchet-Hoeffding bound, the corresponding hierarchical family contains only two elements: all but the greatest element coincide with the Sugeno integral.     

Solution to an open problem: a characterization of conditionally cancellative t-subnorms

In this work we solve an open problem of U. H?hle (Klement et?al. Fuzzy Sets Syst 145:471–479, 2004, Problem 11). We show that the solution gives a characterization of all conditionally cancellative t-subnorms. Further, we give an equivalence condition under which a conditionally cancellative t-subnorm has 1 as its neutral element and hence show that conditionally cancellative t-subnorms whose natural negations are strong are, in fact, t-norms.     

Efficient model for interval goal programming with arbitrary penalty function

Penalty function is a key factor in interval goal programming (IGP), especially for decision makers weighing resources vis-à-vis goals. Many approaches (Chang et al. J Oper Res Soc 57:469–473, 2006; Chang and Lin Eur J Oper Res 199, 9–20, 2009; Jones et al. Omega 23, 41–48, 1995; Romero Eur J Oper Res 153, 675–686, 2004; Vitoriano and Romero J Oper Res Soc 50, 1280–1283, 1999)have been proposed for treating several types of penalty functions in the past several decades. The recent approach of Chang and Lin (Eur J Oper Res 199, 9–20, 2009) considers the S-shaped penalty function. Although there are many approaches cited in literature, all are complicated and inefficient. The current paper proposes a novel and concise uniform model to treat any arbitrary penalty function in IGP. The efficiency and usefulness of the proposed model are demonstrated in several numeric examples.     

Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains

We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980; Knightly in SIAM J. Math. Anal. 3:506–511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633–652, 2012, Quad. Mat. 4:27–68, 1999, Nonlinear Anal. 47:4151–4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159–175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719–732, 2008; Maremonti in Mat. Ves. 61:81–91, 2009, Appl. Anal. 90:125–139, 2011).     

On the Gauss–Newton method

We provide a new semilocal convergence analysis of the Gauss–Newton method (GNM) for solving nonlinear equation in the Euclidean space. Using a combination of center-Lipschitz, Lipschitz conditions, and our new idea of recurrent functions, we provide under the same or weaker hypotheses than before (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982), a finer convergence analysis. The results can be extended in case outer or generalized inverses are used. Numerical examples are also provided to show that our results apply, where others fail (Ben-Israel, J. Math. Anal. Appl. 15:243–252, 1966; Chen and Nashed, Numer. Math. 66:235–257, 1993; Deuflhard and Heindl, SIAM J. Numer. Anal. 16:1–10, 1979; Guo, J. Comput. Math. 25:231–242, 2007; Häußler, Numer. Math. 48:119–125, 1986; Hu et al., J. Comput. Appl. Math. 219:110–122, 2008; Kantorovich and Akilov, Functional Analysis in Normed Spaces, Pergamon, Oxford, 1982).     

Strong NP-hardness of scheduling problems with learning or aging effect

Proofs of strong NP-hardness of single machine and two-machine flowshop scheduling problems with learning or aging effect given in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c; Applied Mathematical Modelling 37:1523–1536, 2013) contain a common mistake that make them incomplete. We reveal the mistake and provide necessary corrections for the problems in Rudek (Computers & Industrial Engineering 61:20–31, 2011; Annals of Operations Research 196(1):491–516, 2012a; Applied Mathematical Modelling 37:1523–1536, 2013). NP-hardness of problems in Rudek (International Journal of Advanced Manufacturing Technology 59:299–309, 2012b; Applied Mathematics and Computations 218:6498–6510, 2012c) remains unknown because of another mistake which we are unable to correct.     

The configurations 123 revisited

As is known, there are 229 symmetric configurations 12₃, (Daublebsky von Sterneck in Monatshefte Math Phys 5:223–255, 1895; Gropp in J Comb Inf Syst Sci 15:34–48, 1990). We use tactical decompositions by automorphism group (TDA) to study these configurations in detail. In (Daublebsky von Sterneck in Monatshefte Math Phys 14, 254–260, 1903) the automorphism groups of the configurations were determined. We find some errors there and correct them. For the configurations with a rather big automorphism group, we give models which display the structure of the group.     

Quasi-stationary stefan problem with values on the front depending on its geometry

The problem presented below is a singular-limit problem of the extension of the Cahn-Hilliard model obtained via introducing the asymmetry of the surface tension tensor under one of the truncations (approximations) of the inner energy [2, 5–8, 10, 12, 13].     

Weyl modules and q-Whittaker functions

Let $G$ be a semi-simple simply connected group over $\mathbb {C}$ . Following Gerasimov et al. (Comm Math Phys 294:97–119, 2010) we use the $q$ -Toda integrable system obtained by quantum group version of the Kostant–Whittaker reduction (cf. Etingof in Am Math Soc Trans Ser 2:9–25, 1999, Sevostyanov in Commun Math Phys 204:1–16, 1999) to define the notion of $q$ -Whittaker functions $\varPsi _{\check{\lambda }}(q,z)$ . This is a family of invariant polynomials on the maximal torus $T\subset G$ (here $z\in T$ ) depending on a dominant weight $\check{\lambda }$ of $G$ whose coefficients are rational functions in a variable $q\in \mathbb {C}^*$ . For a conjecturally the same (but a priori different) definition of the $q$ -Toda system these functions were studied by Ion (Duke Math J 116:1–16, 2003) and by Cherednik (Int Math Res Notices 20:3793–3842, 2009) [we shall denote the $q$ -Whittaker functions from Cherednik (Int Math Res Notices 20:3793–3842, 2009) by $\varPsi '_{\check{\lambda }}(q,z)$ ]. For $G=SL(N)$ these functions were extensively studied in Gerasimov et al. (Comm Math Phys 294:97–119, 2010; Comm Math Phys 294:121–143, 2010; Lett Math Phys 97:1–24, 2011). We show that when $G$ is simply laced, the function $\hat{\varPsi }_{\check{\lambda }}(q,z)=\varPsi _{\check{\lambda }}(q,z)\cdot {\prod \nolimits _{i\in I}\prod \nolimits _{r=1}^{\langle \alpha _i,\check{\uplambda }\rangle }(1-q^r)}$ (here $I$ denotes the set of vertices of the Dynkin diagram of $G$ ) is equal to the character of a certain finite-dimensional $G[[{\mathsf {t}}]]\rtimes \mathbb {C}^*$ -module $D(\check{\lambda })$ (the Demazure module). When $G$ is not simply laced a twisted version of the above statement holds. This result is known for $\varPsi _{\check{\lambda }}$ replaced by $\varPsi '_{\check{\lambda }}$ (cf. Sanderson in J Algebraic Combin 11:269–275, 2000 and Ion in Duke Math J 116:1–16, 2003); however our proofs are algebro-geometric [and rely on our previous work (Braverman, Finkelberg in Semi-infinite Schubert varieties and quantum $K$ -theory of flag manifolds, arXiv/1111.2266, 2011)] and thus they are completely different from Sanderson (J Algebraic Combin 11:269–275, 2000) and Ion (Duke Math J 116:1–16, 2003) [in particular, we give an apparently new algebro-geometric interpretation of the modules $D(\check{\lambda })]$ .     

A multi-dimensional SRBM: geometric views of its product form stationary distribution

We present a geometric interpretation of a product form stationary distribution for a \(d\) -dimensional semimartingale reflecting Brownian motion (SRBM) that lives in the nonnegative orthant. The \(d\) -dimensional SRBM data can be equivalently specified by \(d+1\) geometric objects: an ellipse and \(d\) rays. Using these geometric objects, we establish necessary and sufficient conditions for characterizing product form stationary distribution. The key idea in the characterization is that we decompose the \(d\) -dimensional problem to \(\frac{1}{2}d(d-1)\) two-dimensional SRBMs, each of which is determined by an ellipse and two rays. This characterization contrasts with the algebraic condition of Harrison and Williams (Ann Probab 15:115–137, 1987b). A \(d\) -station tandem queue example is presented to illustrate how the product form can be obtained using our characterization. Drawing the two-dimensional results in Avram et al. (Queueing Syst 37:259–289, 2001), Dai and Miyazawa (Queueing Syst 74:181–217, 2013), we discuss potential optimal paths for a variational problem associated with the three-station tandem queue.     

Exponential polynomials on commutative hypergroups

Polynomials and exponential polynomials play a fundamental role in the theory of spectral analysis and spectral synthesis on commutative groups. Recently several new results have been published in this field [2–4,6]. Spectral analysis and spectral synthesis has been studied on some types of commutative hypergroups, as well. However, a satisfactory definition of exponential monomials on general commutative hypergroups has not been available so far. In [5,7,8] and [9], the authors use a special concept on polynomial and Sturm–Liouville-hypergroups. Here we give a general definition which covers the known special cases.     

On the chaotic properties of the von Foerster-Lasota equation

In this article we present a different approach to some of the results published in our recent paper (Brze?niak and Dawidowicz in Semigroup Forum, 78(1):118–137, 2009). This new approach is based on a deep result from a paper (Ergod. Theory Dyn. Syst. 17(4):793–819, 1997) by Desch Schappacher and Webb.     

Poisson and compound Poisson approximations in conventional and nonconventional setups

It was shown in Kifer (Israel J Math, 2013) that for any subshift of finite type considered with a Gibbs invariant measure the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of almost all points are asymptotically Poisson distributed. Here we not only extend this result to all \(\psi \) -mixing shifts with countable alphabet but actually show that for all points the distributions of these numbers are asymptotically close either to Poisson or to compound Poisson distributions. It turns out that for all nonperiodic points a limiting distribution is always Poisson while at the same time for periodic points there may be no limiting distribution at all unless the shift invariant measure is Bernoulli in which case the limiting distribution always exists. Thus we describe, essentially completely, limiting distributions of multiple recurrence numbers in this setup. As a corollary we obtain also that the first occurence time of the multiple recurrence event is asymptotically exponentially distributed. Most of the results are new also for the widely studied single recurrencies case (see, for instance, Haydn and Vaienti Discret Contin Dyn Syst A 10:589–616, 2004; Probab Theory Relat Fields 144:517–542, 2009; Abadi and Saussol Stoch Process Appl 121:314–323, 2011; Abadi and Vergne Nonlinearity 21:2871–2885, 2008), as well.     

Continuity of the solution mapping to parametric generalized vector equilibrium problems

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511–520, 2010), Chen and Li (Pac J Optim 6:141–152, 2010), Chen et al. (J Glob Optim 45:309–318, 2009), Cheng and Zhu (J Glob Optim 32:543–550, 2005), Gong (J Optim Theory Appl 139:35–46, 2008), Li and Fang (J Optim Theory Appl 147:507–515, 2010), Li et al. (Bull Aust Math Soc 81:85–95, 2010) and Peng et al. (J Optim Theory Appl 152(1):256–264, 2011).     

Dynamical system solutions for homogeneous linear functional equations of higher order

We present the solution of a large class of homogeneous linear functional equations of higher order by using ideas from dynamical systems. A particularly simple example from this class is the functional equation $$f(x) = \frac{1}{2}f \left(\frac{x}{2}\right) + \frac{1}{2}f \left(\frac{x+1}{2}\right), \quad 0 < x < 1.$$ Equations such as these have found important applications in wavelet theory by Hilberdink (Aequa Math 61(1–2):179–189, 2001) where they are called dilation equations and are usually solved by Fourier methods by Daubechies (Comm Pure Appl Math 41(7):909–996, 1988) or iteration methods of Daubechies (SIAM J Math Anal 22(5):1388–1410, 1991). A recent result of Góra (Ergod Theory Dyn Syst 29(5):1549–1583, 2009) allows us to represent the solution as an infinite series that is determined by the dynamics of a map that is defined by the functional equation. In this problem the interplay between dynamical systems and solutions of functional equations is brought into sharp focus.